# Algebra and Algebraic Geometry Seminar Spring 2021

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).

## Spring 2021 Schedule

## Abstracts

### Nir Avni

Title: First order rigidity for higher rank lattices.

Abstract: I'll describe a new rigidity phenomenon for lattices in higher rank simple algebraic groups. Specifically, I'll explain why the first order theories of such groups do not have (finitely generated) deformations and why they are determined (in the class of finitely generated groups) by a single first order axiom.

The results are from joint works with Alex Lubotzky and Chen Meiri.

### Marian Aprodu

Title: Koszul modules, resonance varieties and applications.

Abstract: This talk is based on joint works with Gabi Farkas, Stefan Papadima, Claudiu Raicu, Alex Suciu and Jerzy Weyman. I plan to discuss various aspects of the geometry of resonance varieties, Hilbert series of Koszul modules and applications.

Slides available here [1]

### Dhruv Ranganathan

Title: Logarithmic Donaldson-Thomas theory

Abstract: I will give an introduction to a circle ideas surrounding the enumerative geometry of pairs, and in particular, intersection theory on a new models of the Hilbert schemes of curves on threefolds. These give rise to “logarithmic” DT and PT invariants. I will explain the conjectural relationship between this geometry and Gromov-Witten theory, and give some sense of the role of tropical geometry in the story. The talk is based on joint work, some of it in progress, with Davesh Maulik.

### Philip Engel

Title: Compact K3 moduli

Abstract: This is joint work with Valery Alexeev. A well-known consequence of the Torelli theorem is that the moduli space F_{2d} of degree 2d K3 surfaces (X,L) is the quotient of a 19-dimensional Hermitian symmetric space by the action of an arithmetic group. In this capacity, it admits a natural class of "semitoroidal compactifications." These are built from periodic tilings of 18-dimensional hyperbolic space, and were studied by Looijenga, who built on earlier work of Baily-Borel and Ash-Mumford-Rapaport-Tai. On the other hand, F_{2d} also admits "stable pair compactifications": Choose canonically on any polarized K3 surface X an ample divisor R. Then the works of Kollar-Shepherd-Barron, Alexeev, and others provide for the existence of a compact moduli space of so-called stable pairs (X,R) containing, as an open subset, the K3 pairs.

I will discuss two theorems in the talk: (1) There is a simple criterion on R, called "recognizability" ensuring that the normalization of a stable pair compactification is semitoroidal and (2) the rational curves divisor, generically the sum of geometric genus zero curves in |L|, is recognizable for all 2d. This gives a modular semitoroidal compactification for all degrees 2d.

### Andreas Knutsen

Title: Genus two curves on abelian surfaces

Abstract: Let (S,L) be a general polarized abelian surface of type (d_1,d_2). The minimal geometric genus of any curve in the linear system |L| is two and there are finitely many curves of such genus. In analogy with Chen's results concerning rational curves in primitive linear systems on K3 surfaces, it is natural to ask whether all such curves are nodal. In the seminar I will present joint work with Margherita Lelli-Chiesa (arXiv:1901.07603) where we prove that this holds true if and only if d_2 is not divisible by 4. In the cases where d_2 is a multiple of 4, we show the existence of curves in |L| having a triple, 4-tuple or 6-tuple point, and prove that these are the only types of unnodal singularities a genus 2 curve in |L| may acquire.

### Michael Groechenig

Title: Rigid Local Systems

Abstract: An irreducible representation of a finitely generated group G is called rigid, if it induces an isolated point in the moduli space of representations. For G being the fundamental group of a complex projective manifold, Simpson conjectured that rigid representations should have integral monodromy and more generally, be of geometric origin. In this talk I will give an overview about what is currently known about Simpson’s conjectures and will present a few results joint with H. Esnault.

### Daniele Agostini

Title: Effective Torelli theorem

Abstract: Torelli's theorem is a foundational result of classical algebraic geometry, asserting that a smooth curve can be recovered from its Jacobian. There are many effective proofs of this result, that can even be implemented on a computer. In this talk, I will present this circle of ideas. In particular, I will focus on a method based on the KP equation in mathematical physics, that I have recently implemented together with Türkü Çelik and Demir Eken.

### Gavril Farkas

Title: The Kodaira dimension of the moduli space of curves: recent progress on a century-old problem.

Abstract: The problem of determining the birational nature of the moduli space of curves of genus g has received constant attention in the last century and inspired a lot of development in moduli theory. I will discuss progress achieved in the last 12 months. On the one hand, making essential of tropical methods it has been showed that both moduli spaces of curves of genus 22 and 23 are of general type (joint with D. Jensen and S. Payne). On the other hand I will discuss a proof (joint with A. Verra) of the uniruledness of the moduli space of curves of genus 16.

### Hannah Larson

Title: The rational Chow rings of M_7, M_8, and M_9

Abstract: The rational Chow ring of the moduli space M_g of curves of genus g is known for g \leq 6. In each of these cases, the Chow ring is tautological (generated by certain natural classes known as kappa classes). In recent joint work with Sam Canning, we prove that the rational Chow ring of M_g is tautological for g = 7, 8, 9, thereby determining the Chow rings by work of Faber. In this talk, I will give an overview of our approach, with particular focus on the locus of tetragonal curves (special curves admitting a degree 4 map to P^1).

### Eyal Subag

Title: Algebraic symmetries of the hydrogen atom.

Abstract. In this talk we will examine symmetries of the hydrogen atom from two related algebraic perspectives. The first is in the context of algebraic families of groups. The second comes from a new suggested model for the Schrödinger equation of the hydrogen atom within the algebra of differential operators on a complex null cone. Time permit I will discuss related questions in representation theory of SL(2,R).

This talk is based on joint work with J. Bernstein and N. Higson.

### Gurbir Dhillon

**The Drinfeld--Sokolov reduction of admissible representations of affine Lie algebras**

Abstract: The affine W-algebras are a family of algebras whose representation theory plays an important role in conformal field theory and the geometric Langlands program. In the original paper which introduced W-algebras into mathematics, Feigin and Frenkel conclude with a striking conjecture, joint with Kac and Wakimoto, relating certain irreducible representations of affine Lie algebras and affine W-algebras via a functor since called the `plus' Drinfeld--Sokolov reduction. We have proven this conjecture in forthcoming work. The primary goal of the talk will be to give a motivated introduction to the conjecture, its history, and the objects appearing in it for non-specialists.

### Iordan Ganev

The QR decomposition for radial neural networks

Abstract: We present a perspective on neural networks stemming from quiver representation theory. This point of view emphasizes the symmetries inherent in neural networks, interacts nicely with gradient descent, and has the potential to improve training algorithms. As an application, we establish an analogue of the QR decomposition for radial neural networks, which leads to a dimensional reduction result. This talk is intended for a broad mathematical audience, and we explain all concepts relating to neural networks and machine learning from first principles. It is based on joint work-in-progress with Robin Walters.