# Difference between revisions of "Algebra and Algebraic Geometry Seminar Fall 2020"

Line 49: | Line 49: | ||

===Shamgar Gurevich=== | ===Shamgar Gurevich=== | ||

− | ''' | + | '''‘Harmonic Analysis on GLn over Finite Fields’ |

''' | ''' | ||

− | + | There are many formulas that express interesting properties of a finite group G in terms of sums over | |

+ | its characters. For estimating these sums, one of the most salient quantities to understand is the character ratio: | ||

+ | Trace(ρ(g)) / dim(ρ), for an irreducible representation ρ of G and an element g of G. For example, Diaconis | ||

+ | and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G. Recently, | ||

+ | we discovered that for classical groups G over finite fields there is a natural invariant of representations that | ||

+ | provides strong information on the character ratio. We call this invariant rank. Rank suggests a new | ||

+ | organization of representations based on the very few “Small” ones. This stands in contrast to Harish-Chandra’s | ||

+ | “philosophy of cusp forms”, which is (since the 60s) the main organization principle, and is based on the (huge | ||

+ | collection) of “Large” representations. This talk will discuss the notion of rank for the group GLn over finite | ||

+ | fields, demonstrate how it controls the character ratio, and explain how one can apply the results to verify | ||

+ | mixing time and rate for random walks. This is joint work with Roger Howe (Yale and Texas A&M). The | ||

+ | numerics for this work was carried with Steve Goldstein (Madison) and John Cannon (Sydney). |

## Revision as of 10:48, 13 September 2020

The Virtual Seminar will take place on Fridays at 2:30 pm via Zoom. We will also link to relevant or interesting Zoom talks outside of the seminar.

## Contents

## Algebra and Algebraic Geometry Mailing List

- Please join the AGS Mailing List to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).

## COVID-19 Update

As a result of Covid-19, the seminar for this semester will be held virtually.

## Fall 2020 Schedule

## Abstracts

### Andrei Căldăraru

**Categorical Enumerative Invariants**

I will talk about recent papers with Junwu Tu, Si Li, and Kevin Costello where we give a computable definition of Costello's 2005 invariants and compute some of them. These invariants are associated to a pair (A,s) consisting of a cyclic A∞-algebra and a choice of splitting s of its non-commutative Hodge filtration. They are expected to recover classical Gromov-Witten invariants when A is obtained from the Fukaya category of a symplectic manifold, as well as extend various B-model invariants (solutions of Picard-Fuchs equations, BCOV invariants, B-model FJRW invariants) when A is obtained from the derived category of a manifold or a matrix factorization category.

### Shamgar Gurevich

**‘Harmonic Analysis on GLn over Finite Fields’**

There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For estimating these sums, one of the most salient quantities to understand is the character ratio: Trace(ρ(g)) / dim(ρ), for an irreducible representation ρ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G. Recently, we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank. Rank suggests a new organization of representations based on the very few “Small” ones. This stands in contrast to Harish-Chandra’s “philosophy of cusp forms”, which is (since the 60s) the main organization principle, and is based on the (huge collection) of “Large” representations. This talk will discuss the notion of rank for the group GLn over finite fields, demonstrate how it controls the character ratio, and explain how one can apply the results to verify mixing time and rate for random walks. This is joint work with Roger Howe (Yale and Texas A&M). The numerics for this work was carried with Steve Goldstein (Madison) and John Cannon (Sydney).