# Algebra & Number Theory Seminar

**Spring 2021**

**ANT Zoominar**

**Fridays @ 3pm California Time**

**9045829721**--- password: seminar

For further information please contact

**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 |

**Alicia Lamarche, U of Utah**

* Derived Categories, Arithmetic, and Rationality *

**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.

**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.

**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 T_{p} 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 T_{p --} as in all known quasi-triangular Hopf algebras. This example is the first known bracket computation for a nonquasi-triangular algebra.

**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.