{"id":1498,"date":"2023-07-26T10:37:37","date_gmt":"2023-07-26T10:37:37","guid":{"rendered":"https:\/\/www.uni.lu\/fstm-en\/?post_type=study-programs&#038;p=1498"},"modified":"2025-06-13T14:01:31","modified_gmt":"2025-06-13T12:01:31","slug":"programme","status":"publish","type":"study-programs","link":"https:\/\/www.uni.lu\/fstm-en\/study-programs\/master-in-mathematics-financial-mathematics\/programme\/","title":{"rendered":"Programme"},"content":{"rendered":"<div class=\"section py-0 static\">\n<section class=\"py-0 wp-block-unilux-blocks-free-section section\"><div class=\"container xl:max-w-screen-xl\"><!-- SecondaryNavigationMirror --><div class=\"js-secondary-nav--sticky wp-block-unilux-blocks-secondary-navigation\" id=\"c6b964f8-e0d9-4d65-b889-c0a11e4c3450\">\n    <nav id=\"secondary-navigation\" role=\"navigation\" aria-label=\"Secondary menu\" class=\"secondary-nav\">\n        <button class=\"secondary-nav__button btn btn-solid--primary w-full\" aria-expanded=\"false\" aria-controls=\"secondary-navigation__list\" title=\"The selected page is Programme\" type=\"button\">\n            <span class=\"btn__name mr-auto\">Programme<\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--xs\">\n                <use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\" \/>\n            <\/svg>\n        <\/button>\n\n        <ul id=\"secondary-navigation__list\" class=\"secondary-nav__list\">\n            <li class=\"secondary-nav__item\">\n    <a href=\"https:\/\/www.uni.lu\/fstm-en\/study-programs\/master-in-mathematics-financial-mathematics\/\" target=\"\" rel=\"noreferrer noopener\" class=\"secondary-nav__link \">\n        <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--xs\">\n            <use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-right\" \/>\n        <\/svg>\n        <span class=\"secondary-nav__name\">Overview<\/span>\n    <\/a>\n<\/li>\n<li class=\"secondary-nav__item\">\n    <a href=\"https:\/\/www.uni.lu\/fstm-en\/study-programs\/master-in-mathematics-financial-mathematics\/programme\/\" target=\"\" rel=\"noreferrer noopener\" class=\"secondary-nav__link isactived\" aria-current=\"page\">\n        <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--xs\">\n            <use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-right\" \/>\n        <\/svg>\n        <span class=\"secondary-nav__name\">Programme<\/span>\n    <\/a>\n<\/li>\n<li class=\"secondary-nav__item\">\n    <a href=\"https:\/\/www.uni.lu\/fstm-en\/study-programs\/master-in-mathematics-financial-mathematics\/career\/\" target=\"\" rel=\"noreferrer noopener\" class=\"secondary-nav__link \">\n        <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--xs\">\n            <use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-right\" \/>\n        <\/svg>\n        <span class=\"secondary-nav__name\">Career<\/span>\n    <\/a>\n<\/li>\n<li class=\"secondary-nav__item\">\n    <a href=\"https:\/\/www.uni.lu\/fstm-en\/study-programs\/master-in-mathematics-financial-mathematics\/testimonials\/\" target=\"\" rel=\"noreferrer noopener\" class=\"secondary-nav__link \">\n        <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--xs\">\n            <use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-right\" \/>\n        <\/svg>\n        <span class=\"secondary-nav__name\">Testimonials<\/span>\n    <\/a>\n<\/li>\n<li class=\"secondary-nav__item\">\n    <a href=\"https:\/\/www.uni.lu\/fstm-en\/study-programs\/master-in-mathematics-financial-mathematics\/teaching-staff\/\" target=\"\" rel=\"noreferrer noopener\" class=\"secondary-nav__link \">\n        <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--xs\">\n            <use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-right\" \/>\n        <\/svg>\n        <span class=\"secondary-nav__name\">Teaching staff<\/span>\n    <\/a>\n<\/li>\n<li class=\"secondary-nav__item\">\n    <a href=\"https:\/\/www.uni.lu\/fstm-en\/study-programs\/master-in-mathematics-financial-mathematics\/admissions\/\" target=\"\" rel=\"noreferrer noopener\" class=\"secondary-nav__link \">\n        <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--xs\">\n            <use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-right\" \/>\n        <\/svg>\n        <span class=\"secondary-nav__name\">Admissions<\/span>\n    <\/a>\n<\/li>\n        <\/ul>\n    <\/nav>\n<\/div>\n<!-- end SecondaryNavigationMirror --><\/div><\/section>\n\n\n<section class=\"section section wp-block-unilux-blocks-headline-text-and-image\">\n    \n<div class=\"wp-block-unilux-blocks-wrapper container xl:max-w-screen-xl\">\n<h2 class=\"has-text-align-left wp-block-unilux-blocks-heading\"        id=\"programme\"\n    >\nProgramme<\/h2>\n\n\n\n<div class=\"wp-block-unilux-blocks-wrapper flex flex-wrap lg:-mx-32\">\n<div class=\"wp-block-unilux-blocks-wrapper w-full lg:w-7\/12 lg:order-2 lg:px-32\">\n<div class=\"wp-block-unilux-blocks-image-video-wrapper\">\n    <figure class=\"wp-block-dev4-reusable-blocks-image  object-fit--cover\">\n    \n<img decoding=\"async\" class=\"wp-block-image unilux-custom-image-block\"\n                alt=\"\"\n            src=\"https:\/\/www.uni.lu\/wp-content\/uploads\/sites\/4\/2023\/07\/SuppCom_0226_012-scaled.jpg\"\n                srcset=\"https:\/\/www.uni.lu\/wp-content\/uploads\/sites\/4\/2023\/07\/SuppCom_0226_012-300x200.jpg 300w, https:\/\/www.uni.lu\/wp-content\/uploads\/sites\/4\/2023\/07\/SuppCom_0226_012-1024x683.jpg 1024w, https:\/\/www.uni.lu\/wp-content\/uploads\/sites\/4\/2023\/07\/SuppCom_0226_012-768x512.jpg 768w, https:\/\/www.uni.lu\/wp-content\/uploads\/sites\/4\/2023\/07\/SuppCom_0226_012-1536x1024.jpg 1536w, https:\/\/www.uni.lu\/wp-content\/uploads\/sites\/4\/2023\/07\/SuppCom_0226_012-2048x1365.jpg 2048w\"\n                style=\"object-position: 50.00% 50.00%; font-family: &quot;object-fit: cover; object-position: 50.00% 50.00%;&quot;; aspect-ratio: 3\/2; object-fit: cover; width: 100%;\"\n        loading=\"lazy\"\n\/>    <\/figure><\/div><\/div>\n\n\n\n<div class=\"wp-block-unilux-blocks-wrapper w-full lg:w-5\/12 lg:order-1 lg:px-32\">\n<p>The Financial Mathematics track of the Master in Mathematics is a two-year programme taught in English. To boost their future career, the track includes internships and allows students to write their Master thesis in a professional setting during an internship.<strong> New students are requested to take the <a href=\"https:\/\/www.uni.lu\/wp-content\/uploads\/sites\/4\/2024\/08\/Summer-Refresher-2024.pdf\">Refresher Courses<\/a> scheduled before the start of the first semester.<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-unilux-blocks-custom-buttons btn-list\"><li class=\"wp-block-unilux-blocks-custom-button\"    aria-disabled=\"false\"\n    >\n    <a\n        role=\"link\"\n        aria-disabled=\"false\"\n                    href=\"https:\/\/www.uni.lu\/fstm-en\/study-programs\/master-in-mathematics-financial-mathematics\/admissions\/\"\n                target=\"_self\"\n        class=\"btn btn--primary\"\n            >Admissions<\/a>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n\n\n<section class=\"wp-block-unilux-blocks-free-section section\"><div class=\"container xl:max-w-screen-xl\">\n<h2 class=\"has-text-align-left wp-block-unilux-blocks-heading\"        id=\"academic-contents\"\n    >\nAcademic Contents<\/h2>\n\n\n\n<div id=\"tab-nav\" class=\"tab-nav\">\n    <button\n        type=\"button\"\n        class=\"button-toggle btn btn--white w-full\"\n        aria-expanded=\"false\"\n        aria-controls=\"tab-nav__listing\"\n        title=\"The selected tab is CONTENT OF THE CURRENT LABEL\"\n    >\n        <span class=\"btn__name mr-auto\">Semester 1<\/span>\n        <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--xs\">\n            <use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\" \/>\n        <\/svg>\n    <\/button>\n    <ul\n        id=\"tab-nav__listing\"\n        class=\"tab-nav__listing nav manual\"\n        role=\"tablist\"\n        aria-label=\"About me\"\n    >\n                <li role=\"presentation\" class=\"tab-nav__item\" key={button.id}>\n            <button\n                type=\"button\"\n                id=\"tab-8f71f9d7-bc8c-4fa0-bf98-70ef4245d9e6\"\n                data-bs-toggle=\"tab\"\n                data-bs-target=\"#tabpanel-8f71f9d7-bc8c-4fa0-bf98-70ef4245d9e6\"\n                role=\"tab\"\n                aria-controls=\"tabpanel-8f71f9d7-bc8c-4fa0-bf98-70ef4245d9e6\"\n                aria-selected=\"false\"\n                aria-current=\"false\"\n                class=\"tab-nav__button\"\n                tabIndex=\"-1\"\n            >\n                <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--xs\">\n                    <use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-right\" \/>\n                <\/svg>\n                <span class=\"tab-nav__name\">Semester 1<\/span>\n            <\/button>\n        <\/li>\n                <li role=\"presentation\" class=\"tab-nav__item\" key={button.id}>\n            <button\n                type=\"button\"\n                id=\"tab-72632041-fed8-4a66-b795-326d06533510\"\n                data-bs-toggle=\"tab\"\n                data-bs-target=\"#tabpanel-72632041-fed8-4a66-b795-326d06533510\"\n                role=\"tab\"\n                aria-controls=\"tabpanel-72632041-fed8-4a66-b795-326d06533510\"\n                aria-selected=\"false\"\n                aria-current=\"false\"\n                class=\"tab-nav__button\"\n                tabIndex=\"-1\"\n            >\n                <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--xs\">\n                    <use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-right\" \/>\n                <\/svg>\n                <span class=\"tab-nav__name\">Semester 2<\/span>\n            <\/button>\n        <\/li>\n                <li role=\"presentation\" class=\"tab-nav__item\" key={button.id}>\n            <button\n                type=\"button\"\n                id=\"tab-91d69e16-8af9-453d-8d00-158f5bfb5826\"\n                data-bs-toggle=\"tab\"\n                data-bs-target=\"#tabpanel-91d69e16-8af9-453d-8d00-158f5bfb5826\"\n                role=\"tab\"\n                aria-controls=\"tabpanel-91d69e16-8af9-453d-8d00-158f5bfb5826\"\n                aria-selected=\"false\"\n                aria-current=\"false\"\n                class=\"tab-nav__button\"\n                tabIndex=\"-1\"\n            >\n                <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--xs\">\n                    <use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-right\" \/>\n                <\/svg>\n                <span class=\"tab-nav__name\">Semester 3<\/span>\n            <\/button>\n        <\/li>\n                <li role=\"presentation\" class=\"tab-nav__item\" key={button.id}>\n            <button\n                type=\"button\"\n                id=\"tab-39303298-71a6-445f-952b-72604c08c906\"\n                data-bs-toggle=\"tab\"\n                data-bs-target=\"#tabpanel-39303298-71a6-445f-952b-72604c08c906\"\n                role=\"tab\"\n                aria-controls=\"tabpanel-39303298-71a6-445f-952b-72604c08c906\"\n                aria-selected=\"false\"\n                aria-current=\"false\"\n                class=\"tab-nav__button\"\n                tabIndex=\"-1\"\n            >\n                <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--xs\">\n                    <use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-right\" \/>\n                <\/svg>\n                <span class=\"tab-nav__name\">Semester 4<\/span>\n            <\/button>\n        <\/li>\n            <\/ul>\n<\/div>\n<div class=\"tab-content\">\n    <section\n    id=\"tabpanel-8f71f9d7-bc8c-4fa0-bf98-70ef4245d9e6\"\n    aria-labelledby=\"tab-8f71f9d7-bc8c-4fa0-bf98-70ef4245d9e6\"\n    class=\"tab-panel fade hidden is-hidden\"\n>\n    <div class=\"wp-block-unilux-blocks-course-offer\">\n    <h3>Course offer for Financial Mathematics, Semestre 1 (2025-2026 Winter)<\/h3>\n    \n<div id=\"accordion-module-6a141e2796ce9\" class=\"accordion accordion--theme\">\n            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e2796d0f\"\n        >\n            <span class=\"accordion__title\">\n                Discrete-time stochastic processes                 <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            6                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_FM-1                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e2796d0f\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e2796ce9\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Discrete-time stochastic processes <\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 6<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_FM-1<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 1<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> Yes<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Objectives<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>\n                            Introduction to basic concepts of modern probability theory                        <\/p>\n                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>On successful completion of the course, the student should be able to:<\/p><br><ul><br><li>Understand and use concepts of modern probability theory (e.g., filtrations, martingales, stopping times)<\/li><br><li>Apply the notion of martingale to model random evolutions<\/li><br><li>Know and apply classical martingale convergence theorems<\/li><br><li>Describe and manipulate basic properties of Brownian motion\u00a0<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Radon-Nikodym Theorem, conditional expectations, martingales, stopping times, optional stopping theorems, Doob\u2019s inequalities, martingale convergence theorems, martingale central limit theorem, Brownian motion.                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Exam modalities for the first session\u00a0<br>Combined Assessment (40% Homeworks, 60% Written In-person Final Exam).<\/p><br><p><br>Exam modalities for the retake exam<br>Combined Assessment (40% Homeworks, 60% Written In-person Final Exam).<\/p><br><p><br>Absence Plan<br>In the case of justified absence in the final exam a replacement exam during the same semester will be offered.<\/p><br><p>\u00a0<\/p><br><p>\u00a0<\/p>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul><br><li>Rick Durrett: Probability Theory and Examples<\/li><br><li>Yuval Peres, Peter Morters: Brownian Motion<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e2796d2f\"\n        >\n            <span class=\"accordion__title\">\n                Functional Analysis                <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            6                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_MMCS2-2                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e2796d2f\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e2796ce9\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Functional Analysis<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 6<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_MMCS2-2<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 1<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Objectives<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>\n                            The aim of the course is to provide students with efficient tools to study linear operators between infinite dimensional vector spaces. In particular, this class will deal with normed vector spaces, Banach and Hilbert spaces, with bounded linear operators on normed vector spaces, fundamental principles such as Fourier analysis, Lebesgue integral, Han-Banach Theorem, Uniform Boundedness Principle or Closed Graph Theorem, and with spectral theory of compact (self-adjoint) linear operators.\u00a0                        <\/p>\n                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Students will acquire a solid understanding of functional analysis, its fundamental results and basic techniques. In particular, students will understand applications to measure theory, Fourier theory, and the spectral theorem for (unbounded) operators. Students will know the relevance of a theorem, its underlying motivation and a precise idea of its proof. Hopefully, students will demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from functional analysis.\u00a0                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Functional analysis aims to study infinite-dimensional spaces of functions and features of linear operators on these spaces. Though, from the point of view of Mathematics, functional analysis has its own interest, it plays a crucial role in many related areas, in particular in Physics, Engineering or Finance. Roughly speaking, most \u201creal-life problems\u201d involve non-linear partial differential equations with infinite-dimensional solution spaces of functions (or distributions). This course provides Master students with the basic tools and the fundamental results to develop skills to solve such problems.                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Exam modality for the first session<br><br><\/p><br><ul><br><li>Written test on week of October 28, duration 1 hour, 30% of the final grade,<\/li><br><li>Final written exam (scheduled by the administration), 60% of the final grade,<\/li><br><li>Written homework (problems to be determined), 10% of the final grade.<\/li><br><\/ul><br><p><br>Exam modality for the retake exam<br><br>Retake exam (scheduled by the administration) will be in written form.<\/p><br><p><br>Absence<br><br>A make-up will be possible only for an unavoidable reason supported by a proof.<\/p>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Note \/ Literature \/ Bibliography\u00a0<br><br>Introductory Functional Analysis with Applications, by Erwin Kreyszig.<br>Functional Analysis, Spectral Theory and Applications, by Manfred Einsiedler and Thomas Ward.<br>An Introduction to Fourier Analysis, by Russell L. Herman                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e2796d47\"\n        >\n            <span class=\"accordion__title\">\n                Riemann Surfaces                <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            8                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_GM-13                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e2796d47\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e2796ce9\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Riemann Surfaces<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 8<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_GM-13<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 1<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Objectives<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>\n                            Understand and be able to use and apply the key notions related to Riemann surfaces: definitions, key examples, maps between Riemann surfaces, integration on Riemann surfaces, divisors and meromorphic functions, algebraic curves and the Riemann-Roch theorem.\u00a0                        <\/p>\n                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        At the end of the course the students should have a basic understanding of what Riemann surfaces are and how they can be studied from different points of view, especially the complex and topological one. They should be able to understand and solve simple problems involving the material from the course.                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Riemann surfaces lie at the crossroads of many fields of research of modern mathematics: topology, complex and algebraic geometry, number theory and harmonic analysis, among others.\u00a0<br>The course will cover the basic notions of the theory, starting from the definition of a complex manifold, and move on to the classification of Riemann surfaces via the study of their divisors. This will allow the students to acquire familiarity with some ubiquitous tools from complex geometry, such as the Riemann-Roch theorem.\u00a0<br>We will put some focus on examples along the way. If time allows, we will see how some constructions from the complex analytic theory can be reinterpreted in the language of modern algebraic geometry, bridging the gap between the two worlds, and we will touch some more advanced topics that could inspire the students\u2019 further work.<br>The course will mostly follow the first chapters of the book by R. Miranda, Algebraic curves and Riemann surfaces.\u00a0<br>                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>Combined evaluation (final exam and continuous evaluation)<\/p><br><p><br>First session<br>Weekly exercise will be provided and the students will be asked to present some of them orally. This will provide for continuous evaluation that will make up for 1\/3 of the grade. The remaining 2\/3 will be given by the result of a final oral exam.<\/p><br>Retake exam\u00a0<br>Students who fail the exam can do a retake exam at the end of the following semester.<br><p>\u00a0<\/p>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Note \/ Literature \/ Bibliography<br><\/p><br><ul><br><li>Rick Miranda, Algebraic Curves and Riemann Surfaces.<\/li><br><li>Martin Schlichenmaier, An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces.<\/li><br><li>Wilhelm Schlag, A concise course in complex analysis and Riemann surfaces<\/li><br><li>Curtis McMullen, Courses notes, Complex analysis and Riemann surfaces (available online)<\/li><br><li>J. Jost, Compact Riemann surfaces<br>The above texts are available either online or on a-z.lu. More references will be provided during the course.<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e2796d5b\"\n        >\n            <span class=\"accordion__title\">\n                Partial Differential Equations I                <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            7                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_GM-3                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e2796d5b\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e2796ce9\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Partial Differential Equations I<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 7<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_GM-3<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 1<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Objectives<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>\n                            The goal of the course it to get acquainted with Partial differential equations (PDE) as a powerful tool for modeling problems in science, providing functional analytic techniques in order to deal with PDE.                        <\/p>\n                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>On successful completion of the course the student should be able to:\u00a0<\/p><br><ul><br><li>Apply methods of Fourier Analysis to the discussion of constant coefficient differential equations<\/li><br><li>Work freely with the classical formulas in dealing with boundary value problems for the Laplace equation<\/li><br><li>Prove acquaintance with the basic properties of harmonic functions (maximum principle, mean value property) and solutions of the wave equation (Huygens property)<\/li><br><li>Solve Cauchy problems for the heat and the wave equations<\/li><br><li>Give a pedagogic talk for peers on a related topic\u00a0<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Fourier transform, the classical equations, spectral theory of unbounded operators, distributions, fundamental solutions.                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Written exam                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Literatur<br><\/p><br><ol><br><li>Rudin: Functional analysis<\/li><br><li>Jost: Postmodern analysis<\/li><br><li>Folland: Introduction to partial differential equations.<\/li><br><li>Reed-Simon: Methods of mathematical physics I-IV<\/li><br><\/ol>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e2796d71\"\n        >\n            <span class=\"accordion__title\">\n                Probabilistic Models in Finance                 <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            6                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_FM-3                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e2796d71\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e2796ce9\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Probabilistic Models in Finance <\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 6<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_FM-3<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 1<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> Yes<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Objectives<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>\n                            Introductory course to basic concepts of Mathematical Finance, also suitable for students who are not going to choose their specialization in Finance. The goal is to deepen the knowledge of modern probability theory by studying applications of general interest in an actual field of applied mathematics.                        <\/p>\n                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>On successful completion of the course, the student should be able to:<\/p><br><ul><br><li>Derive and apply formulas for option pricing and hedging strategies<\/li><br><li>Carry out calculations based on arbitrage arguments<\/li><br><li>Calculate the price of European call and put options using the Cox, Ross and Rubinstein model<\/li><br><li>Apply the techniques of Snell envelopes to evaluate American options<\/li><br><li>Derive the classical Black-Scholes formulas as limiting case of a sequence of CRR markets<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Discrete financial markets, the notion of arbitrage, discrete martingale theory, martingale transforms, complete markets, the fundamental theorem of asset pricing,\u00a0 European and American options, hedging strategies, optimal stopping, Snell envelopes, the model of Cox, Ross and Rubinstein.                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Written exam                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Literatur<br><\/p><br><ul><br><li>D. Lamberton, B. Lapeyre: Introduction au calcul stochastique appliqu\u00e9 \u00e0 la finance. Ellipses,\u00a0 1997<\/li><br><li>\u00a0S. E. Shreve: Stochastic calculus for finance. I: The binomial asset pricing model. Springer Finance, 2004\u00a0<\/li><br><li>H. F\u00f6llmer, A. Schied: Stochastic finance. An introduction in discrete time. 2nd ed., de Gruyter, 2004\u00a0<\/li><br><li>F. Delbaen, W. Schachermayer: The mathematics of arbitrage. Springer Finance, 2006<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e2796d87\"\n        >\n            <span class=\"accordion__title\">\n                Numerical Analysis                 <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            6                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_MMCS2-5                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e2796d87\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e2796ce9\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Numerical Analysis <\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 6<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_MMCS2-5<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 1<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Objectives<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>\n                            <p>The objective of the course is to provide the main tools in numerical analysis to manage rigorously the computer solution of practical problems in physics and engineering.\u00a0<\/p>                        <\/p>\n                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>At the end of the course, a student will be able<\/p><br><ul><br><li>to understand the main methods, the rigorous mathematical analysis and algorithms developed in numerical analysis<\/li><br><li>to master their concrete implementation on a computer<\/li><br><li>to formulate some basic physics or engineering applications in view of their treatment by a numerical method and its computer solution.<br><\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>The main results about the theory of numerical methods will be explained. During the course, the student will implement algorithms in Python to propose some concrete, robust and efficient scientific computing solutions to concrete problems. The course content is the following:<br><\/p><ol><li>Notion of error in numerical analysis<\/li><li>Polynomial interpolation and approximation<\/li><li>Numerical derivation and integration<\/li><li>Numerical solution of Ordinary Differential Equations<\/li><li>Direct and iterative methods for solving linear systems<\/li><li>Nonlinear equations in one and several variables<\/li><li>Project presentation + Complementary topics (optimization, eigenvalues problems)<br><\/li><\/ol>                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        The evaluation will be based on continuous assessment.<br>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ol><br><li>X. Antoine, Numerical Analysis, course at the 8xav福利导航 of Luxembourg<\/li><br><li>Ascher, Uri M and Greif, Chen, A First Course in Numerical Methods, SIAM, 2011.<\/li><br><li>Heath, Michael T, Scientific computing: an introductory survey, revised second edition SIAM, 2018.<\/li><br><\/ol>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e2796d9e\"\n        >\n            <span class=\"accordion__title\">\n                Mathematical Modelling                <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            5                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_MMCS2-3                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e2796d9e\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e2796ce9\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Mathematical Modelling<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 5<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_MMCS2-3<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 1<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Objectives<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>\n                            Develop competencies in the mathematics of advanced finite element methods with a focus on polynomial (p) type methods. More specifically, students will be able to prove mathematical results related to interpolation estimates, convergence, and stability of p-finite element methods.<br>Develop an understanding of the implementation of a p-finite element method, including appropriate algorithms and data structures. This will be achieved by the students implementing missing pieces of a provided finite element code.<br>If there is sufficient time students will also be exposed to advanced topics such as mixed finite element methods and preconditioning of finite element linear systems.<br>                        <\/p>\n                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>At the end of the course students should have a thorough understanding of the mathematical underpinnings of the polynomial (p) finite element method. Students will have a deep knowledge of the data structures and algorithms present in a modern finite element code such as Firedrake or the FEniCS Project.<br><\/p>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>Please see https:\/\/jhale.github.io\/finite-elements\/ for a full syllabus.\u00a0<\/p><p>\u00a0<\/p>                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Combined evaluation (final exam and continuous evaluation)<br><br><\/p><br><p><br>First session<br><\/p><br><ul><br><li>50% written examination<\/li><br><li>50% coursework.<\/li><br><\/ul><br><p><br>Retake exam<br>Retake exam possible if coursework component passed, otherwise course must be retaken.<\/p><br>Absence\u00a0<br>If justified absence during the written exam, an oral examination.<br>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        https:\/\/jhale.github.io\/finite-elements\/\u00a0<br>The Mathematical Theory of Finite Element Methods, Brenner S.C. and Scott, L. R., Springer 2008. https:\/\/doi.org\/10.1007\/978-0-387-75934-0 (available digitally via the uni.lu library)<br>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e2796db4\"\n        >\n            <span class=\"accordion__title\">\n                Stochastic Analysis                <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            6                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_FM-2                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e2796db4\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e2796ce9\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Stochastic Analysis<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 6<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_FM-2<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 1<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> Yes<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Objectives<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>\n                            Introduction to basic concepts of Stochastic Analysis.                        <\/p>\n                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Continuous martingales, stochastic integration, quadratic variation, It\u00f4 calculus, theorem of Girsanov, stochastic differential equations, Markov property of solutions, connection of stochastic differential equations and partial differential equations, martingale representation theorems, chaotic expansions, Feynman-Kac formula.                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Written exam                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul><br><li>I. Karatzas, S. Shreve: Brownian motion and stochastic calculus. 2nd edition. Springer, 1991<\/li><br><li>B. Oksendal:\u00a0 Stochastic differential equations. Springer, 2003<\/li><br><li>D. Revuz, M. Yor: Continuous Martingales and Brownian Motion. Springer Grundl., 1999<br><\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e2796dcb\"\n        >\n            <span class=\"accordion__title\">\n                Scientific Python                <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            1                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_MMCS2-4                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e2796dcb\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e2796ce9\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Scientific Python<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 1<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_MMCS2-4<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 1<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Objectives<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>\n                            <ul><br><li>Introduction<\/li><br><li>Getting started<\/li><br><li>Basics of Python<\/li><br><li>Array computations with numpy<\/li><br><li>Array computations with numpy (cont.)<\/li><br><li>Plotting with matplotlib<\/li><br><li>Tabular data manipulation with pandas<\/li><br><li>Tabular data manipulation with pandas (cont.)<\/li><br><li>Writing good quality and robust Python code<\/li><br><\/ul>                        <\/p>\n                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        First session<br><p>You should feel comfortable writing basic scientific programs in Python, and be able to participate fully in future courses that require an element of programming.<\/p><br><p><br>Retake exam<br><br>Retake exam not possible, course must be retaken.<\/p>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        This course covers the basics of scientific programming with Python. It is aimed at people who have done some programming before, perhaps on an undergraduate course, but need a refresher before starting their Masters or Doctoral degrees at the 8xav福利导航.<br>https:\/\/jhale.github.io\/scientific-python\/\u00a0                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        A coursework will be distributed at the end of the class. To pass the course and receive the ECTS credits you must complete the coursework.                    <\/div>\n                <\/li>\n                \n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e2796de1\"\n        >\n            <span class=\"accordion__title\">\n                The mathematics of networks: theory and algorithms                <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            3                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            MA_DS-8                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e2796de1\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e2796ce9\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> The mathematics of networks: theory and algorithms<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 3<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> MA_DS-8<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 1<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Objectives<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>\n                            <p>On successful completion of the course the students should be able to<\/p><br><ul><br><li>understand the relevance of the topics covered in the course for their applications,<\/li><br><li>master the proofs of the main results of the course,<\/li><br><li>solve problems using the toolkit developed in the course<\/li><br><li>be autonomous in learning in the field of Graph Theory.<\/li><br><\/ul>                        <\/p>\n                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>Through a presentation of selected topics, the course aims to be an introduction to graph theory, its applications and its algorithmic aspects. It is designed as a self-contained course and focused on problems pertaining to Data Science. <\/p><p>Possible topics for the course include, but are not limited to<\/p><ul><li>Graphs and digraphs, degree and the degree sequence algorithm<\/li><li>Connectedness, distance, shortest paths and connected components algorithms<\/li><li>Graph matching problems and algorithms<\/li><li>Elements of algebraic graph theory and PageRank algorithm<\/li><li>Graph traversal algorithms<\/li><li>Trees and applications<\/li><li>Minimum spanning tree algorithms<\/li><li>Network flow, min cut \u2013 max flow theorem and Ford\u2013Fulkerson algorithm<\/li><li>Centrality and betweness measures<\/li><li>Cluster analysis<\/li><li>Random Graphs<\/li><\/ul>                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>First session<br><\/p>Written exam and homework, and possibly algorithm implementation project during the semester.\u00a0\u00a0<br>Retake exam\u00a0<br>Writen exam<p>\u00a0<\/p>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Note \/ Literature \/ Bibliography<br><\/p><br><ul><br><li>R. Diestel, Graph Theory, Springer, 2017<\/li><br><li>D. Jungnickel, Graphs, Networks and algorithms, Springer 2017<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e2796df7\"\n        >\n            <span class=\"accordion__title\">\n                Student project 1                <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            4                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_GM-7                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e2796df7\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e2796ce9\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Student project 1<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 4<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_GM-7<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 1<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> <\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>On successful completion of the project the student should be able to:<\/p><br><ul><br><li>analyse complex tasks<\/li><br><li>propose solution strategies,<\/li><br><li>break up a longer project into subsequent steps,<\/li><br><li>apply a variety of methods in one project,<\/li><br><li>present a task and its solution in a scientific way.<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        The ability to work on a project, either individually or as part of a team, is one of the key skills expected of a mathematician, in and outside academia. The Student Project offers the student the opportunity to improve this skill by allowing them to work on a well-defined and well chosen project, either individually or as a team, under the guidance of a researcher.<br>The Student Project consists of project work that is carried out under the supervision of a researcher. The work is either individual or group work. At the beginning of the project, supervisor and student(s) define tasks to be carried out by the student(s), corresponding to the volume of 100 working hours (4 ECTS) for each student. The student(s) need to submit the planned topic to the course responsible for approval at the latest on 15 October (Student Project 1,3) or 15 March (Student Project 2). The project outcome is a pdf document written by the student(s). Additional outcomes (such as computer code, images, videos) can be asked for. The required outcome has to be handed in on Moodle at the latest on 31 December (Student Project 1,3) or 31 May (Student Project 2). The project file will undergo an automatic plagiarism and AI check.<br>                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        The students are marked for their project work.<br>In the case of a retake exam, a new project has to be done. The retake can be done with other supervisors.<br>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Note \/ Literature \/ Bibliography<br>Depends on the project.                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e2796e0c\"\n        >\n            <span class=\"accordion__title\">\n                Student seminar 1                <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            1                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_GM-8                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e2796e0c\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e2796ce9\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Student seminar 1<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 1<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_GM-8<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 1<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>On successful completion of the Student Seminar 1, the students should be able to:<\/p><br><ul><br><li>Acquire good insight into a field by means of individual work<\/li><br><li>Deliver a mathematical lecture on a topic of their choice<\/li><br><li>Share their knowledge with others<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Every participant chooses a supervisor among the academic staff of the Department of Mathematics or among the instructors of the Master in Mathematics and, jointly with the supervisor, a topic for a talk.<br>The audience of the talk consists at least of the participants of the Student Seminar 1, the supervisor and one other academic staff member of the Department of Mathematics (e.g. the supervisor of another talk). The duration of the talk is 40 minutes (time for questions included).                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        The evaluation is based on the quality of the talk delivered by the student                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Note \/ Literature \/ Bibliography\u00a0<br>Depends on the subject and is communicated by the chosen supervisor                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>    <\/div><\/div><\/section>\n<section\n    id=\"tabpanel-72632041-fed8-4a66-b795-326d06533510\"\n    aria-labelledby=\"tab-72632041-fed8-4a66-b795-326d06533510\"\n    class=\"tab-panel fade hidden is-hidden\"\n>\n    <div class=\"wp-block-unilux-blocks-course-offer\">\n    <h3>Course offer for Financial Mathematics, Semestre 2 (2025-2026 Summer)<\/h3>\n    \n<div id=\"accordion-module-6a141e279c1a4\" class=\"accordion accordion--theme\">\n            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e279c1d4\"\n        >\n            <span class=\"accordion__title\">\n                Advanced Stochastic Models and Financial Applications                 <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            5                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_FM-4                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e279c1d4\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e279c1a4\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Advanced Stochastic Models and Financial Applications <\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 5<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_FM-4<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 2<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> Yes<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Objectives<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>\n                            <ul><br><li>Learning the concept of stable convergence<\/li><br><li>Reviewing the theory of semimartingales<\/li><br><li>Proving the law of large numbers for power variation of semimartingales<\/li><br><li>Showing the stable central limit theorem for power variation of semimartingales<\/li><br><li>Discuss statistical applications<\/li><br><\/ul>                        <\/p>\n                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        This course gives an introduction to statistics of high frequency data. The main focus is on power variation statistics of semimartingales, and in particular on the law of large numbers and a stable limit theorem.                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Written exam at the end of the course                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Literatur<br><\/p><br><ul><br><li>Jean Jacod and Philip Protter \u201cDiscretization of Processes\u201d, Springer 2012.<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e279c1fa\"\n        >\n            <span class=\"accordion__title\">\n                Numerical Linear Algebra                 <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            5                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_MMCS2-6                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e279c1fa\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e279c1a4\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Numerical Linear Algebra <\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 5<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_MMCS2-6<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 2<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>Topics : matrices, linear systems, eigenvalues\/eigenvectors.\u00a0<br>The course is devided into three parts.\u00a0<\/p><ul><li>Part one is devoted to &#8220;reminders&#8221; about matrix theory: matrix norms, triangularization, diagonalization, spectral decomposition, perturbation theory, complexity of some matrix operations\u00a0<\/li><li>Part two is concerned with the numerical analysis of several algorithms\u00a0<ul><li>Solving linear systems<ul><li>Gaussian elimination, LU and Cholesky methods,\u2026\u00a0<\/li><li>The conjugate gradient method\u00a0<\/li><\/ul><\/li><li>The least squares problem\u00a0<ul><li>Normal equations, QR and Householder algorithms,\u2026\u00a0<\/li><\/ul><\/li><li>The eignevalues problem\u00a0<ul><li>Power methods, Jacobi, Givens&#8211;Householde, Lanczos methods,\u2026\u00a0<\/li><\/ul><\/li><\/ul><\/li><ul><li>Part three is devide to computer sessions to program some of the previous numerical methods. Programming language : Python.<\/li><\/ul><\/ul>                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul><br><li>written final exam (50%)\u00a0<\/li><br><li>project (50%) carried out by groups of 2 to more students\u00a0<br><\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Note<br><\/p><br><p>Course<\/p><br><ul><br><li><br>Part 1 (draft version)<br><\/li><br><\/ul><br><p>Exercises<\/p><br><ul><br><li><br>Exercices 1 (Preliminaries (linear algebra))<br><\/li><br><li><br>Exercices 2 (Norms, condition number)<br><\/li><br><\/ul><br><p>Computer Sessions<\/p><br><ul><br><li><br>Exercices 1 Gram-Scmidt process<br><\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e279c20a\"\n        >\n            <span class=\"accordion__title\">\n                Riemannian Geometry                <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            8                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_GM-2                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e279c20a\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e279c1a4\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Riemannian Geometry<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 8<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_GM-2<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 2<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> Yes<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>On successful completion of the course, the student should be able to:<br><\/p><br><ul><br><li>compute with tensors, such as metrics and curvature, in coordinates or coordinate-free<\/li><br><li>demonstrate detailed knowledge of the exponential map, including criteria for completeness, conjugate points, and Jacobi fields<\/li><br><li>understand multiple interpretations of curvature, for example, as the obstruction to a local parallel framing, or how it influences the spreading of geodesics via the Jacobi equation<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        This foundational Master-level course is centered around the concepts of connection on a manifold and curvature. These are investigated in further depth in the setting of Riemannian or Lorentzian metrics, where the geometry of geodesics and its relation to curvature are studied.\u00a0                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>First session<br>Oral or written exam<\/p>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Literatur<br><\/p><br><ul><br><li>M. P. do Carmo, Riemannian Geometry, Birkh\u00e4user (1992)<\/li><br><li>B. O\u2019Neill, Semi-Riemannian Geometry with applications to Relativity, Elsevier (1983)<\/li><br><li>John M. Lee, Introduction to Smooth Manifolds (2nd edition), Springer (2012)<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e279c217\"\n        >\n            <span class=\"accordion__title\">\n                Continuous Time Models in Mathematical Finance                 <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            8                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_FM-5                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e279c217\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e279c1a4\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Continuous Time Models in Mathematical Finance <\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 8<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_FM-5<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 2<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> Yes<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Objectives<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>\n                            Introduction to continuous time models in mathematical finance (Black-Scholes model).                        <\/p>\n                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Arbitrage, risk-neutral measures, option pricing, hedging, Black-Scholes-Merton equation, call-put parity, connections with partial differential equations, forwards and futures, american options, exotic options, change of num\u00e9raire, Garman-Kohlhagen formula, term-structure models, Vasicek model, Heath-Jarrow-Morton model, forward LIBOR model.                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Written exam<br>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Literatur<br><br><\/p><br><ul><br><li>M. Baxter, A. Rennie: Financial calculus. An introduction to derivative pricing. CUP, 1996\u00a0<\/li><br><li>F. E. Benth: Option Theory with Stochastic Analysis. Springer, 2004\u00a0<\/li><br><li>R. J. Elliot, P. E. Kopp: Mathematics of Financial Markets. 2nd ed., Springer Finance, 2004\u00a0<\/li><br><li>J. C. Hull:\u00a0 Options, futures, and other derivatives. 6th ed., Prentice-Hall, 2005\u00a0<\/li><br><li>S. E. Shreve: Stochastic calculus for finance. II: Continuous-time models. Springer, 2004\u00a0<\/li><br><li>J. Michael Steele: Stochastic Calculus and Financial Mathematics. Springer, 2001\u00a0<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e279c225\"\n        >\n            <span class=\"accordion__title\">\n                Mathematical Statistics                 <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            5                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            MA_DS-35                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e279c225\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e279c1a4\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Mathematical Statistics <\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 5<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> MA_DS-35<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 2<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        In parametric models, a successful student should be is able to establish the main properties (consistency, asymptotic normality, etc) of the most classical estimators, provide (possibly asymptotic) confidence regions and tests between two hypotheses.\u00a0\u00a0                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        The main problems of mathematical statistics: estimation, testing, the notion of risk. Construction of confidence intervals and tests. False discovery rate. The empirical measure and its applications: the moment method, estimation by empirical quantiles, the Kolmokorov-Smirnov Test. The maximum likelihood estimator. Exponential families in one dimension. Introduction to the Bayes paradigm and elements of decision theory.                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>First session<br>A partial written exam and a final written exam.<\/p><br><p><br>Retake exam<br><\/p><br>The retake\u00a0exam will consist of a written test that will be the same as that taken by other students sitting the exam\u00a0for the first time.\u00a0\u00a0                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Literatur<br><\/p><br><ul><br><li><br>For probability theory: Real Analysis and Probability, R.M. Dudley<\/li><br><li>For statistics: Mathematical Statistics, P. Bickel and K. Doksum<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e279c233\"\n        >\n            <span class=\"accordion__title\">\n                Advanced Graph Theory                 <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            6                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_GM-16                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e279c233\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e279c1a4\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Advanced Graph Theory <\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 6<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_GM-16<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 2<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Objectives<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>\n                            <p>\u00a0<\/p>                        <\/p>\n                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>On successful completion of the course, the student should be able to\u00a0<\/p><br><ul><br><li>Illustrate the main results and concepts with well-chosen examples<\/li><br><li>Master the proofs and techniques of the theory\u00a0<\/li><br><li>Solve exercises related to the topics covered in class\u00a0<\/li><br><li>Give an overview of the course content, focusing on his\/her own taste and favorite topics\u00a0<\/li><br><li>Communicate his\/her own pleasure in solving mathematical problem\u00a0<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>Graphs are structures that are ubiquitous in Mathematics and its applications. Through a presentation of selected topics, the course aims to be an introduction to certain modern aspects of graph theory. <br>Some basic knowledge of the concept of graphs is a prerequisite for this course.<br>According to time and taste, topics covered will be chosen among (and are not limited to) the following ones:<br><\/p><ul><li>Matching, covering and packing problems<\/li><li>Edge and vertex colorings<\/li><li>Graph polynomials<\/li><li>Ramsey theory for graphs<\/li><li>Infinite Graph Theory<\/li><\/ul>\u00a0                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Oral exam or written exam according to number of participants<br>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Literatur<br><\/p><br><ul><br><li>Reinhard Diestel. Graph Theory, 5th edition. Graduate texts in mathematics 173, Springer, 2017<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e279c240\"\n        >\n            <span class=\"accordion__title\">\n                Statistical Methods for High Dimensional Data                <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            5                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            MA_DS-11                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e279c240\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e279c1a4\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Statistical Methods for High Dimensional Data<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 5<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> MA_DS-11<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 2<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Objectives<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>\n                            The goal of the course is a detailed introduction to modern techniques in high dimensional estimation of linear models and covariance matrices. After successful completion the student should be able to establish the main properties of Lasso and related estimators in the setting of linear regression models, asymptotic results for large random matrices and non-asymptotic properties for high dimensional estimators of covariance matrices.                        <\/p>\n                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul><li>High dimensional regression model <\/li><li>Parameter estimation under constraints: Lasso and relates estimation procedures <\/li><li>Variable selection <\/li><li>Introduction to principal component analysis <\/li><li>Basic elements of random matrix theory: semi-circle law, Marcenko-Pastur distribution, Tracy-Widom distribution <\/li><li>Estimation of high dimensional covariance and precision matrices; hard and soft thresholding<\/li><\/ul>                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>First session<br><\/p><br><p>Combined evaluation<br>Midterm + final written exam\u00a0<\/p>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Bibliography<br><\/p><br><ul><br><li>P. B\u00fchlmann and S. van de Geer \u201eStatistics for High-Dimensional Data\u201c<\/li><br><li>C. Giraud \u201cIntroduction to High-Dimensional Statistics\u201d<\/li><br><li>G.W. Anderson, A. Guoinnet and O. Zeitouni \u201eAn Introduction to Random Matrices\u201c<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e279c24d\"\n        >\n            <span class=\"accordion__title\">\n                Sustainable Scientific Software                 <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            3                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_MMCS2-8                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e279c24d\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e279c1a4\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Sustainable Scientific Software <\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 3<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_MMCS2-8<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 2<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Objectives<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>\n                            During this course you will learn foundational skills that will enable you to<br>develop numerical and scientific software that is sustainable and reproducible.<br>                        <\/p>\n                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Students should be able to:<br><br><\/p><br><ul><br><li>Understand the motivation for writing scientific software to a high standard.<\/li><br><li>Work effectively as a team to write consistent and clean code, and share that code using tools, e.g. version control.<\/li><br><li>Understand appropriate methodologies for improving code quality, robustness, and performance, e.g. linting, writing unit tests, static analysis tools, debuggers, performance profilers.<\/li><br><li>Effectively plan a small piece of software, including the choice of and combination of different programming paradigms.<\/li><br><li>Reflect on the tradeoffs in terms of time to write vs overall quality. and the long term sustainability implications.<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul><li>Definitions of sustainability in a software context<\/li><li>Introduction to the Unix shell with bash<\/li><li>Introduction to version control with git<\/li><li>Clean code, style guidelines and working in a team<\/li><li>Tools for improving code.<\/li><li>Writing documentation<\/li><li>Test driven development<\/li><li>Containers<\/li><li>Continuous integration<\/li><\/ul>                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Two courseworks are set during the semester with equal weight towards the final mark.<br>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Note \/ Literature<br><\/p><br><p><br>https:\/\/jhale.github.io\/sustainable-scientific-software\/ for further links.<\/p>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e279c258\"\n        >\n            <span class=\"accordion__title\">\n                Algebraic Topology                 <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            8                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_GM-12                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e279c258\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e279c1a4\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Algebraic Topology <\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 8<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_GM-12<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 2<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>On successful completion of the course, the student should be able to:<\/p><br><ul><br><li>Explain the main definitions and results of Algebraic Topology;<\/li><br><li>Comment on new concepts;<\/li><br><li>Apply the new techniques and solve problems;<\/li><br><li>Structure the acquired abilities and summarize essential aspects adopting a higher standpoint;<\/li><br><li>Give a talk for peers or students on a related topic and write scientific texts or lecture notes, observing modern standards in scientific writing, in Didactics and in Pedagogy;<\/li><br><li>Provide evidence for the mastery of the Mathematical Method.\u00a0<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>Topology is a field of mathematics that deals with the fundamental properties of spaces. In the topological world, two spaces are considered identical if they can be continuously transformed into one another or more generally, if they are related by what topologists call homotopy equivalence. For example, a sphere and a pyramid are identical, as are a donut and a coffee cup\u2026 A main goal is therefore to classify spaces up to homotopy equivalence.<\/p><p>Algebraic topology uses algebraic methods to solve this and other topological problems. An important method is the search for algebraic homotopy invariants. In particular, we try to assign an algebraic object such as a group to each topological space and to prove that this group is invariant if we replace the space with a homotopy equivalent space. One of the proofs of the fundamental theorem of algebra uses an algebraic invariant.<\/p><p>The main homotopy invariants we discuss are the \u2018singular homology functor\u2019 and the \u2018homotopy functor\u2019. An amazing result called the Hurewicz isomorphism, shows that the first singular homology group and the (abelianized) first homotopy group of a space are two variants of the same information. Van Kampen&#8217;s theorem allows us to practice our understanding of category theory and to compute homotopy groups of larger spaces from those of smaller spaces from which they are constructed. The final chapter will highlight the relationship between the coverings of a space and subgroups of its first homotopy group.<\/p><p>The objective is to allow the student to familiarize himself with a very active field of mathematics, with broad applications throughout science. Beyond this goal, special emphasizes will be put on the Mathematical Method, i.e., the optimal technique to learn and apply mathematics. This method is the most important of the objectives of any study program in mathematics.<\/p><p>Fundamental concepts of homological algebra<br>Chain complexes, chain maps, chain homotopies, homology functor, connecting homomorphism, K\u00fcnneth isomorphism, simplicial and singular homology, homotopy invariance, examples and applications.<\/p><p>Homotopy groups<br>Algebra and topology (revision), first homotopy group, Hurewicz isomorphism, homotopy functor, homotopy invariance, typical examples and applications.<\/p><p>Van Kampen theorem<br>Free product with amalgamation, Van Kampen-Seifert theorem (basic, general and groupoid version), examples and applications (in particular: first homotopy group of a product and of a coproduct).<\/p>Covering space theory<br>Coverings of spaces, lifting property, coverings of groupoids, orbit categories, classifications and constructions of coverings, connections with the preceding chapters, examples and applications.<p>\u00a0<\/p>                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Written exam\u00a0                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Literatur<br><br><\/p><br><ul><br><li>Allen Hatcher, Algebraic Topology, Cambridge 8xav福利导航 Press, 2002 &#8211; Mathematics &#8211; 544 pages, ISBN-13: 978-0521795401\u00a0<\/li><br><li>Peter May, A concise course in Algebraic Topology, 8xav福利导航 of Chicago Press, 1999 &#8211; Mathematics &#8211; 243 pages, ISBN-13: 978-0226511832<\/li><br><li>Charles Weibel, An introduction to homological algebra, Cambridge studies in advanced mathematics 38, 1997, ISBN: 0-521-55987-1<\/li><br><li>Saunders Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics, 5(2), Springer, 1978, ISBN: 978-0-387-98403-2<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e279c263\"\n        >\n            <span class=\"accordion__title\">\n                Partial Differential Equations II                 <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            8                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_GM-15                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e279c263\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e279c1a4\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Partial Differential Equations II <\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 8<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_GM-15<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 2<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Objectives<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>\n                            Learning tools in order to deal with PDE, understanding the interplay between local and global problems and techniques.                        <\/p>\n                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Distributions as generalized functions continued, Sobolev spaces, elliptic regularity, elliptic operators on compact manifolds, some non-linear equations.\u00a0                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Written exam                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Literatur<br><\/p><br><ul><br><li>Jost: Postmodern analysis<\/li><br><li>Folland: Introduction to partial differential equations\u00a0<\/li><br><li>Reed-Simon: Methods of mathematical physics I-IV\u00a0<\/li><br><li>Aubin: Nonlinear analysis on manifolds<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e279c26e\"\n        >\n            <span class=\"accordion__title\">\n                Numerical solution of partial differential equations and applications                 <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            6                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_MMCS2-7                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e279c26e\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e279c1a4\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Numerical solution of partial differential equations and applications <\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 6<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_MMCS2-7<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 2<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Objectives<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>\n                            This course is an introduction to the numerical solution of partial differential equations (PDEs). It contains a theoretical part setting the mathematical foundations necessary for some important numerical methods used to obtain solutions to some classical PDEs, in particular the finite element method and the finite difference method.\u00a0<br>The theoretical part of the course is supported by the development of a one-dimensional Galerkin finite element code for the Poisson problem and a one-dimensional finite difference code for a scalar hyperbolic transport problem.<br>We largely follow the reference [Qua09] which is available via the National Library\u2019s Website as well as on some less official websites. In particular, we cover the contents of [Qua09, Chapters 1, 2, 3, 4, and 14].                        <\/p>\n                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>A brief survey of partial differential equations<br>This chapter briefly introduces the notion of linear PDEs and their classification into elliptic, parabolic and hyperbolic equations. We mention some classical examples, mainly issued from physics and engineering, such as the transport equation, the Laplace equation, the heat equation and the wave equation.<br><\/p><p>Elements of functional analysis<br>We introduce the main notions and theoretical results of functional analysis that are extensively used in the numerical analysis of partial differential equations. We consider the Riesz representation theorem regarding representation of continuous linear forms on a Hilbert space, and we survey the notions of bilinear form, of continuous injection of a Hilbert space into another, the notion of derivative in the sense of Fr\u00e9chet, some elements of the theory of distributions, the basic properties of the Lebesgue and Sobolev spaces, and the notion of adjoint operator.<\/p><p>Elliptic equations<br>We illustrate boundary-value problems for elliptic equations (in one and several dimensions), present their variational reformulations, treat the boundary conditions and analyze their well-posedness. Several examples of physical interest are introduced, in particular the Poisson equation, starting with the one-dimensional case, for various boundary conditions. We consider some variational formulations of these problems, and then turn to the boundary-value problems associated to the Poisson equation in the two-dimensional case. We establish that under some regularity condition the weak formulation is equivalent to the strong one. For general elliptic problems, the Lax-Milgram theorem ensures that the weak formulation is well-posed.<\/p><p>The Galerkin finite element method for elliptic problems<br>We formulate Galerkin\u2019s method for the numerical discretization of elliptic boundaryvalue problems and analyze its existence, uniqueness, stability and convergence features in an abstract functional setting. We then introduce the Galerkin finite elements method, first in one dimension, and then in several dimensions.<\/p><p>The Galerkin finite element method \u2013 a numerical example<br>This section introduces students to the development of a one-dimensional Galerkin finite element solver for the Poisson problem using Python. We focus on building the solver within a provided Jupyter notebook, providing hands-on experience with the computational and algorithmic aspects of the finite element method.<\/p><p>Finite differences for hyperbolic equations<br>The aim of this final chapter is to study classical finite differences methods for approximating solutions of first-order hyperbolic equations. We start with exposing some hyperbolic equations starting with the scalar transport problem in one dimension which we analyze by the method of characteristics. We also establish an a priori estimate by the energy method. We then turn to systems of linear hyperbolic equations in one dimension and give the example of the wave equation. Then we introduce the finite difference method, together with its variants: the forward\/centered Euler scheme, the Lax-Friedrichs scheme, the Lax-Wendroff scheme, in the case of the scalar transport problem, before to move to more general cases, for which we analyze the consistency, stability, convergence, dissipation and dispersion properties of the finite difference methods.<\/p>Finite differences \u2013 a numerical example<br>This section introduces students to the development of a one-dimensional finite difference solver for a hyperbolic transport problem using Python. We focus on building the solver within a provided Jupyter notebook, providing hands-on experience with the computational and algorithmic aspects of the finite difference method.                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>First session<br>Assessment is via coursework (30%) and final examination (70%). The due date for the coursework will be set at a later time.<\/p><br><p><br>Retake policy<br>Failure on the coursework can be compensated through the examination, and vice versa, as long as the final mark for the course is greater than or equal to 10.<br>In the event that the final mark is less than 10:<\/p><br><ul><br><li>The student may request a retake the examination once.<\/li><br><li>The student may not resubmit the coursework.<\/li><br><\/ul>If the final mark remains less than 10 the student has failed, and can take the entire course again in the next semester that it is offered.                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Reference<br>[Qua09] Alfio Quarteroni. Numerical models for differential problems. Vol. 2. Springer, 2009.<\/p>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e279c279\"\n        >\n            <span class=\"accordion__title\">\n                Student Project 2                 <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            4                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_GM-18                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e279c279\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e279c1a4\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Student Project 2 <\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 4<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_GM-18<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 2<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>On successful completion of the Student Project 2, the student should be able to:<br><\/p><br><ul><br><li>analyse complex tasks,<\/li><br><li>propose solution strategies,<\/li><br><li>break up a longer project into subsequent steps,<\/li><br><li>apply a variety of methods in one project,<\/li><br><li>present a task and its solution in a scientific way.<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        The student project consists of project work that is carried out under the supervision of a professor or a postdoc. The work is either individual or group work. Group work needs the explicit approval of the Study Director. At the beginning of the project, supervisor and student(s) define tasks to be carried out by the student(s), corresponding to the volume of 100 working hours (4 ECTS). The student(s) need to notify the Study Director of the project and the tasks at the latest on 15 March. The project outcome is a pdf document written by the student. Additional outcomes (such as computer code, images, videos) can be asked for. The required outcome has to be handed in on Moodle at the latest on 31 May. The project file will undergo an automatic plagiarism check.                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        The students are marked for their project work. In the case of a retake exam, a new project has to be done. The retake can be done with other supervisors.                    <\/div>\n                <\/li>\n                \n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e279c284\"\n        >\n            <span class=\"accordion__title\">\n                Student seminar 2                 <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            2                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_GM-19                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e279c284\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e279c1a4\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Student seminar 2 <\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 2<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_GM-19<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 2<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>On successful completion of the Student Seminar 2, the students should be able to:<br><\/p><br><ul><br><li>Fully benefit from seminar talks<\/li><br><li>Acquire good insight into a field by means of individual work<\/li><br><li>Give themselves lectures on specific topics<\/li><br><li>Share their knowledge with others<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Every participant chooses a supervisor among the academic staff of the Department of Mathematics and, jointly with the supervisor, a topic for a talk.<br>The audience of the talk consists at least of the participants of the seminar, the supervisor and one other academic staff member of the Department of Mathematics (e.g. the supervisor of another talk). The duration of the talk is 75 minutes (time for questions not included).<br>A typewritten version of this (these) lecture(s) is requested.<br>The students&#8217; seminar talks will take place during April and May. The exact timetable depends on the number of registered students and is made during the semester.<br>The supervisor together with the course responsible marks the students\u2019 performance and the written text.<p>\u00a0<\/p><p><br><\/p>                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Continuous evaluation. The mark is based on the talk and the written text delivered by the student.<br>If a student fails, a new talk and a new text can be delivered in the following semester. A supervisor change is allowed.                    <\/div>\n                <\/li>\n                \n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>    <\/div><\/div><\/section>\n<section\n    id=\"tabpanel-91d69e16-8af9-453d-8d00-158f5bfb5826\"\n    aria-labelledby=\"tab-91d69e16-8af9-453d-8d00-158f5bfb5826\"\n    class=\"tab-panel fade hidden is-hidden\"\n>\n    <div class=\"wp-block-unilux-blocks-course-offer\">\n    <h3>Course offer for Financial Mathematics, Semestre 3 (2025-2026 Winter)<\/h3>\n    \n<div id=\"accordion-module-6a141e27a1be6\" class=\"accordion accordion--theme\">\n            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e27a1c0e\"\n        >\n            <span class=\"accordion__title\">\n                Lie Algebras and Lie Groups                <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            6                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_GM-20                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e27a1c0e\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e27a1be6\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Lie Algebras and Lie Groups<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 6<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_GM-20<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 3<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Objectives<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>\n                            The purpose of this course is to give an introduction to the theory of finite dimensional Lie groups and Lie algebras, assuming knowledge of linear algebra and differentiable topology.                        <\/p>\n                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>On successful completion of the course, the student should be able to:<\/p><br><ul><br><li>Explain the deep interrelations between Lie groups and Lie algebras, as well as the technical tools behinds these interrelations.<\/li><br><li>Simplify mathematical problems admitting symmetry Lie groups actions to problems admitting symmetry actions of their Lie algebras.<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Lie groups, Lie algebra of a Lie group, the exponential map and the Lie group-Lie algebra correspondence; the adjoint representation; structure theory and classification results, with focus on simple Lie algebras.                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Combined evaluation (final exam and continuous evaluation)<\/p><br><p><\/p><br><ol><br><li>If justified absent during midterms, a make-up oral exam.<\/li><br><li>If justified absence during the written exam, a make-up written exam.<\/li><br><\/ol>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ol><br><li>Representation Theory by W. Fulton and J. Harris<\/li><br><li>Notes on Lie Algebras by H. Samelson<\/li><br><li>\u201cPrerequisites from Differential Geometry\u201d by Sergei Merkulov (free to download)<\/li><br><\/ol>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e27a1c2f\"\n        >\n            <span class=\"accordion__title\">\n                Nonparametric Statistics                <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            5                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            MA_DS-18                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e27a1c2f\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e27a1be6\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Nonparametric Statistics<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 5<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> MA_DS-18<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 3<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Objectives<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>\n                            <p>Upon completion a successful student should be able to<\/p><br><ul><br><li>Construct classical estimators of unknown probability distribution<\/li><br><li>Derive asymptotic properties of empirical distribution functions<\/li><br><li>Perform tests for unknown distribution functions<\/li><br><li>Construct non-parametric estimators of the density<\/li><br><li>Estimate components of a non-linear regression model<\/li><br><li>Understand the concept of minimax estimation <\/li><br><\/ul>                        <\/p>\n                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul><br><li>Understanding the concepts of empirical distribution functions, kernel estimation and minimax theory<\/li><br><li>Performing derivation of standard estimation and testing methods in nonparametric statistics <br><\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul><li>Estimation of probability measures<\/li><li>Weak limit theorems for empirical measures<\/li><li>Kolmogorov-Smirnov and Cramer-von Mises tests<\/li><li>Estimation of the density<\/li><li>Nadaraja-Watson estimator and general kernel estimators<\/li><li>Non-linear regression models<\/li><li>Minimax theory<\/li><\/ul>                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        First session<br>Solving bi-weekly exercises (40%) and a final exam (60%) will build the overall grade.<br>Retake exam<br>Written exam.<br>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Literatur<br><\/p><br><ul><br><li>A.B. Tsybakov (2009): \u201cIntroduction to Nonparametric Estimation\u201d, Springer.<\/li><br><li>A. Van der Vaart and J. Wellner\u00a0 (1996): \u201cWeak Convergence and Empirical Processes\u201d, Springer.<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e27a1c47\"\n        >\n            <span class=\"accordion__title\">\n                Numerical Methods for Variational Problems                 <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            5                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_MMCS2-9                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e27a1c47\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e27a1be6\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Numerical Methods for Variational Problems <\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 5<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_MMCS2-9<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 3<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> <\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        At the end of the course students should have a thorough understanding of the mathematical underpinnings of the polynomial (p) finite element method and its implementation in Python code. If there is sufficient time students will also be exposed to advanced topics such as mixed finite element methods and preconditioning finite element linear systems.<br>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Please see https:\/\/jhale.github.io\/finite-elements\/\u00a0for a full syllabus.\u00a0                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>First session<br>50% written examination.<br>50% coursework.<\/p><br><p><br>Retake examRetake exam possible if coursework component passed, otherwise course must be retaken.<\/p><br><p><br>Absence<br>Absence at final exam treated via procedure in R\u00e8glement des \u00e9tudes.<\/p>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Note \/ Literature \/ Bibliography<br><br>https:\/\/jhale.github.io\/finite-elements\/\u00a0<br>The Mathematical Theory of Finite Element Methods, Brenner S.C. and Scott, L. R., Springer 2008. https:\/\/doi.org\/10.1007\/978-0-387-75934-0\u00a0(available digitally via the uni.lu library)                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e27a1c5e\"\n        >\n            <span class=\"accordion__title\">\n                Advanced Econometrics                <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            6                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            QTECOFIN-2                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e27a1c5e\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e27a1be6\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Advanced Econometrics<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 6<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> QTECOFIN-2<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 3<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Objectives<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>\n                            This course is intended for students enrolled in the Master of Science program in Quantitative Economics and Finance.<br>Ph.D. students interested in learning this material are also welcome. The objective of this course is to familiarize students with microeconometric models and methods that are widely used in economics, finance, and social science research. We will focus on linear models. Nonlinear models are covered in a subsequent course.<br>On completion of the course unit successful students will be able to:<br> Have a good understanding of some widely used econometric models and techniques used by economists to answer policy related questions<br> Understand how these models are identified, estimated, and tested, how the asymptotic distributions of the various estimators and test statistics are obtained, and the fundamental assumptions underlying these results. This will not only enable the students to process and interpret empirical data and test whether they are in accordance with economic theory, but should also help them read, understand, and critically evaluate the econometrics articles in peer-reviewed journals encountered during the course of their own research.<br>                        <\/p>\n                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>COURSE CONTENT<br>1. Basics.<br>_ What is a statistical model?<br>_ Interpreting and estimating models of causal relationships with observational data.<br>_ Exogenous and endogenous variables.<br>_ Reference: Chapter 1 and class-notes.<br>2. Statistics review.<br>_ Best linear predictors and best predictors.<br>_ Conditional expectations and their properties.<br>_ Basic asymptotic theory.<br>_ Applications of asymptotic theory to estimation and testing.<br>_ Reference: Chapters 2, 3 and class-notes.<br>3. Single equation linear models.<br>_ Ordinary least squares.<br>_ Consistency and asymptotic normality of OLS estimators.<br>_ Estimating heteroscedasticity-robust standard errors.<br>_ Problems associated with endogenous regressors.<br>_ Prevalence of endogenous regressors in econometric models.<br>_ The instrumental variables (IV) solution to endogenous regressors.<br>_ Reference: Chapter 4 and class-notes.<br>4. IV estimation of single equation models.<br>_ The two-stage least squares (2SLS) estimator and its properties.<br>_ The control function estimator and its properties.<br>_ Testing for endogeneity.<br>_ Relationship between 2SLS and control function estimators.<br>_ Reference: Chapters 5, 6 and class-notes.<br>5. Generated regressors and instruments.<br>_ OLS with generated regressors.<br>_ 2SLS with generated instruments.<br>_ Effect of generated regressors and instruments on the properties of estimators.<br>_ Specification testing in linear models with generated regressors.<br>_ Reference: Chapter 6 and class-notes.<br>6. Generalized method of moments (GMM)<br>_ Econometric models defined in terms of moment conditions.<br>_ Under-identified, just-identified and over-identified moment-condition models.<br>_ Estimation of just- and over-identified moment-condition models.<br>_ Asymptotic properties of GMM estimators.<br>_ Two-step optimal GMM estimators.<br>_ GMM based hypothesis testing.<br>_ GMM based specification testing.<br>_ Reference: Chapter 14 and class-notes.<br>7. Systems of equations with exogenous regressors.<br>_ Seemingly unrelated regressions (SUR).<br>_ Optimal GMM estimation of SUR models.<br>_ Imposing cross-equation restrictions.<br>_ Reference: Chapter 7 and class-notes.<br>8. Systems of equations with endogenous regressors.<br>_ GMM estimation of SUR with varying instruments.<br>_ GMM versus 3SLS.<br>_ Reference: Chapter 8 and class-notes.<br>9. Linear simultaneous equations.<br>_ Classical supply and demand models.<br>_ Natural instruments: identification via exclusion restrictions.<br>_ Identification via cross-equation and covariance restrictions.<br>_ Reference: Chapter 9 and class-notes.<br>10. Linear panel data models.<br>_ Linear panel data models without unobserved effects.<br>_ Linear panel data models with unobserved effects.<br>_ Estimating fixed effects models under strict exogeneity.<br>_ The random effects estimator.<br>_ Testing the random effects hypothesis.<br>_ Correlated random effects: Hausman-Taylor type models.<br>_ Dynamic panel data models and sequential exogeneity.<\/p><p>_ Reference: Chapters 10, 11 and class-notes.<\/p>                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul><br><li>60% of the course grade is based on the final exam, which is closed book and closed notes. The final exam is comprehensive, i.e., it includes all material covered in the class, the homeworks, and the TA sessions<\/li><br><li>40% of the course grade is based on 5 homework assignments. The completed homeworks have to be handwritten and handed in individually<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Bibliography:<br><\/p><br><p><br>The required text for this course is<br>1. Econometric analysis of cross section and panel data, 2nd ed., J. M. Wooldridge (2010).<br>All chapter references in the syllabus are to this book. <br><\/p><br><p>Students interested in additional reading may consider the following references:<br> \u201cEconometrics,\u201d B. Hansen, 2022.<br> \u201cEconometric theory and methods,\u201d J. G. MacKinnon and R. Davidson, 2009.<br> \u201cMostly harmless econometrics: An empiricist&#8217;s companion,\u201d J. D. Angrist and J. S. Pischke, 2008.<br> \u201cMicroeconometrics: Methods and applications,\u201d A. C. Cameron and P. K. Trivedi, 2005. Also by the same authors, \u201cMicroeconometrics using STATA,\u201d 2009.<br> \u201cPanel data econometrics,\u201d M. Arellano, 2003.<br> \u201cEconometric analysis,\u201d W. H. Greene, 2003.<br> \u201cEconometrics,\u201d F. Hayashi, 2000.<br> \u201cAn introduction to classical econometric theory,\u201d P. Ruud, 2000.<br> \u201cStatistics and econometric models,\u201d vol. 1 and 2, C. Gourieroux and A. Monfort, 1995.<br> \u201cA course in econometrics,\u201d A. S. Goldberger, 1991.<br> \u201cLarge sample estimation and hypothesis testing,\u201d W. K. Newey and D. McFadden, in Handbook of Econometrics, vol. 4, 1994.<br> \u201cAdvanced econometrics,\u201d T. Amemiya, 1985.<br> \u201cHandbooks of econometrics,\u201d vol. 1 &#8211; vol. 6B, 1983 &#8211; 2007.<br><\/p>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e27a1c77\"\n        >\n            <span class=\"accordion__title\">\n                American Options: Optimal stopping theory and numerical methods                 <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            5                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_FM-7                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e27a1c77\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e27a1be6\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> American Options: Optimal stopping theory and numerical methods <\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 5<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_FM-7<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 3<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> Yes<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Objectives<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>\n                            The lectures provide an introduction to American options and optimal stopping problems (theoretical foundation and numerical implementation).                        <\/p>\n                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>On successful completion of the course, the student should be able to:<\/p><br><ul><br><li>Explain pricing and hedging of \u2018Vanilla\u2019 American options in a multi-asset Black-Scholes model using bi- or multinomial trees<\/li><br><li>Extend multinomial tree methods to the pricing and hedging of American options in local volatility or stochastic volatility models (CEV, SABR, Heston\u2026)<\/li><br><li>Implement efficiently a regression method \u2018\u00e0 la Longstaff-Schwarz\u2019 for solving multi-dimensional optimal stopping problems like the pricing and the hedging of multi-asset American options or energy derivatives like swing options (take-or-pay gas con-tracts)<\/li><br><li>Implement an optimal quantization based numerical scheme to price and hedge multi-asset American style derivatives<\/li><br><li>Analyze these numerical methods and their respective ranges of efficiency, especial-ly compared to PDE methods<br><\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul><li>Discrete time optimal stopping theory: Snell envelope, Rogers&#8217; dual representation, optimal stopping times\u00a0<\/li><li>Backward dynamical programming principle\u00a0<\/li><li>American options pricing and hedging in a complete market, with and without div-idends\u00a0<\/li><li>Bermuda options (discrete time)\u00a0<\/li><li>Swing options on energy markets\u00a0<\/li><li>Numerical methods: variational inequality (PDE method), regression methods (Longstaff-Schwarz), Quantization methods with a priori error bounds\u00a0<\/li><li>Continuous time optimal stopping theory and applications to complete markets\u00a0<\/li><li>R\u00e9duites, variational inequalities, free boundary\u00a0<\/li><\/ul>                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>First session<br><br>Written or oral exam (depending on the number of enrolled students)<\/p><br><p><br>Retake exam<br><br>Written or oral exam (depending on the number of enrolled students)<\/p>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Note \/ Literature \/ Bibliography<br><br>Support\u00a0<br>provided in class<br><br>Literature<br><\/p><br><ul><br><li>L. Carassus, G. Pag\u00e8s (2015). Finance de march\u00e9, Mod\u00e8les math\u00e9matiques \u00e0 temps discret, Vuibert, 400p.<\/li><br><li>V. Bally, G. Pag\u00e8s, J. Printems (2005). A quantization tree method for pricing and hedging multidimensional American options, Mathematical Finance, 15(1):119-168.\u00a0<\/li><br><li>\u00c9. Cl\u00e9ment, P. Protter, D. Lamberton (2002). An analysis of a least squares regression method for American option pricing, Finance Stochastics, 6(2):449-47\u00a0<\/li><br><li>N. El Karoui (1981). Les aspects probabilistes du contr\u00f4le stochastique, Lecture Notes in Mathematics 876, 72-238, Springer Verlag.\u00a0<\/li><br><li>D. Lamberton, B. Lapeyre (1996). Introduction to stochastic calculus applied to fi-nance. Chapman Hall, London, 185 pp. ISBN: 0-412-71800-6.\u00a0<\/li><br><li>F.A. Longstaff, E.S. Schwarz (2001). Valuing American options by simulation: a simple least-squares approach, Review of Financial Studies, 14:113-148.\u00a0<\/li><br><li>J. Neveu (1964). Bases math\u00e9matiques du calcul des Probabilit\u00e9s, Masson, Paris; Eng-lish translation Mathematical Foundations of the Calculus of Probability (1965), Holden Day, San Francisco.\u00a0<\/li><br><li>A. N. Shyriaev (2008). Optimal stopping rules. Translated from the 1976 Russian sec-ond edition by A. B. Aries. Reprint of the 1978 translation. Stochastic Modeling and Applied Probability, 8, Springer-Verlag, Berlin, 217 pp.\u00a0\u00a0<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e27a1c8e\"\n        >\n            <span class=\"accordion__title\">\n                Numerical methods in Finance                 <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            5                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_FM-8                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e27a1c8e\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e27a1be6\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Numerical methods in Finance <\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 5<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_FM-8<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 3<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> Yes<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Objectives<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>\n                            The objective of this course is to present numerical methods currently used in finance, especially in option pricing and portfolio optimization, and to introduce basic skill programming to tackle problems encountered in mathematical modelling of finance.<br>                        <\/p>\n                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>On successful completion of the course, the student should be able to:<\/p><br><ul><br><li>Explain and apply relevant numerical methods currently used in finance.<\/li><br><li>Carry out calculations based on Monte Carlo methods, especially in option pricing and portfolio optimization.<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>First, we will introduce mathematical problems occurring in finance, mainly in option pricing and portfolio optimization in financial markets following stochastic models. These formulations call upon partial differential equations, conditional expectation, optimal time, dynamic programming and control. Next, we shall describe methods to approach numerical simulations of these objects. Finally, in practical sessions, we will use recent software and Python programming to illustrate efficiency of these methods.<\/p><p>The course will be organized in three parts:<\/p><ul><li>PDE methods for option pricing and numerical methods in stochastic control: 15h (A. Sulem)<\/li><li>Monte Carlo methods (tree method, regression, machine learning) : 8h (Ludovic Gouden\u00e8ge)<\/li><li>Applied sessions with computer using the computational finance software &#8220;Premia&#8221; (www.premia.fr) and Python programming: 7h (Ludovic Gouden\u00e8ge)<br><\/li><\/ul>                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>First session<br><br>Written exam with possibly an applied project<\/p><br><p><br>Retake exam<br><br>Written exam with possibly an applied project<\/p>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Note \/ Literature<br><\/p><br><p>D. Lamberton and B. Lapeyre. Introduction to Stochastic Calculus Applied to Finance. Chapman and Hall, 2nd edition, 2007.\u00a0<\/p><br><ul><br><li>B. Lapeyre, A. Sulem and D.Talay. Simulation of Financial Models: Mathematical Foundations and Applications, to appear. Cambridge 8xav福利导航 Press.\u00a0<\/li><br><li>B. Jourdain. Probabilit\u00e9s et statistique. Ellipes, 2009.<\/li><br><li>T. M. Mitchell. Machine learning. WCB\/McGraw-Hill, 1997.<\/li><br><li>G. Pag\u00e8s. Introduction to numerical methods in probability in finance. 3em cycle Hanoi. Vietnam, 2007.<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e27a1ca4\"\n        >\n            <span class=\"accordion__title\">\n                Bayesian Statistics                <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            5                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            MA_DS-22                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e27a1ca4\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e27a1be6\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Bayesian Statistics<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 5<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> MA_DS-22<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 3<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Objectives<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>\n                            Provide students with the necessary knowledge of Bayesian statistics, both from a theoretical as well as practical viewpoint.\u00a0                        <\/p>\n                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Handle of the concepts from Bayesian statistics, Bayesian inference, Bayesian modelling, practical handle of Bayesian techniques on Python.                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        This course will provide a thorough introduction and overview on the most important concepts from Bayesian inference, starting with the Bayesian philosophy in contrast to frequentist statistics. We will discuss various choices of prior distributions, Bayesian inference, Bayesian modelling, model checking and comparison (in particular introducing the concept of Bayes factor), and advanced computation. The latter aspect will especially be dealt with in the practicals, where Markov Chain Monte Carlo methods shall be treated.                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        First session<br>Written exam with open questions, both theory and exercises, in particular there will be computational exercises<br>Retake exam<br>Written exam with open questions, both theory and exercises, in particular there will be computational exercises                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Note \/ Literature \/ Bibliography\u00a0<br><\/p><br><ul><br><li>Box, GEP and Tiao, GC (1992) Bayesian Inference in Statistical Analysis. John Wiley and Sons<\/li><br><li>Gelman, A, Carlin, JB, Stern, HS, Dunson, DB, Vehtari, A and Rubin, DB (2013) Bayesian Data Analysis 3rd edition. Chapman Hall\/CRC.<\/li><br><li>Brooks, S, Gelman, A, Jones, GL, Meng, X-L (2011) Handbook of Markov Chain Monte Carlo. Chapman Hall\/CRC<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e27a1cbb\"\n        >\n            <span class=\"accordion__title\">\n                Gaussian processes and applications                 <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            5                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_FM-9                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e27a1cbb\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e27a1be6\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Gaussian processes and applications <\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 5<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_FM-9<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 3<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Objectives<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>\n                            The objective is to learn the basic concepts and techniques associated with Gaussian processes, that are omnipresent in modelling random phenomena (finance, physics, machine learning, statistics, etc.).<br>                        <\/p>\n                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>On successful completion of the course, the student should be able to:\u00a0\u00a0<\/p><br><ul><br><li>Explain the language, basic concepts and techniques associated with Gaussian variables, vectors, and processes<\/li><br><li>Identify, analyse, and prove relevant properties of models based on a Gaussian structure<\/li><br><li>Solve exercises involving a Gaussian structure<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul><li>Gaussian random variables (characteristic function, CLT, stability properties, Stein&#8217;s lemma)<\/li><li>Gaussian random vectors (definition, characteristic function, existence, uniqueness in law, multivariate CLT, density, Hermite polynomials)<\/li><li>Gaussian random processes (definition, modifications, uniqueness in law, function of positive type, existence, Brownian motion, continuity)<\/li><li>Fractional Brownian motion (definition, existence, H\u00f6lder regularity)<\/li><\/ul>                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>First session<br>Written exam during the exam period (January\/February).<\/p><br><p><br>Retake exam<br>Written exam\u00a0during the exam period July).<\/p>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Note \/ Literature \/ Bibliography<br>Will be discussed in class                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e27a1cd1\"\n        >\n            <span class=\"accordion__title\">\n                Data Science                 <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            5                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_GM-24                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e27a1cd1\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e27a1be6\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Data Science <\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 5<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_GM-24<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 3<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Objectives<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>\n                            The successful candidate will understand the fundamental theoretical concepts of data science and be able to work with real data. The aim is to contribute to data-driven AI as outlined in the German government&#8217;s AI strategy paper.                        <\/p>\n                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>On successful completion of the course the student should be able to:<\/p><br><ul><br><li>Explain and apply basic theoretical concepts on selected aspects of data science.<\/li><br><li>Develop appropriate solutions in the scope of a data science towards a data-driven AI<\/li><br><li>Consolidate acquired competences in the subject area<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        In this course, the term \u2018data\u2019 is seen centric and we will look at data from different perspectives in order to serve a data-driven AI. We will discuss selected aspects of Data Preparation and Preprocessing, Data Statistics, Data Security, Data Privacy, Data Management, Big and Small Data, Data Retrieval, Data Visualization, and Data Analytics. In the project part, each participant contributes to a public poster workshop, in which the results of the project has to be shown.\u00a0                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>\u00a0<\/p><br><p><br>Poster Workshop Attendance and Presentation of the project and final report<br><\/p><br><p><br>First session<br>50% written report + 50% presentation of the project results as poster at the AICaf\u00e9 2025:\u00a0<br>The candidate will participate in the AICaf\u00e9 2025 \u2013 AI Competition (4 December 2025, 18h00 \u2013 21h00, Cercle Cit\u00e9 Luxembourg, Auditoire Cit\u00e9) and show the project work in the field of a Data-driven AI. This is a mandatory examination requirement.<\/p><br><p><br>Retake exam<br>Retakes are offered in Summer. Candidates who wish to do are pleased to get in contact with Prof. Schommer.<\/p><br><p>\u00a0<\/p><br><p>\u00a0<\/p>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Note \/ Literature \/ Bibliography\u00a0<br><\/p><br><ul><br><li>Elmasri, Navathe: Fundamentals of Database Systems. Pearson Addison Wesley. 2006.<\/li><br><li>Han, Kamber: Data Mining \u2013 Concepts and Techniques. Morgan Kaufmann. 2011.<\/li><br><li>Manning, Raghavan, Sch\u00fctze: Introduction to Information Retrieval. Cambridge 8xav福利导航 Press.<br>Ware: Information Visualization. Morgan Kaufmann. 2012.<\/li><br><li>Witten, Kamber: Data Mining: Practical Machine Learning Tools and Techniques. Morgan Kaufmann.<\/li><br><li>Aggarwal, Yu: Privacy-Preserving Data Mining \u2013 Models and Algorithms. Springer. 2008.<\/li><br><li>Marz: Big Data: Principles and best practices of scalable realtime data systems. Manning. 2015.<br>as well as different articles, reports, and journals contributions.<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e27a1ce7\"\n        >\n            <span class=\"accordion__title\">\n                Continuous-Time Stochastic Calculus and Interest Rate Models                 <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            5                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_FM-6                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e27a1ce7\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e27a1be6\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Continuous-Time Stochastic Calculus and Interest Rate Models <\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 5<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_FM-6<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 3<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> Yes<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>On successful completion of the course, the student should be able to:<\/p><br><ul><br><li>Calculate probabilities and expectations related to the semi-martingale models presented in the lectures<\/li><br><li>Carry out calculations based on change of num\u00e9raire and no-arbitrage pricing<\/li><br><li>Compute the prices of interest rate derivatives<\/li><br><li>Apply stochastic volatility models to deal with implied volatility surfaces<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        Basic Notions of Fixed Income Markets; Semimartingale Modeling; Stochastic Differential Equations; No-Arbitrage Pricing; Change of Num\u00e9raire; Short Rate Models; Heath-Jarrow-Morton Framework; Market Models; Stochastic Volatility.<br>                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>First session<br>Written exam<\/p><br><p><br>Retake exam<br>Written exam<\/p>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Note<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>Note \/ Literature \/ Bibliography\u00a0<br><\/p><br><ul><br><li>D. Brigo and F. Mercurio (2006) Interest Rate Models : Theory and Practice. Springer Verlag\u00a0<\/li><br><li>Damir Filipovic (2009). Term-structure models. A graduate course. Springer Verlag.<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e27a1cfe\"\n        >\n            <span class=\"accordion__title\">\n                Internship (in a company)                 <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            5                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_MMCS2-13                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e27a1cfe\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e27a1be6\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Internship (in a company) <\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 5<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_MMCS2-13<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 3<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>On successful completion of the Internship in a Company, the student should be able to:<br><\/p><br><ul><br><li>Identify employment opportunities in the local job market and possible career tracks;<\/li><br><li>Apply insights gained during the internship into the functioning and the operational business of companies and institutions outside the academic world;<\/li><br><li>Perform work in an institution outside the academic world at a level adequate to a Master student in mathematics;<\/li><br><li>Apply and adapt the theoretical skills and techniques learned in the Master programme to the practical needs of the chosen company or institution.<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        The objective of the Internship in a Company is to allow the student to get acquainted with a professional work situation outside the academic context.<br>The Internship is typically carried out in the summer break between the 1st and the 2nd year and counts for the 3rd semester. A typical duration is 2 months.                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>The course is either succeeded or failed, but not marked.<br>For succeeding, the student needs to get an overall positive evaluation by the company supervisor (Annexe 1 to the convention), and the student needs to hand in a report after the internship, which must be judged satisfactory by the academic supervisor and the SPD. The report needs to cover at least the following topics:<\/p><br><ul><br><li>Short description of the company and the work environment;<\/li><br><li>Description of the tasks, objectives and the performed work (possibly with examples);<\/li><br><li>How the student was integrated in the company\/institution;<\/li><br><li>An indication if the work performed by the student was successful with respect to the objectives;<\/li><br><li>How the performed work fits in her\/his Master studies;<\/li><br><li>How the student liked it, if (s)he would recommend other students to choose this internship, possibly with recommendations.<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e27a1d15\"\n        >\n            <span class=\"accordion__title\">\n                Student Project 3                 <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            4                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_GM-25                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e27a1d15\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e27a1be6\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Student Project 3 <\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 4<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_GM-25<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Financial Mathematics 3<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Course learning outcomes<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p>On successful completion of the Student Project 1 (or 3) the student should be able to:<\/p><br><ul><br><li>analyse complex tasks,<\/li><br><li>propose solution strategies,<\/li><br><li>break up a longer project into subsequent steps,<\/li><br><li>apply a variety of methods in one project,<\/li><br><li>present a task and its solution in a scientific way.<\/li><br><\/ul>                    <\/div>\n                <\/li>\n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Description<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        The ability to work on a project, either individually or as part of a team, is one of the key skills expected of a mathematician, in and outside academia. The Student Project offers the student the opportunity to improve this skill by allowing them to work on a well-defined and well chosen project, either individually or as a team, under the guidance of a researcher.<br>The Student Project consists of project work that is carried out under the supervision of a researcher. The work is either individual or group work. At the beginning of the project, supervisor and student(s) define tasks to be carried out by the student(s), corresponding to the volume of 100 working hours (4 ECTS) for each student. The student(s) need to submit the planned topic to the course responsible for approval at the latest on 15 October (Student Project 1,3) or 15 March (Student Project 2). The project outcome is a pdf document written by the student(s). Additional outcomes (such as computer code, images, videos) can be asked for. The required outcome has to be handed in on Moodle at the latest on 31 December (Student Project 1,3) or 31 May (Student Project 2). The project file will undergo an automatic plagiarism and AI check.<br><p>\u00a0<\/p><p>\u00a0<\/p>                    <\/div>\n                <\/li>\n                \n                \n                                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Assessment<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <p><br>First session<br>The students are marked for their project work. In the case of a retake exam, a new project on the same or a different topic has to be done.\u00a0<\/p><br><p><br>Retake exam<br>The retake can be done with a different supervisor.<\/p>                    <\/div>\n                <\/li>\n                \n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>    <\/div><\/div><\/section>\n<section\n    id=\"tabpanel-39303298-71a6-445f-952b-72604c08c906\"\n    aria-labelledby=\"tab-39303298-71a6-445f-952b-72604c08c906\"\n    class=\"tab-panel fade hidden is-hidden\"\n>\n    <div class=\"wp-block-unilux-blocks-course-offer\">\n    <h3>Course offer for Financial Mathematics, Semestre 4 (2025-2026 Summer)<\/h3>\n    \n<div id=\"accordion-module-6a141e27a7224\" class=\"accordion accordion--theme\">\n            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e27a7242\"\n        >\n            <span class=\"accordion__title\">\n                Master Thesis                <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            30                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_FM-16                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e27a7242\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e27a7224\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Master Thesis<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 30<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_FM-16<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Master Thesis<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> EN<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                \n                \n                \n                \n                \n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>            \n<div class=\"accordion__item\">\n    <h4 class=\"accordion__header\">\n        <button\n                type=\"button\"\n                class=\"accordion__button collapsed\"\n                aria-expanded=\"false\"\n                data-bs-toggle=\"collapse\"\n                data-bs-target=\"#accordion-panel-6a141e27a7256\"\n        >\n            <span class=\"accordion__title\">\n                Master Thesis with internship                <span class=\"accordion__data\">\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">ECTS credits:<\/span>\n                        <span class=\"accordion__data__value\">\n                            30                        <\/span>\n                    <\/span>\n                    <span class=\"accordion__data__item\">\n                        <span class=\"accordion__data__name\">Course code:<\/span>\n                        <span class=\"accordion__data__value\">\n                            F1_MA_MAT_FM-11                        <\/span>\n                    <\/span>\n                <\/span>\n            <\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-down icon--lg\"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\"><\/use><\/svg>        <\/button>\n    <\/h4>\n\n    <div id=\"accordion-panel-6a141e27a7256\" class=\"accordion__collapse collapse\" data-bs-parent=\"#accordion-module-6a141e27a7224\">\n        <div class=\"accordion__body p-0\">\n            <ul class=\"list-highlighted\">\n                <li class=\"list-highlighted__item items-start\">\n                    <header class=\"list-highlighted__header\">\n                        <h5 class=\"text-18\"><strong>Details<\/strong><\/h5>\n                    <\/header>\n                    <div class=\"list-highlighted__body\">\n                        <ul>\n                            <li><span class=\"text-dark font-heading font-bold\">Course title:<\/span> Master Thesis with internship<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Number of ECTS:<\/span> 30<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Course code:<\/span> F1_MA_MAT_FM-11<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Module(s):<\/span> Master Thesis with internship<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Language:<\/span> FR<\/li>\n                            <li><span class=\"text-dark font-heading font-bold\">Mandatory:<\/span> No<\/li>\n                        <\/ul>\n                    <\/div>\n                <\/li>\n\n                <!--<li class=\"list-highlighted__item items-start\">-->\n                <!--    <header class=\"list-highlighted__header\">-->\n                <!--        <h5 class=\"text-18\"><strong>--><!--<\/strong><\/h5>-->\n                <!--    <\/header>-->\n                <!--    <div class=\"list-highlighted__body\">-->\n                <!--        <div class=\"speaker-group\">-->\n                <!--            -->                <!--        <\/div>-->\n                <!--    <\/div>-->\n                <!--<\/li>-->\n\n                \n                \n                \n                \n                \n                \n            <\/ul>\n        <\/div>\n    <\/div>\n<\/div>    <\/div><\/div><\/section>\n<\/div>\n<\/div><\/section>\n\n\n<section class=\"section section wp-block-unilux-blocks-quick-link-discover-section\">\n    <div class=\"container xl:max-w-screen-xl\">\n        \n<h2 class=\"has-text-align-left wp-block-unilux-blocks-heading\"        id=\"important-information\"\n    >\nImportant information<\/h2>\n\n<ul class=\"wp-block-unilux-blocks-quick-link-discover quick-link-list\">\n<li class=\"wp-block-unilux-blocks-quick-link-discover-item\">\n    <a\n                    href=\"https:\/\/www.uni.lu\/fstm-en\/study-programs\/master-in-mathematics-financial-mathematics\/new-students\/\"\n                    class=\"quick-link\"\n            target=\"\"\n    >\n            <span class=\"quick-link__container\">\n                <span class=\"quick-link__text\">\n                    Refresher Courses                <\/span>\n                <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-right \"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-right\"><\/use><\/svg>            <\/span>\n    <\/a>\n<\/li>\n\n\n\n<li class=\"wp-block-unilux-blocks-quick-link-discover-item\">\n    <a\n                    href=\"https:\/\/www.uni.lu\/life-en\/financial-support\/scholarships\/#scholarships-for-master-programmes\"\n                    class=\"quick-link\"\n            target=\"\"\n    >\n            <span class=\"quick-link__container\">\n                <span class=\"quick-link__text\">\n                    FSTM Scholarships                <\/span>\n                <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-right \"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-right\"><\/use><\/svg>            <\/span>\n    <\/a>\n<\/li>\n<\/ul>\n    <\/div>\n<\/section><\/div>","protected":false},"excerpt":{"rendered":"<p>The Financial Mathematics track of the Master in Mathematics is a two-year programme taught in English. To boost their future career, the track includes internships and allows students to write their Master thesis in a professional setting during an internship. New students are requested to take the Refresher Courses scheduled before the start of the first semester.<\/p>\n","protected":false},"author":33,"featured_media":0,"parent":1497,"menu_order":0,"template":"","meta":{"featured_image_focal_point":[],"show_featured_caption":false,"ulux_newsletter_groups":"","uluxPostTitle":"Master in Mathematics - Financial Mathematics","uluxPrePostTitle":"Study Programme","_trash_the_other_posts":false,"_price":"","_stock":"","_tribe_ticket_header":"","_tribe_default_ticket_provider":"","_tribe_ticket_capacity":"0","_ticket_start_date":"","_ticket_end_date":"","_tribe_ticket_show_description":"","_tribe_ticket_show_not_going":false,"_tribe_ticket_use_global_stock":"","_tribe_ticket_global_stock_level":"","_global_stock_mode":"","_global_stock_cap":"","_tribe_rsvp_for_event":"","_tribe_ticket_going_count":"","_tribe_ticket_not_going_count":"","_tribe_tickets_list":"[]","_tribe_ticket_has_attendee_info_fields":false,"study_program_identifiers":"MA_MAT_FM__F1_MA_MAT_FM__F1_MA_MAT","override_seo_title":false},"type-of-studies":[],"teaching-languages":[],"field-of-interest":[],"organisation":[],"authorship":[33,276,277,278],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.3 (Yoast SEO v22.3) - 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