# Corona Outbreak-Response Omnipresent (Noncommutative) Algebraic Geometry Seminar

Happens on Thursdays at 6—7:30pm ET.

David Mumford once said that algebraic geometry is subject whose intent was to take over all of mathematics. Our goal in this seminar is to combat one virus with another :)
But this is not a seminar like any other. We can't just have chaos. We must have rules. Here are the rules:
1. This is a preprint/paper seminar. I know you would love to share your Newtonian discoveries during the plague but let us stick with the available literature. If the speaker really wants to illustrate the talk with some exciting result of their own, they should take a shot right before the formulation.
2. The speaker should pick a topic of interest in algebraic geometry (the noncommutative part was just to fit the acronym) - it can be very general or very specific, come up with the reading list and give a talk on it.
3. Each talk is 1.5 hours. But there is catch - it is a seminar like any other for the first hour but the next half an hour follows the zoooooom drinking game rules.

Organizers: Elden, Joaquín, Sveta

## August 20

Title: A toy model for the Drinfeld-Lafforgue shtuka construction. Part 2

Speaker: Dennis Gaitsgory

Abstract: The goal of this talk is to provide a categorical framework that leads to the definition of shtukas à la Drinfeld and of excursion operators à la V. Lafforgue. We take as the point of departure the Hecke action of Rep(G^L) on the category Shv(Bun_G) of sheaves on Bun_G, and also the endofunctor of the latter category, given by the action of the geometric Frobenius. The shtuka construction will be obtained by applying (various versions of) categorical trace.

## August 13

Title: A toy model for the Drinfeld-Lafforgue shtuka construction

Speaker: Dennis Gaitsgory

Abstract: The goal of this talk is to provide a categorical framework that leads to the definition of shtukas à la Drinfeld and of excursion operators à la V. Lafforgue. We take as the point of departure the Hecke action of Rep(G^L) on the category Shv(Bun_G) of sheaves on Bun_G, and also the endofunctor of the latter category, given by the action of the geometric Frobenius. The shtuka construction will be obtained by applying (various versions of) categorical trace.

## August 6

Title: Local structure of algebraic stacks and applications

Speaker: David Rydh

Abstract: Some natural moduli problems, such as moduli of sheaves and moduli of singular curves, give rise to stacks with infinite stabilizers that are not known to be quotient stacks. The local structure theorem states that many stacks locally look like the quotient of a scheme by the action of a stabilizer group. This is related to Luna's slice theorem in equivariant geometry. I will explain the statement of the local structure theorem and some of the techniques that are used in the proof such as cohomological affineness, complete stacks, Tannaka duality and Artin algebraization.
I will also mention some applications such as Nisnevich neighborhoods (useful in K-theory), Bialynicki-Birula decompositions for stacks, Kirwan partial desingularization, criteria for the existence of good moduli spaces and compact generation of derived categories.
This is joint work with Jarod Alper, Jack Hall and Daniel Halpern-Leistner.

## July 23

Title: Non-commutative resolutions and semi-orthogonal decompositions for quotient singularities

Abstract: I will talk about results of Spenko and van den Bergh who construct non-commutative resolutions of quotient singularities. These NCR also appear as summands in semi-orthogonal decompositions. I will also discuss the connections between the Spenko- van den Bergh results and semi-orthogonal decompositions in the literature inspired by geometric invariant theory.

## July 9 (5pm ET = 11pm Copenhagen)

Title: Cubical chains, Adams cobar construction, and cohomology of the loop space

Speaker: Hippie (Sergei Arkhipov)

Abstract: We start from the classical exposition of an explicit model for cohomology of the loop space of a topological space due to Adams. Starting from the usual singular chains for a topological space X, Adams produced a dg-algebra calculating cohomology of the loop space of X. Algebraically this was the famous cobar construction. Topologically, the construction hides a beautiful passage from singular chains on X to cubical chains on the loop space of X. There was a strong restriction on X: it should be connected simply connected for the construction to work. Recently Zeinalian and coauthors revealed a relation between necklicial sets, Lurie strictification functor, Lurie dg nerve construction, and cubical sets. This led the authors to a generalization of Adams result for non-simply connected topologicla spaces. (Nothing of this is my own work).

## July 2 (6pm ET)

Title: From bar to Barr-Beck

Speaker: Rina Anno

Abstract: I am going to talk about the classic Barr-Beck theorem from category theory, then introduce its analogue for triangulated categories with a DG enhancement, and then talk about bar categories of modules -- the technique that makes the latter available. Based on a joint work with Tim Logvinenko and Sergey Arkhipov.

## June 25 (6pm ET = 11pm London)

Note unusual time!

Title: P^n-functors

Speaker: Timothy Logvinenko

Abstract: I will give a survey talk introducing the notion of P^n functors and charting out their development from its mirror symmetric origins to the present day.

## June 18

Title: K-stability of Fano varieties

Speaker: Ziquan Zhuang

Abstract: K-stability is a notion in algebraic geometry that has close relation with the existence of K\"ahler-Einstein metric. In this talk, I will survey some recent understanding of K-stability through birational geometry and valuation theory.

## June 11 (5pm ET)

Note unusual time!

Title: Categorification of 2-dimensional K-theoretical Hall algebras

Speaker: Mauro Porta

Abstract: In this talk I will survey the results obtained in my joint paper with Francesco Sala, arXiv 1903.07253. Using techniques from derived geometry, I will explain how to find a natural categorification of K-theoretical Hall algebras associated to 2-dimensional objects. Among the examples, I will discuss the Hall algebra attached to a surface and the ones attached to Higgs bundles and flat vector bundles on a curve. If time permits, I will sketch how to obtain the categorification of certain natural representations.

## June 4

Title: Stable Representation Categories

Speaker: Chris Ryba

Abstract: Representations of certain families of groups (e.g. GL_n, O_n, S_n) exhibit a variety of stability properties. We will give some examples, and then show how certain categories constructed by Sam and Snowden provide an explanation for these phenomena. These categories have interesting homological algebra, and also provide new insights into the representation categories of the original groups.

## May 28

Title: K3 surfaces: hyperkahler structures and motives

Speaker: Ziquan Yang

Abstract: I will give a survey talk on K3 surfaces, with an emphasis on how their hyperkahler structures are related to Torelli type theorems and motivic questions. I will give a sketch of Buskin's proof of the Shafarevich conjecture, which asserts that every rational Hodge isometry between K3 surfaces is algebraic. If time permits, I will talk about Huybrechts' second proof of the conjecture which uses only algebraic methods.

## May 21 (6pm ET = 11pm London)

Note unusual time!

Title: Spherical and P^n-functors

Speaker: Timothy Logvinenko

Abstract: I will give a survey talk introducing the notions of spherical and P^n functors and charting out their development from their mirror symmetric origins to our present day state of knowledge.

## May 14

Title: Does the following situation ever occur?

Speaker: Craig Westerland

Abstract: There are many questions in number theory and arithmetic geometry of the sort “Does the following situation ever occur?” For instance, the inverse Galois problem asks whether every finite group occurs as the Galois group of an extension of the rationals. Similarly, one might ask whether one expects the rank of elliptic curves to be unbounded.

Arithmetic statistics, broadly speaking, pursues the more quantitative question of how often such situations occur. The extension of the inverse Galois problem to this setting is a conjecture of Malle’s, which predicts an asymptotic formula for the number of occurrences of a given finite group G as the Galois group of a number field, as a function of the discriminant.  There are analogous statistical conjectures regarding the distribution of class groups ordered by discriminant (e.g., the Cohen-Lenstra heuristics), or the rank of elliptic curves ordered by height (Katz-Sarnak).

In this talk, we will give an introduction to these sort of questions, focusing on Malle’s conjecture. Additionally, we will explain how to formulate function field analogues of this conjecture and transform this conjecture into a problem in algebraic topology (about the homology of certain moduli spaces of branched covers of P^1). In joint work with Ellenberg and Tran, we partially solved this problem, giving the upper bound in Malle’s conjecture.

## May 7 (5pm ET = 11pm Copenhagen)

Note unusual time!

Title: Operadic diagonals and tensor products

Speaker: Dasha Polyakova

Abstract: Having two DG-algebras, one can easily construct their tensor product, another DG-algebra. Tensoring two A-infinity algebras is more tricky. To do that, one needs to construct a diagonal in A-infinity operad. I will explain how one such diagonal can be obtained using cubical subdivision of associahedra (the original construction is due to Saneblidze-Umble, but I will be following Markl-Schnider).

## April 30

Title: The stable maps limit of a rational function

Speaker: Dori Bejleri

Abstract: A rational function in one variable induces a map $\mathbb{P}^1 \to \mathbb{P}^1$ and a generic such map of degree n has n poles. Marking these n poles gives an embedding of the space of degree n maps, unramified over infinity, into the Kontsevich space $\bar{M}_{0,n}(\mathbb{P}^1,n)$ of n-pointed genus 0 stable maps of degree n. The question we will address in this talk is what is the closure of this locus? Phrased another way, which configurations of pointed trees of rational curves can appear as the limit of a family of degree n rational functions as the n poles collide? In the process we will review the relevant notions of stable curves, stable maps, and their deformations. If time permits we will explain the motivation for this question coming from the theory of elliptic surfaces.

## April 23

Title: The minimal model program

Speaker: Joaquín Moraga

Abstract: First, I will sketch the aim of the minimal model program. Then, I will show some major theorems in the field and some open problems. Finally, I will discuss some recent results towards the understanding of the singularities of the minimal model program.

## April 17, Friday

Note unusual day!

Title: Some examples of Koszul duality via dg algebras

Speaker: Figlio di Guglielmo (Geordie Williamson)

Abstract: I will explain the basics of Morita theory for derived categories and give some examples, including the classic BGG duality between symmetric and exterior algebras. This is standard stuff but should perhaps be better known. No new results or even recent research will be discussed as where I am it’s the morning, and I’d prefer not to drink...

Time: same 8pm ET 4/17 = 10am Syd 4/18

## April 9

Title: Good moduli spaces for Artin stacks

Speaker: murmuno (Sveta Makarova)

Abstract: The talk is based on Alper's paper "Good moduli spaces for Artin stacks". I will briefly remind definitions of moduli problems and stacks and then proceed to explaining Alper's results. After that, I will focus on giving various examples and providing some intuition about how one could use good moduli spaces.

## April 2

Title: Some non-noetherian sorcery

Speaker: Elden Elmanto

Abstract: Before the pandemic broke, I was working on Milnor excision problems for motivic cohomology which presents a lot of non-noetherian trapdoors. I will survey the sequence of words one needs to say to live through these trapdoors after the work of SGA, Thomason, Lurie and Temkin.

## March 26

https://harvard.zoom.us/j/652411376

Title: The nonabelian Hodge correspondence

Speaker: Skd (Sanath Devalapurkar)

Abstract: We'll talk about a nonabelian version of Hodge theory, due to Simpson. This provides a correspondence between representations of the fundamental group pi_1 and (certain) vector bundles equipped with a "Higgs field". We'll also talk about the associated moduli spaces (de Rham, Higgs, and Dolbeaut).