Anders Södergren is an Associate Professor at the Division of Algebra and Geometry, and the main aim of his research project is to make detailed studies of problems at the intersection of analytic number theory and arithmetic statistics. Of particular interest is the distribution of zeros of L-functions, which are central functions within analytic number theory. The most well-known of all billions of different L-functions is the Riemann zeta function, which holds many secrets regarding how prime numbers are distributed among the natural numbers. The famous Riemann hypothesis states that the distribution of prime numbers is totally random, but nobody has yet managed to prove it. Many mathematicians are convinced that the path to proving the Riemann hypothesis is through families of L-functions.

One way to explore the zeros of L-functions was described about twenty years ago by the Katz-Sarnak heuristic. This conjecture combines methods from algebraic geometry with the theory of random matrices, producing accurate predictions of the statistical distribution of zeros in certain families of L-functions. The hope is that this project will lead to precise results that confirm several different aspects of the Katz-Sarnak heuristic. As yet, some of the necessary techniques are still missing, so developing suitable tools is an important part of this project.

– The analytic theory of L-functions holds a central position in number theory. The number theory group at the department is rather large, but I am relatively alone in working on such problems. The grant from KAW provides a great opportunity to change that. I hope this will allow me to intensify my work on statistical problems in families of L-functions. Unfortunately, the uncertain situation that the corona virus has put us in makes it impossible to say how difficult it will be to recruit someone for the postdoctoral position.

More detailed description of Anders Södergren’s research at the KAW web >>

Professor Gerard Freixas i Montplet is a researcher at the Department of Mathematics, Jussieu – Paris Rive Gauche, France. He will work with Dennis Eriksson, Associate Professor at the Department of Algebra and Geometry, among others. The current project focuses on better understanding both the geometric and the number theoretical properties of moduli spaces. These are at the heart of modern algebraic geometry and were designed about 50 years ago as a way of organising and classifying a large number of geometric objects.

A moduli space can also be regarded as a map. While each point on the familiar geographical maps corresponds to a location, the points in a moduli space correspond to different geometric objects. However, moduli spaces are very complicated maps, so different approaches have been developed to read them. Often, attempts to understand the geometry of the moduli space use properties that are inspired by physics, for example how a kind of temperature varies between different points. From the basis of various energy levels found in the space a special type of function, analytic torsion, is built up, so studies of how this function varies across the moduli space might lead to the desired charting.

– It is a great opportunity to be able to work directly with our projects here in Gothenburg for a whole year. We already have an ongoing collaboration that can now be further deepened. In addition, Gerard already has many other links to the research at the department, mainly in algebraic and complex geometry but also in analysis, so that more researchers can benefit from his presence, Dennis Eriksson concludes.

More detailed description of Gerard Freixas i Montplet’s research at the KAW web >>

The Knut and Alice Wallenberg Foundation started the ten-year mathematics programme in 2014 and has since granted an average of SEK 25 million every year for positions and scholarships. The programme is a partnership with the Royal Swedish Academy of Sciences, which evaluates all nominated candidates. Sixteen prominent mathematicians will receive grants this year.

Read more about all researchers and projects on the KAW web >>**Text**: KAW, Anders Södergren, Dennis Eriksson**Photos**: Setta Aspström, Gerard Freixas i Montplet**Equation**: The Riemann zeta function**Illustration**: 2-holed torus with significant equations, Dennis Eriksson