Algebra and Algebraic Geometry Seminar Fall 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.

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COVID-19 Update

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

Fall 2020 Schedule

date	speaker	title	link to talk
September 14 @ 10am, Madison time	Andrei Căldăraru (Madison)	Categorical Enumerative Invariants	Talk 1/4 in lecture series at Imperial College - Register here to get the link to the talk!
September 18 @ 2:30pm	Dima Arinkin (Madison)
September 21 @ 10am	Andrei Căldăraru (Madison)	Categorical Enumerative Invariants	Talk 2/4 in lecture series at Imperial College
September 28 @ 10am	Andrei Căldăraru (Madison)	Categorical Enumerative Invariants	Talk 3/4 in lecture series at Imperial College
October 5 @ 10am	Andrei Căldăraru (Madison)	Categorical Enumerative Invariants	Talk 4/4 in lecture series at Imperial College
October 7 @ 8pm	Shamgar Gurevich (Madison)	Harmonic Analysis on GLn over Finite Fields	Register here to get link to talk at University of Sydney
October 16 @ 2:30pm	Ruijie Yang (Stony Brook)
October 30 @ 2:30pm	Reimundo Heluani (IMPA, Rio de Janeiro)	Rogers Ramanujan type identities coming from representation theory

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.

Dima Arinkin

TBD

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).

Ruijie Yang

TBD

Reimundo Heluani

A Rogers-Ramanujan-Slater type identity related to the Ising model

Abstract: We prove three new q-series identities of the Rogers-Ramanujan-Slater type. We find a PBW basis for the Ising model as a consequence of one of these identities. If time permits it will be shown that the singular support of the Ising model is a hyper-surface (in the differential sense) on the arc space of it's associated scheme. This is joint work with G. E. Andrews and J. van Ekeren and is available online at [4]https://arxiv.org/abs/2005.10769