# Difference between revisions of "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.

## Algebra and Algebraic Geometry Mailing List

• Please join the AGS mailing list by sending an email to ags+join@g-groups.wisc.edu 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

date speaker title link to talk
September 14 @ 10am 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 21 @ 10am Andrei Căldăraru (Madison) Categorical Enumerative Invariants Talk 2/4 in lecture series at Imperial College
September 25 Aleksandra Sobieska (Madison) Toward Free Resolutions Over Scrolls
September 28 @ 10am Andrei Căldăraru (Madison) Categorical Enumerative Invariants Talk 3/4 in lecture series at Imperial College
October 2 Robert Scherer (UC Davis) A Criterion for Asymptotic Sharpness in the Enumeration of Simply Generated Trees
October 5 @ 10am Andrei Căldăraru (Madison) Categorical Enumerative Invariants Talk 4/4 in lecture series at Imperial College
October 7 @ 7pm Shamgar Gurevich (Madison) Harmonic Analysis on GLn over Finite Fields Register here to get link to talk at University of Sydney
October 9 German Stefanich (Berkeley) TBA
October 16 Ruijie Yang (Stony Brook)
October 23 Nadia Ott (Mittag-Leffler Institute) The Supermoduli Space of Genus Zero SUSY Curves with Ramond Punctures TBA
October 30 Reimundo Heluani (IMPA, Rio de Janeiro) Rogers Ramanujan type identities coming from representation theory
November 6 Ben Bakker (UIC) TBA
November 13 Katrina Honigs (Oregon) TBA

## 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

Singular support of categories

In many situations, geometric objects on a space have some kind of singular support, which refines the usual support. For instance, for smooth X, the singular support of a D-module (or a perverse sheaf) on X is as a conical subset of the cotangent bundle; there is also a version of this notion for coherent sheaves on local complete intersections. I would like to describe a higher categorical version of this notion.

Let X be a smooth variety, and let Z be a closed conical isotropic subset of the cotangent bundle of X. I will define a 2-category associated with Z; its objects may be viewed as `categories over X with singular support in Z'. In particular, if Z is the zero section, this gives the notion of categories over Z in the usual sense.

The project is motivated by the local geometric Langlands correspondence; I will sketch the relation with the Langlands correspondence without going into details.

### Aleksandra Sobieska

Toward Free Resolutions Over Scrolls

Free resolutions over the polynomial ring have a storied and active record of study. However, resolutions over other rings are much more mysterious; even simple examples can be infinite! In these cases, we look to any combinatorics of the ring to glean information. This talk will present a minimal free resolution of the ground field over the semigroup ring arising from rational normal $2$-scrolls, and (if time permits) a computation of the Betti numbers of the ground field for all rational normal $k$-scrolls.

### Robert Scherer

A Criterion for Asymptotic Sharpness in the Enumeration of Simply Generated Trees

We study the identity $y(x)=xA(y(x))$, from the theory of rooted trees, for appropriate generating functions $y(x)$ and $A(x)$ with non-negative integer coefficients. A problem that has been studied extensively is to determine the asymptotics of the coefficients of $y(x)$ from analytic properties of the complex function $z\mapsto A(z)$, assumed to have a positive radius of convergence $R$. It is well-known that the vanishing of $A(x)-xA'(x)$ on $(0,R)$ is sufficient to ensure that $y(r)<R$, where $r$ is the radius of convergence of $y(x)$. This result has been generalized in the literature to account for more general functional equations than the one above, and used to determine asymptotics for the Taylor coefficients of $y(x)$. What has not been shown is whether that sufficient condition is also necessary. We show here that it is, thus establishing a criterion for sharpness of the inequality $y(r)\leq R$. As an application, we prove, and significantly extend, a 1996 conjecture of Kuperberg regarding the asymptotic growth rate of an integer sequence arising in the study of Lie algebra representations.

### 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).

TBA