Algebra & Number Theory Seminar


Spring 2021 ANT Zoominar
Fridays @ 3pm California Time 
Meeting ID:  9045829721  --- password: seminar
For further information please contact
Professor Beren Sanders

Schedule (click dates for title and abstract) 

4/23 Alicia Lamarche University of Utah
4/30 Joe Kramer-Miller UC Irvine
5/7 @ 10am Serge Bouc University of Picardie
5/14 NO TALK
5/21 Frank Thorne U of South Carolina
5/28 Tekin Karadag Texas A&M
6/4 Andrew Kobin UCSC
Friday, April 23, at 3pm PST

Alicia Lamarche, U of Utah

Derived Categories, Arithmetic, and Rationality

When trying to apply the machinery of derived categories in an arithmetic setting, a natural question is the following: for a smooth projective variety X, to what extent can Db(X) be used as an invariant to answer rationality questions? In particular, what properties of Db(X) are implied by X being rational, stably rational, or having a rational point? On the other hand, is there a property of Db(X) that implies that X is rational, stably rational, or has a rational point? In this talk, we will examine a family of arithmetic toric varieties for which a member is rational if and only if its bounded derived category of coherent sheaves admits a full etale exceptional collection. Additionally, we will discuss the behavior of the derived category under twisting by a torsor, which is joint work with Mattew Ballard, Alexander Duncan, and Patrick McFaddin.

Friday, April 30, at 3pm PST

Joe Kramer-Miller, UC Irvine

The ramification of p-adic representations coming from geometry

A classical theorem of Sen describes a close relationship between the p-adic Lie filtration and the ramification filtration for a p-adic Galois representation of a p-adic field. Unfortunately, things are much too *wild* in the positive characteristic case to have an analogue of Sen's theorem. In general the ramification invariants can behave arbitrarily bad. However, there is hope if we restrict to certain 'geometric' representations! These are p-adic representations that come from p-adic etale cohomology of smooth proper fibrations. We prove an analogue of Sen's theorem for the relative p-adic cohomology of a smooth proper and ordinary fibration of varieties.

Friday, May 7, 2021 @ 10am PST *** (rescheduled)

Serge Bouc, University of Picardie

A functorial resolution of units of Burnside rings

Most of the structural properties - prime spectrum, species,  idempotents, ... - of the Burnside ring of a finite group have been precisely described a few years after its introduction in 1967. An important missing item in this list is its group of units. After a - non exhaustive - review of this subject, I will present some recent results on the functorial aspects of this group.

Friday, May 21, at 3pm PST

Frank Thorne, U of South Carolina

Enumerating Number Fields

How many number fields are there of fixed degree and bounded discriminant? I will start off with an overview of what is expected and what is known -- often in the case where the Galois group is specified. In the second part I will give an overview of recent work with Robert Lemke Oliver, which combines ideas from the geometry of numbers with a smidgen of algebraic geometry to improves upon the best known general upper bounds.

Friday, May 28, at 3pm PST

Tekin Karadag, Texas A&M

Gerstenhaber bracket on Hopf algebra cohomology

It is known that the graded Lie bracket (Gerstenhaber bracket) structure on Hopf algebra cohomology of a quasitriangular algebra is abelian. We calculate the graded Lie bracket (Gerstenhaber bracket) on Hochschild and Hopf algebra cohomologies of the Taft algebra Tp for any integer p > 2 which is a nonquasi-triangular Hopf algebra. We show that the bracket is indeed zero on Hopf algebra cohomology of Tp -- as in all known quasi-triangular Hopf algebras. This example is the first known bracket computation for a nonquasi-triangular algebra.

Friday, June 4, at 3pm PST

Andrew Kobin, UC Santa Cruz

Zeta functions and decomposition spaces

Zeta functions show up everywhere in math these days. While several works over the years have brought homotopical methods into the theory of zeta functions, there is in fact a lesser-known zeta function that is native to homotopy theory. Namely, every suitably finite decomposition space (aka 2-Segal space) admits an abstract zeta function as an element of its incidence algebra. In this talk, I will show how many 'classical' zeta functions from number theory and algebraic geometry can be realized in this homotopical framework, and outline some preliminary work in progress with Bogdan Krstic towards a motivic version of the above story.