A question of uniqueness: Nathaniel Sagman's work on minimal surfaces

8xav��

During his time at the 8xav�� of Luxembourg, Nathaniel Sagman helped resolve a long-standing question in geometry. His work has been accepted for publication in the , one of the field’s leading journals.

When Nathaniel Sagman arrived at the 8xav�� of Luxembourg as a postdoctoral researcher, he was stepping into a new academic environment, and into a mathematical problem that had been puzzling researchers for years. In his first year in Luxembourg, working closely with collaborators, he helped resolve a long-standing question in differential geometry, a field concerned with understanding shapes, spaces, and their properties.

A collaborative research environment at Uni.lu

Although the project had begun during his PhD at , it was in Luxembourg that the work truly matured and that Nathaniel became a leader in his research area. “Luxembourg was the place where I really came to terms with all of the mathematics,” he says. “I more carefully developed ideas and thought about future questions and conjectures.”

Nathaniel was drawn to the 8xav�� of Luxembourg in part by the research environment and the opportunity to work with Prof. Jean-Marc Schlenker, whose work closely relates to his own. He also highlights the strength of the broader network surrounding the research group, which allowed him to exchange ideas with other mathematicians and explore new directions in his work.

Within the Department of Mathematics, Nathaniel found a culture that encouraged collaboration and exchange. “We were encouraged to have reading groups on specialised topics, participate in seminars, and interact with visiting scholars.”

In this collaborative environment, his research project fully took shape and led to new insights. This also led Nathaniel to collaborate with other postdoctoral researchers and PhD students on several projects during his time in Luxembourg.

A simple question with subtle answers

At the center of this research lies a deceptively simple question. Given a closed loop in space, what is the surface of the smallest possible area that spans it, and is that surface unique? In everyday terms, it’s a question about whether there is always a single “best” solution, or whether several equally optimal answers can exist.

Consider the simple challenge of finding the quickest way home. In a flat, open field, there is only one shortest path: a straight line. But as soon as you introduce hills, valleys, or obstacles, the “best” route becomes less obvious. There might be two different paths that are exactly the same length, giving an option to choose between them. This is the essence of a unique “minimisation”&�Բ��;problem.

Mathematicians have known for nearly a century that such minimal surfaces exist. But whether the most efficient solution is unique depends on the setting: sometimes there is only one, and sometimes there are several.

From classical geometry to modern structures

This question becomes more subtle in modern geometry, where mathematicians move beyond simple shapes and begin studying more abstract spaces.

Nathaniel’s work connects this classical question to a modern area of mathematics known as higher Teichmüller theory. Here, researchers study geometric objects called Hitchin representations, which help describe rich and still not fully understood structures.

A conjecture by the mathematician states that, in this setting, the minimal surfaces associated with these representations should always be unique.

“��&�Բ��;�ǳܰ�&�Բ��;�貹��,&�Բ��;�ɱ�&�Բ��;�ڴǳܲԻ�&�Բ��;��ǳܲԳٱ��&�Բ��;��油��&�Բ��;�ٴ�&�Բ��;��ǳܰ��’s&�Բ��;��ǲ��𳦳ٳܰ��,”&�Բ��;Nathaniel says. “We constructed Hitchin representations that come with multiple minimal surfaces.”&�Բ��;

This matters within the mathematical theory because uniqueness is often what makes a geometric object predictable: it guarantees that it is defined in a consistent way. When uniqueness fails, the same underlying data can give rise to several different, equally valid solutions, showing that there is still something further to understand about the geometry involved.

Looking ahead

Now based at the 8xav�� of North Carolina at Chapel Hill, Nathaniel continues to work on topics connected to the publication while exploring new research directions. For him, one of the most exciting aspects of mathematics is that solving one problem often leads to many more questions. “The publication opened up new perspectives and raised questions that will probably occupy me for years to come,” he says.

Looking back on his time in Luxembourg, he also emphasises the importance of the research environment and community he found at the 8xav��. “Luxembourg remains an important place for me, both for the mathematics developed there and for the community that helped shape it.”