Algebra and Algebraic Geometry Seminar Spring 2024

The seminar normally meets 2:30-3:30pm on Fridays, in the room Van Vleck B139. Current semester

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Spring 2024 Schedule

date	speaker	title	host/link to talk
February 16	Sean Cotner (Michigan)	Schemes of homomorphisms	Josh
February 23	Lingfei Yi (Minnesota)	Slices in the loop spaces of symmetric varieties	Dima/Josh
March 1	Shravan Patankar (UIC)	The absolute integral closure in equicharacteristic zero	Dima/Josh
March 18 (Monday) 2:30-3:30pm in B123	Marton Hablicsek (Leiden University)	A formality result for logarithmic Hochschild (co)homology	Dima
April 19	Teresa Yu (Michigan)	Standard monomial theory modulo Frobenius in characteristic two	Dima/Jose
May 2 (Thursday) 3:30-4:30pm in B321	John Cobb (UW)	Multigraded Stillman’s Conjecture	local

Abstracts

Sean Cotner

Schemes of homomorphisms

Given two algebraic groups G and H, it is natural to ask whether the set Hom(G, H) of homomorphisms from G to H can be parameterized in a useful way. In general, this is not possible, but there are well-known partial positive results (mainly due to Grothendieck). In this talk I will describe essentially optimal conditions on G and H under which Hom(G, H) is a scheme. There will be many examples, and we will see how a geometric perspective on Hom(G, H) can be useful in studying concrete questions. Time permitting, I will discuss some aspects of the theory of Hom schemes over a base.

Lingfei Yi

Slices in the loop spaces of symmetric varieties

Let X be a symmetric variety. J. Mars and T. Springer constructed conical transversal slices to the closure of Borel orbits on X and used them to show that the IC-complexes for the orbit closures are pointwise pure. This is an important geometric ingredient in their work providing a more geometric approach to the results of Lusztig-Vogan. In the talk, I will discuss a generalization of Mars-Springer's construction of transversal slices to the setting of the loop space LX of X where we consider closures of spherical orbits on LX. I will also explain its applications to the formality conjecture in the relative Langlands duality. If time permits, I will discuss similar constructions for Iwahori orbits. This is a joint work with Tsao-Hsien Chen.

Shravan Patankar

The absolute integral closure in equicharacteristic zero

In spite of being large and non noetherian, the absolute integral closure of a domain R, R^{+}, carries great importance in positive characteristic commutative algebra and algebraic geometry. Recent advances due to Bhatt hint at a similar picture in mixed characteristic. In equicharacteristic zero however, this object seems largely unexplored. We answer a series of natural questions which suggest that it might play a similar central role in the study of singularities and algebraic geometry in equicharacteristic zero. More precisely, we show that it is rarely coherent, and facilitates a characterization of regular rings similar to Kunz's theorem. Both of these results, have in turn, applications back to positive characteristics.

Marton Hablicsek

A formality result for logarithmic Hochschild (co)homology

Hochschild homology is a foundational invariant for associate algebras, schemes, stacks, etc. For smooth and proper varieties X over a field of characteristic 0, Hochschild homology and its variants, like cyclic homology, are closely related to Hodge cohomology and to de Rham cohomology. For affine schemes, the Hochschild invariants are, in general, infinite dimensional. In this talk, we extend Hochschild homology to logarithmic schemes, in particular to compactifications, i.e, to pairs (X,D) where X is a smooth and proper variety and D is a simple normal crossing divisor. Using the formality theorem of Arinkin and Căldăraru, we recover an HKR isomorphism for logarithmic schemes relating logarithmic Hochschild homology to logarithmic differential forms. I will also discuss simple applications of our framework. This is a joint work with Francesca Leonardi and Leo Herr.

Teresa Yu

Standard monomial theory modulo Frobenius in characteristic two

Over a field of characteristic zero, standard monomial theory and determinantal ideals provide an explicit decomposition of polynomial rings into simple GL_n-representations, which have characters given by Schur polynomials. In this talk, we present work towards developing an analogous theory for polynomial rings over a field of characteristic two modulo a Frobenius power of the maximal ideal generated by all variables. In particular, we obtain a filtration by modular GL_n-representations whose characters are given by certain truncated Schur polynomials, thus proving a conjecture by Gao-Raicu-VandeBogert in the characteristic two case. This is joint work with Laura Casabella.

John Cobb

Multigraded Stillman’s Conjecture

In 2000, Mike Stillman conjectured that the projective dimension of a homogeneous ideal in a standard graded polynomial ring can be bounded just in terms of the number and degrees of its generators. I’ll describe the Ananyan-Hochster principle important to its proof, how to package this up using ultraproducts, and use this to characterize the polynomial rings graded by any abelian group that possess a Stillman bound.