The ECM is held every four years, alternating at two-yearly intervals with the International Congress of Mathematicians, at which the illustrious Fields Medal is presented. “So it’s a bit like the World Cup and European Championships in soccer,” Jessica Fintzen explains. Whilst the Fields Medal is awarded to mathematicians under 40 from anywhere in the world, the EMS Prize sets an age limit of 35. It is open to any mathematician who holds European citizenship or is currently conducting research in Europe. In a way, therefore, the EMS Prize can be seen as the “little sister” of the Fields Medal, especially since no fewer than 11 of its winners (including the University of Bonn’s very own Peter Scholze) have gone on to be awarded what is the most famous prize for mathematics in the world, the Fields Medal.

Jessica Fintzen is being honored for “her groundbreaking work on the representation theory of p-adic groups.” She is regarded as one of the world’s leading mathematician in this field, which connects number theory and representation theory, and has already won numerous awards including the Frank Nelson Cole Prize in Algebra at the start of this year and the London Mathematical Society’s Whitehead Prize back in 2022. She was recently awarded a medal for giving the Cours Peccot lectures at the Collège de France. “These prizes are a huge motivation for me,” Jessica Fintzen says. “I’m always looking ahead; there’s still a lot to do. This particular prize is very special for me, because it can be awarded for any discipline in mathematics.” The €5,000 prize money is of secondary importance as far as she is concerned.

Jessica Fintzen's research is in pure mathematics and concerns groups and so-called p-adic numbers. Groups are sets whose elements can be composed so that certain rules apply, such as the associative law. An example of a group is the symmetry group of a cube, which contains all operations that keep the cube invariant. In representation theory, groups are described by matrices, i.e. by linear maps between vector spaces, which are mathematically very well understood. If you start with the field of rational numbers, you can extend and complete it in various different ways to obtain larger fields. A very well-known way of doing this results in the field of real numbers known from school. Another possibility leads to the so-called p-adic numbers for each prime number p. In Jessica Fintzen's research, the aim is to describe groups over these fields of p-adic numbers with the help of matrices, i.e., to "represent" them. The class of groups over all p-adic numbers can be visualized as a large ocean, where each prime number p stands for a specific location in the ocean and where an infinite chain of groups extends deep into the ocean. In recent work, Jessica Fintzen, together with other scientists from all over the world, has now succeeded in bringing everything from the infinite depth of this ocean back to the surface, where the general linear groups over finite fields are located, which are already very well known. These "depth-zero representations" of p-adic groups have a great influence on the well-known Langlands program, in which far-reaching conjectures are made that link number theory and representation theory of groups, and which is considered one of the most important mathematical programs in modern mathematics.

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Biography

**Jessica Fintzen** obtained a bachelor’s degrees in mathematics and one in physics from the international Jacobs University Bremen before moving on to Harvard University for her doctorate. Following posts as a postdoctoral researcher at the University of Michigan, the Institute for Advanced Study in Princeton and Trinity College Cambridge, she worked as a lecturer and Royal Society University Research Fellow at the University of Cambridge and as an assistant professor and later full professor at Duke University. She took up a professorship at the University of Bonn in 2022 and has been a member of its Hausdorff Center for Mathematics Cluster of Excellence ever since.

About the award

The **EMS Prizes** were introduced in 1992. Up to 10 are presented at every ECM to researchers who are no older than 35 when they are nominated and who either hold European citizenship or work in Europe in recognition of outstanding contributions made to mathematics. “European” here means any country or part of a country that is located in Europe in geographical terms or in which a corporate member of the EMS is headquartered. The age limit can be raised to 38 in certain circumstances (generally on family or health grounds).