# 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.

## Contents

## 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. The default Zoom link for the seminar is https://uwmadison.zoom.us/j/9502605167 (sometimes we will have to use a different meeting link, if Michael K cannot host that day).

## 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.

### 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).

### German Stefanich

**Categorified sheaf theory and the spectral Langlands TQFT**

It is expected that the Betti version of the geometric Langlands program should ultimately be about the equivalence of two 4-dimensional topological field theories. In this talk I will give an overview of ongoing work in categorified sheaf theory and explain how one can use it to describe the categories of boundary conditions arising on the spectral side.

### Ruijie Yang

**Decomposition theorem for semisimple local systems**

In complex algebraic geometry, the decomposition theorem asserts that semisimple geometric objects remain semisimple after taking direct images under proper algebraic maps. This was conjectured by Kashiwara and is proved by Mochizuki and Sabbah in a series of very long papers via harmonic analysis and $D$-modules. In this talk, I would like to explain a simpler proof in the case of semisimple local systems using a more geometric approach. This is joint work in progress with Chuanhao Wei.

### Nadia Ott

**The Supermoduli Space of Genus Zero SUSY Curves with Ramond Punctures**

Super Riemann surfaces (SUSY curves) arise in the formulation of superstring theory, and their moduli spaces, called supermoduli space, are the integration spaces for superstring scattering amplitudes. I will focus specifically on genus zero SUSY curves. As with ordinary curves, genus zero SUSY curves present a certain challenge, as they have an infinitesimal group of automorphisms, and so in order for the moduli problem to be representable by a Deligne-Mumford superstack, we must introduce punctures. In fact, there are two kinds of punctures on a SUSY curve of Neveu-Schwarz or Ramond type. Neveu-Schwarz punctures are entirely analogous to the marked points in ordinary moduli theory. By contrast, the Ramond punctures are more subtle and have no ordinary analog. I will give a construction of the moduli space M_{0,n}^R of genus zero SUSY curves with Ramond punctures as a Deligne-Mumford superstack by an explicit quotient presentation (rather than by an abstract existence argument).

### Reimundo Heluani

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

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 https://arxiv.org/abs/2005.10769

### Katrina Honigs

**An obstruction to weak approximation on some Calabi-Yau threefolds**

The study of Q-rational points on algebraic varieties is fundamental to arithmetic geometry. One of the few methods available to show that a variety does not have any Q-points is to give a Brauer-Manin obstruction. Hosono and Takagi have constructed a class of Calabi-Yau threefolds that occur as a linear section of a double quintic symmetroid and given a detailed analysis of them as complex varieties in the context of mirror symmetry. This construction can be used to produce varieties over Q as well, and these threefolds come tantalizingly equipped with a natural Brauer class. In work with Hashimoto, Lamarche and Vogt, we analyze these threefolds and their Brauer class over Q and give a condition under which the Brauer class obstructs weak approximation, though it cannot obstruct the existence of Q-rational points.