Algebra and Algebraic Geometry Seminar Spring 2024
The seminar normally meets 2:30-3:30pm on Fridays, in the room Van Vleck B139.
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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 |
| April 18 | Teresa Yu (Michigan) | TBA | Dima/Jose |
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.