## The Number Theory Seminar

## at the University of Alberta

# Fall 2019

Unless stated otherwise, the **Fall 2019 number theory seminar meets from 2:30-3:30pm on Fridays, at 563 CAB.**

## Upcoming Seminars

**Friday 2019-12-06, 2:30-3:30pm**

**Speaker: **Raphaël Belliard (Hamburg University, University of Alberta)

**Title: **Introduction to topos quantum theory

**Abstract: **In this talk, we will review the motivation and setup for considering topoï for describing quantum physics in the context of realistic quantum gravity, building up on the content of the paper "Topos methods in the foundations of physics" by C. Isham in 2010.

## Past Seminars

**Friday 2019-10-18, 2:30-3:30pm**

**Speaker:** Seidon Alsaody (University of Alberta)

**Title: **Exceptional Groups and Exceptional Algebras over Rings

**Abstract: **The study of algebraic groups over rings that are not fields is of wide interest, not the least for various applications in number theory. The exceptional groups arise as symmetry groups of various remarkable objects, such as octonion algebras, triality and exceptional Jordan algebras. I will talk about an approach to these objects combining classical algebraic constructions with the powerful machinery of Grothendieck topologies, torsors, descent and cohomology. This combination has led to some recent insight into the rich and intricate structure that these objects have when considered over an arbitrary base ring.

**Friday 2019-10-25, 2:30-3:30pm**

**Speaker:** Eric Primozic (University of Alberta)

**Title: **Computations of de Rham cohomology rings of classifying stacks

**Abstract: **Recent work in p-adic Hodge theory gives comparison results between de Rham cohomology for proper schemes over a base field of characteristic p and mod p etale cohomology over a base field of characteristic 0. The same results have not yet been obtained for stacks. Building on the work of Totaro, we compute the de Rham cohomology rings of some classifying stacks BG over a field of characteristic p where p is a torsion prime for G. Our main tool (due to Totaro) for our calculations is an analogue of the Serre spectral sequence from topology for Hodge cohomology.

**Friday 2019-11-01, 2:30-3:30pm**

**Speaker: **Adam Topaz (University of Alberta)

**Title: **Parameterization of divisors and applications

**Abstract: **In this talk, I will discuss two cohomological methods which parameterize all prime divisors on regular models of a given geometric function field. The technical core of these methods is fairly elementary, using mainly ideas from classical projective geometry. Time permitting, I will discuss how these methods can be used to prove new results in the context of birational anabelian geometry.

**Friday 2019-11-08, 2:30-3:30pm**

**Speaker: **Jeongseok Oh (Fields Institute)

**Title: **Counting sheaves on Calabi-Yau 4-folds

**Abstract: **We define a localised Euler class for isotropic sections, and isotropic cones, in SO(N) bundles. We use this to give an algebraic definition of Borisov-Joyce's sheaf counting invariants on Calabi-Yau 4-folds. When a torus acts, we prove a localisation result. This talk is based on the joint work with R. P. Thomas.

**Friday 2019-11-29, 2:30-3:30pm**

**Speaker:** Jack Klys (University of Calgary)

**Title: **Densities of eigenspaces of Frobenius and Distributions of R-Modules

**Abstract: **We will discuss density theorems for cokernels of Tate modules and Jacobians of hyperelliptic curves over finite fields (in the large genus limit), viewed as modules over the Frobenius. Our work builds on recent advances in this area of Ellenberg-Venkatesh-Westerland and Lipnowski-Tsimerman. The methods involve the study of measures on R-modules over rings R which are finite Z_p-algebras. As an application, for any polynomial p(x) over F_p we determine an explicit formula for the asymptotic density of hyperelliptic curves for which p(x) divides the characteristic polynomial of Frobenius acting on the p-torsion of the Jacobian. Joint work with Jacob Tsimerman.