Algebra Seminar
(Mon 3:30-4:30pm, Kap 265)
Organizer: Cris Negron
Topics include (but are not limited to):
Algebra, representation theory, algebraic and arithmetic geometry, homotopy theory, and mathematical physics.
Fall 2024
| Date | Location | Speaker | Title and abstract |
|---|---|---|---|
| Sep 9 | |||
| Sep 16 | |||
| Sep 23 | Robert Cass (U Michigan) | Integral motives on affine flag varieties Abstract: A result of Bezrukavnikov gives two categorical incarnations of the affine Hecke algebra, and can be viewed as a geometric Langlands correspondence in the tamely ramified case. I will discuss progress toward an enhancement of this result to mixed sheaves with integral coefficients. This includes motivic versions of the geometric Satake equivalence and Gaitsgory's central functor, which categorify the center of the affine Hecke algebra. This is joint work with T. van den Hove and J. Scholbach. |
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| Sep 30 | [canceled] Vyjayanthi Chari (UC Riverside) | ||
| [Friday!] Oct 4 | 2:30 pm, KAP 265 | Eugene Gorsky (UC Davis) | Trace of the affine Hecke category Abstract: The affine Hecke category, defined using affine Soergel bimodules, categorifies the affine Hecke algebra. I will compare the derived horizontal trace of the affine Hecke category with the elliptic Hall algebra, and with the derived category of the commuting stack. In particular, I will describe certain explicit generators for the trace category and some categorical commutation relations between these. This is a joint work with Andrei Negut. |
| Oct 7 | no talk | ||
| Oct 14 | Jacob Greenstein (UC Riverside) | Hecke and Artin monoids and their homomorphisms Abstract: In the present talk we discuss relations between various types of homomorphisms between Hecke and Artin monoids. Our original motivation was based on a striking observation that parabolic projections of Hecke monoids preserve parabolic elements which appeared naturally in the framework of actions of cacti on geometric crystals. We will present several new infinite families of homomorphisms which in some cases provide a complete classification. |
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| Oct 21 | Yu Fu (Cal Tech) | Algebraic Independence of Special Points on Shimura Varieties Abstract: The subject of special subvarieties in Shimura varieties has been of longstanding interest in algebraic geometry and number theory – major questions include the Zilber--Pink Conjecture and the consequential André--Oort Conjecture, which was recently proved by Pila, Tsimerman and Shankar. In this talk, I will introduce new results and applications in the direction of unlikely intersections in the recent joint paper with Roy Zhao. Especially, given some correspondence V between a connected Shimura variety S and a commutative connected algebraic group G, and a positive integer n, we prove that the V-images of any n special points on S outside a proper Zariski closed subset are algebraically independent. Our result unifies previous unlikely intersection results on multiplicative independence and linear independence. We also give an application to abelian varieties by proving that the special points of S whose V-images lie in a finite-rank subgroup of G are contained in a finite union of proper special subvarieties of S, only dependent on the rank of the subgroup. In this way, our result is a generalization of the works of Pila--Tsimerman and Buium--Poonen. Also, I will talk about our results on multiplicative independence of differences of singular moduli, generalizing previous results by Pila--Tsimerman, and Aslanyan--Eterovic--Fowler. |
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| Oct 28 | Alexandra Utiralova (UC Berkeley) | Representations of general linear groups in the Verlinde category Abstract: The Verlinde category Verp is the semisimplification of the category of finite-dimensional representations of Z/pZ in characteristic p. In arXiv:2107.02372 Coulembier, Etingof, and Ostrik showed that all pre-Tannakian categories of moderate growth with additional nice property (Frobenius exactness) admit a fiber functor into Verp. Consequently, the study of representations of affine group schemes in such categories reduces to studying representations of affine group schemes in Verp. I will talk about what is known about the category of representations of the affine group scheme GL(X) for an object X in Verp, and about the highest weight structure on it. |
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| Nov 4 | Jon Brundan (U Oregon) | Classical representation theory via categorification The standard approach to representation theory in categories like Rep(G) for a reductive group over an algebraically closed field, or the BGG category O attached to a semisimple Lie algebra, is based on the combinatorics of the underlying Weyl group (and its Hecke algebra). In Cartan type A, there is another approach exploiting combinatorics of an underlying Kac-Moody algebra (or its quantized enveloping algebra). This was developed in examples over many decades, and fits into a unified general framework which we now call "Heisenberg categorification". I will explain the general setup and some of its consequences. Analogous approaches are slowly emerging for the other families OSp, P and Q of classical supergroups, and I will say a bit more about the type Q case which is the next most complete. |
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| [Friday!] Nov 15 | KAP 414, 2:00 pm | David Green (Ohio State) | Levin-Wen is a Gauge Theory In this talk we will examine a class of SPT's protected by the tube algebra of a unitary fusion category C and show that gauging this symmetry produces the standard Levin-Wen model associated to C. Mathematically, the main tool we introduce to accomplish this goal are the spine coral skein modules which corresponds graphically to the Day convolution products of tube algebra representations. This talk is based off of arXiv:2401.13838, joint with Kyle Kawagoe, Corey Jones, Sean Sanford, and David Penneys. |
| Nov 18 | Shubhodip Mondal (UBC) | Unipotent homotopy theory of schemes Building on Toen’s work on affine stacks, I will discuss a notion of homotopy theory for schemes, which we call "unipotent homotopy theory". Over a field of characteristic p>0,I will explain how the unipotent homotopy group schemes recover (1) unipotent completion of the Nori fundamental group scheme, (2) p-adic etale homotopy groups, and (3) certain formal group laws arising from algebraic varieties constructed by Artin and Mazur. Time permitting, I will discuss unipotent homotopy types of Calabi–Yau varieties and show that the unipotent homotopy group schemes πiU(X) of Calabi--Yau varieties (of dimension n) are derived invariant for all i; the case i=n corresponds to a recent result of Antieau–Bragg. This is a joint work with Emanuel Reinecke. |
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| [Friday!] Nov 22 | KAP 265, 3:30 pm | Ivan Losev (Yale) | Harish-Chandra center for affine Kac-Moody algebras in positive characteristic This talk is based on a joint work in progress with Gurbir Dhillon. A remarkable theorem of Feigin and E. Frenkel from the early 90's describes the center of the universal enveloping algebra of an (untwisted) affine Kac-Moody Lie algebra at the so called critical level proving a conjecture of Drinfeld: the center in question is the algebra of polynomial functions on an infinite dimensional affine space known as the space of opers. In our work we study a part of the center in positive characteristic p at an arbitrary non-critical level. Namely, we prove that the loop group invariants in the completed universal enveloping algebra is still the algebra of polynomials on an infinite dimensional affine space that is "p times smaller than the Feigin-Frenkel center". In my talk I will introduce all necessary notions, state the result, explain motivations and examples. |
Spring 2025
| Date | Location | Speaker | Abstract |
|---|---|---|---|
| Jan 13 | Jennifer Brown (U Edinburgh) | ||
| Feb 10 | |||
| Feb 17 | off | ||
| Feb 24 | |||
| March 3 | |||
| March 10 | |||
| March 17 | spring break | ||
| March 24 | Nathan Geer (Utah State) | ||
| March 31 | |||
| April 7 | |||
| April 14 | |||
| April 21 | |||
| April 28 |
Mailing list: Contact me if you want to be added to the seminar mailing list.
See our friends at the Geometry seminar (M 2:00 pm) and the Combinatorics seminar (W 2:00 pm).