Algebra and Algebraic Geometry Seminar Fall 2023

The seminar normally meets 2:30-3:30pm on Fridays, in the room VV B135.

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Fall 2023 Schedule

date	speaker	title	host/link to talk
September 15	Joshua Mundinger	Quantization in positive characteristic	local
September 22	Andrei Negut	Computing K-HA's of quivers	local
October 6	Daniel Bragg (Utah)	A Stacky Murphy’s Law for the Stack of Curves	Josh
October 13	Xinchun Ma (UChicago)	Filtrations on the finite dimensional representations of rational Cherednik algebras	Josh
November 3	Junyan Zhao	Moduli of curves and K-stability	Peter W
November 17	Purnaprajna Bangere	Syzygies of adjoint linear series on projective varieties	Michael K
December 1	Ekaterina Bogdanova (Harvard)	Sheaves of categories on the moduli stack of local systems on the formal punctured disk via factorization	Dima
December 8	Wanchun (Rosie) Shen (Harvard)	Du Bois singularities, rational singularities, and beyond	Andrei

Abstracts

Joshua Mundinger

Quantization in positive characteristic

In order to answer basic questions in modular and geometric representation theory, Bezrukavnikov and Kaledin introduced quantizations of symplectic varieties in positive characteristic. These are certain noncommutative algebras over a field of positive characteristic which have a large center. I will discuss recent work describing how to construct an important class of modules over such algebras.

Andrei Negut

Computing K-HA's of quivers

Many interesting moduli stacks M in geometric representation theory admit interesting K-theoretic Hall algebras (K-HAs), defined by endowing the algebraic K-theory of M with an appropriate convolution product. While these algebras are notoriously hard to compute, they have an interesting relative called the shuffle algebra S. When M is a moduli stack of quiver representations, S is given by a collection of ideals inside polynomial rings, and their study can be reduced to commutative algebra. Fortunately/unfortunately, the commutative algebra in question is challenging, and we do not yet know of a complete description for a general quiver. In this talk, I will explain the general framework behind this problem, and survey results for the following special cases of quivers:

• double quivers arising in the theory of Nakajima quiver varieties
• quivers corresponding to symmetric Cartan matrices, yielding simply laced quantum loop groups
• quivers associated to toric Calabi-Yau threefolds in mathematical physics

Daniel Bragg

A Stacky Murphy’s Law for the Stack of Curves

We show that every Deligne-Mumford gerbe over a field occurs as the residual gerbe of a point of the moduli stack of curves. Informally, this means that the moduli space of curves fails to be a fine moduli space in every possible way. We also show the same result for a list of other natural moduli problems. This is joint work with Max Lieblich.

Xinchun Ma

Filtrations on the finite dimensional representations of rational Cherednik algebras

Under the Gordon-Stafford functor, every filtered representation of the type A rational Cherednik algebra corresponds to an equivariant coherent sheaf on the Hilbert scheme of points on the plane. Under the decategorification of this functor, the images of the finite-dimensional representations are conjectured to be closely related to the torus knot superpolynomials (with some special cases proved). There are several candidates for the filtrations coming from algebraic or geometric formulations conjectured to coincide with each other. I'll talk about recent developments on these conjectures including my own work in progress.

Junyan Zhao

Moduli of curves and K-stability

The K-moduli theory provides us with an approach to study moduli of curves. In this talk, I will introduce the K-moduli of certain log Fano pairs and how it relates to moduli of curves. We will see that the K-moduli spaces interpolate between different compactifications of moduli of curves. In particular, the K-moduli gives the last several Hassett-Keel models of moduli of curves of genus six.

Purnaprajna Bangere

Syzygies of adjoint linear series on projective varieties

Syzygies of algebraic varieties have long been a topic of intense interest among algebraists and geometers alike. After the pioneering work of Mark Green on curves, numerous attempts have been made to extend some of these results to higher dimensions. It has been proposed that the syzygies of adjoint linear series L=K+mA, with A ample is a natural analogue for higher dimensions to explore. The very ampleness of adjoint linear series is not known for even threefolds. So the question that has been open for many years is the following (Question): If A is base point free and ample, does L satisfy property N_p for m>=n+1+p? Ein and Lazarsfeld proved this when A is very ample in 1991. In a joint work with Justin Lacini, we give a positive answer to the original question above.

Ekaterina Bogdanova

Sheaves of categories on the moduli stack of local systems on the formal punctured disk via factorization

Given a DG category acted on by the category of quasi-coherent sheaves on LocSys_G(D°) (the stack of G-local systems on the punctured formal disk D°), one can define a factorization Rep(G)-module category. Following ideas of Beilinson and Drinfeld, Gaitsgory conjectured that this construction loses no information: that it gives a fully faithful 2-functor QCoh(LocSys_G(D°))-mod(DGCat)→Rep(G)-mod^fact(DGCat). I will give a quick introduction to the local Geometric Langlands program, discuss preliminaries, and the role of the above conjecture in this context. If time permits, we will discuss a partial result in the direction of the conjecture. Namely, the fully faithfulness for QCoh(LocSys_G(D°))-modules set-theoretically supported over the stack of local systems with restricted variation on the formal punctured disk.

Wanchun (Rosie) Shen

Du Bois singularities, rational singularities, and beyond

We survey some extensions of the classical notions of Du Bois and rational singularities, known as the k-Du Bois and k-rational singularities. By now, these notions are well-understood for local complete intersections (lci). We explain the difficulties beyond the lci case, and propose new definitions in general to make further progress in the theory. This is joint work (in progress) with Mihnea Popa, Matthew Satriano, Sridhar Venkatesh and Anh Duc Vo.