(Rapinchuk talk) Over the last few years, the analysis of algebraic groups with good reduction has come to the forefront in the emerging arithmetic theory of linear algebraic groups over higher-dimensional fields. Current efforts are focused on finiteness conjectures for forms of reductive algebraic groups with good reduction that share some similarities with the famous Shafarevich Conjecture in the study of abelian varieties. Most results on these conjectures obtained so far have ultimately relied on finiteness properties of appropriate unramified cohomology groups. However, quite recently, methods based on building-theoretic techniques have emerged as a promising alternative approach. I will showcase some of these developments by sketching a new proof of a theorem of Raghunathan-Ramanathan concerning torsors over the affine line. Time-permitting, I will also discuss some connections to the genus problem, which deals with simple algebraic groups having the same isomorphism classes of maximal tori.

(Joshua talk) This concerns the following conjecture of Fabien Morel. Given a linear algebraic group G defined over a field k, with a maximal torus T, the conjecture was that a suitable form of the Euler Characteristic of G/N(T), where N(T) is the normalizer of T in G, is 1, when viewed as a class in the Grothendieck-Witt group of the field k. In a paper in 2023, together with P. Pelaez, we settled this conjecture in the affirmative for split groups G, provided the field k has a square root of -1. We also showed that this implies the same Euler characteristic is a unit in general. This has numerous applications to splitting in various forms of Borel-style equivariant cohomology theories, for example, equivariant K-theory, equivariant (higher) Chow groups, equivariant Brauer groups etc., and greatly facilitates the computation of all such equivariant cohomology theories. The talk will sketch a proof of the conjecture and outline some of the applications.

(Haburcak talk) The moduli space of algebraic curves is a central object of study, and its cohomology ring is deeply related to spaces of modular forms. The tautological ring is a subring of the cohomology of the moduli space of stable curves which contains most cohomology classes of geometric interest and is simpler to study. A question of interest is therefore how different the tautological ring is from the full cohomology. The first known example of a non-tautological algebraic class was discovered by Graber and Pandharipande, which was significantly generalized by van Zelm to find an infinite family of non-tautological classes. On the other hand, very little is known about non-tautological classes on the moduli space of smooth curves. In joint work with Arena, Canning, Clader, Li, Mok, and Tamborini, we show that there are non-tautological classes on the moduli space of smooth marked curves for an infinite family of genera.

(Scully talk)

(Berg talk)