Algebra and Algebraic Geometry Seminar Spring 2024

The seminar normally meets 2:30-3:30pm on Fridays, in the room Van Vleck B317.

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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)	Marton Hablicsek	TBA	Andrei/Dima
March 29	TBA	TBA	Josh

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.