GQT
Day 1 (Tuesday 22 April) @ 13:45–15:15
Peter Hochs (Radboud University)
Equivariant analytic torsion and an equivariant Ruelle dynamical zeta function
Analytic torsion was introduced by Ray and Singer as a way to realise Reidemeister-Franz torsion analytically. (The equality was independently proved by Cheeger and Müller.) The Ruelle dynamical zeta function is a topological way to count closed curves of flows on compact manifolds. The Fried conjecture states that, for a suitable class of flows, the Ruelle dynamical zeta function has a well-defined value at zero, and that the absolute value of this value equals analytic torsion. With Hemanth Saratchandran, we define equivariant versions of analytic torsion and of the Ruelle dynamical zeta function, which incorporate group actions. This leads to the question under what conditions the resulting equivariant version of Fried’s conjecture is true. With Chris Pirie, we have recently obtained positive results on a basic class of flows: suspension flows of isometries.
Photocredits: Bert Beelen
Marta Pieropan (Utrecht University)
Rational points and Campana points on Fano varieties
Diophantine geometry is the study of integer and rational solutions of polynomial equations. On its geometric side, the equations define algebraic varieties and the solutions correspond to integer or rational points on the variety. Geometric properties such as hyperbolicity conjecturally determine how many rational points a variety can have. Fano varieties form the class of varieties with the highest density of rational points. On Fano varieties rational points and integer points coincide. Campana points are rational points whose integer coordinates satisfy certain arithmetic conditions, such as being divisible by a sufficiently high power of their radical. Campana points are important to study images of rational points under morphisms of algebraic varieties. This talk will introduce the main conjectures in the field of rational points and of Campana points on Fano varieties, including joint work with Smeets, Tanimoto, and Várilly-Alvarado, and then discuss the state of the art and future directions.
Michal Wrochna (Utrecht University)
Lorentzian Dirac operators and geometric invariants
Dirac operators on Riemannian manifolds play a fundamental role in linking differential geometry, topology, and analysis through index theory. They are also essential in developments at the interface of non-commutative geometry and high energy physics. Establishing a theory with similar features on Lorentzian manifolds has been a long-term challenge, it has however undergone major progress in the last few years thanks to incorporating tools from microlocal analysis. The talk will present the current state of the art and new perspectives, discussing in particular an index theorem due to Christian Bär and Alexander Strohmaier as well as recent results in non-compact settings obtained in collaboration with Nguyen Viet Dang, Dawei Shen and András Vasy.
