Algebra Seminar
(Mon 3:30-4:30pm, Kap 245)
Organizer: Cris Negron
Topics include (but are not limited to):
Algebra, representation theory, algebraic and arithmetic geometry, homotopy theory, and mathematical physics.
Fall 2023
| Date | Location | Speaker | Abstract |
|---|---|---|---|
| Aug 28 | Harold Williams (USC) | Canonical Bases for Coulomb Branches
Abstract: Following work of Kapustin-Saulina and Gaiotto-Moore-Neitzke, one expects half-BPS line defects in certain supersymmetric 4d quantum field theories to form monoidal categories with rich structures (for example, a monoidal cluster structure in many cases). This talk explains an algebro-geometric approach to defining this category in the case of gauge theories with polarizable matter. This definition takes the form of a nonstandard t-structure on the dg category of coherent sheaves on the derived Braverman-Finkelberg-Nakajima space of triples. We refer to objects in this category as Koszul-perverse coherent sheaves, as this t-structure interpolates between the perverse coherent t-structure and certain t-structures appearing in the theory of Koszul duality (specializing to these in the case of a pure gauge theory and an abelian gauge theory, respectively). As a byproduct, this defines a canonical basis in the associated quantized Coulomb branch by passing to classes of irreducible objects. This is joint work with Sabin Cautis. |
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| holiday | |||
| Sept 11 | Joaquín Morage (UCLA) | Coregularity of Fano varieties
Abstract: Fano varieties are one of the three important blocks of algebraic varieties. In this talk we will review the history of Fano varieties Then, we will discuss some recent invariant, called the coregularity of the Fano variety that measures certain combinatorial complexity of the variety. We will explain how this invariant reflects in the geometry of the Fano variety and how it can help for the classification of such varieties. |
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| Sept 18 | |||
| Sept 25 | Daniele Garzoni (USC) | Hilbert's irreducibility theorem via random walks
Abstract: Let G be a linear algebraic group over a number field K (e.g., SLn), and let f: V → G be a cover of finite degree. Perform a long random walk on a Cayley graph of a fixed finitely generated subgroup of G(K) (e.g., SLn(Z)). Should you expect to hit elements g whose fibre f-1(g) is K-irreducible? After giving motivation, we will see that the answer is Yes, under some conditions on G and f. This represents a quantitative version of Hilbert's irreducibility theorem for linear algebraic groups. Joint work with Lior Bary-Soroker. |
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| Oct 2 | |||
| Oct 9 | Ben Elias (U Oregon) | Demazure operators for G(m,m,n) and quantum geometric Satake at a root of unity
Abstract: The reflection representation of the affine Weyl group in type A admits a q-deformation, which plays a role in quantum geometric Satake. Specializing q to a primitive 2m-th root of unity, one obtains a reflection representation of the complex reflection group G(m,m,n). In this way, a category of "singular Soergel bimodules" for G(m,m,n) is related to representations of quantum groups at roots of unity. To properly study these bimodules, additional algebraic background is needed. Associated to a reflection representation one has an action of a reflection group W on a polynomial ring R. A classic result of Demazure for Coxeter groups states that the inclusion of invariant polynomials into all polynomials is a Frobenius extension. A key tool in Demazure's proof is the nilCoxeter algebra, which acts on R by certain operators now called Demazure operators. Demazure proves that the Frobenius trace agrees with the Demazure operator associated to the longest element. These facts are what make the theory of Soergel bimodules tick. In joint work with Juteau and Young, we study the affine reflection representation of G(m,m,n). We prove the corresponding Frobenius extension result. We investigate an exotic nilCoxeter algebra with Demazure operators associated to elements of the affine Weyl group, acting on the ring R. When n=3, we determine which elements of the affine Weyl group are "longest elements" for G(m,m,n). |
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| Oct 16 | |||
| Oct 23 | Agustina Czenky (U Oregon) | Low rank symmetric fusion categories in positive characteristic
Abstract: In this talk, we look at the classification problem for symmetric fusion categories in positive characteristic. We recall the second Adams operation on the Grothendieck ring and use its properties to obtain some classification results. In particular, we show that the Adams operation is not the identity for any non-trivial symmetric fusion category. We also give lower bounds for the rank of a (non-super-Tannakian) symmetric fusion category in terms of the characteristic of the field. As an application of these results, we classify all symmetric fusion categories of rank 3 and those of rank 4 with exactly two self-dual simple objects. |
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| Oct 30 | Pablo S. Ocal (UCLA) | Quantum symmetries via twisted tensor products and their Balmer spectrum
Abstract: In this talk I will present an approach to understand the representation theory of Hopf algebras. The ideas are inspired by the Balmer spectrum of a symmetric tensor triangulated category, a topological tool analogous to the usual spectrum of a commutative ring. Our main tool will be twisted tensor products, a noncommutative generalization of the coordinate ring of a variety. After introducing them, I will give some properties of their module categories, and set up how to understand their tensor-triangulated geometry using cohomology. |
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| Nov 6 | Peter Haine (UC Berkeley) | On Voevodsky’s reconstruction theorem
Abstract: In Grothendieck’s 1983 letter to Faltings, he made a number of conjectures about reconstructing schemes—initiating the field of anabelian geometry. The first conjecture is about reconstructing schemes from their étale fundamental groups. The second is about reconstructing schemes from their étale topoi. In 1990, Voevodsky proved that normal schemes of finite type over a finitely generated field of characteristic zero can be reconstructed from their étale topoi. However, this is not the optimal result; one expects to be able to reconstruct seminormal schemes from their étale topoi. In this talk, we’ll discuss joint work with Sebastian Wolf that generalizes Voevodsky’s theorem. In particular, in characteristic zero, we prove the optimal result. |
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| Nov 11 (Saturday!) |
Multiple Speakers | Southern California Algebraic Geometry Seminar
sites.google.com/site/socalags/fall-2023 |
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| Nov 13 | Tony Feng (UC Berkeley) | Mirror symmetry and the Breuil-Mezard Conjecture
Abstract: I will introduce the Breuil-Mezard Conjecture, which predicts the existence of hypothetical “Breuil-Mezard cycles” in the moduli space of representations of Galois groups of p-adic numbers. I will talk about a new approach to the Breuil-Mezard Conjecture, joint with Bao Le Hung, which is based on the intuition that it is analogous to homological mirror symmetry. In particular, we construct Breuil-Mezard cycles by combining certain instances of homological mirror symmetry with Bezrukavnikov-Mirkovic-Vilonen localization and a bit of the theory of quantum groups. This talk will be aimed at a general audience of algebraists. |
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| Nov 20 | early holiday | ||
| Nov 27 | Monica Vazirani (UC Davis) | Skeins on Tori
Abstract: We study skeins on the 2-torus and 3-torus via the representation theory of the double affine Hecke algebra of type A and its connection to quantum D-modules. As an application we can compute the dimension of the generic SLN- and GLN-skein module of the 3-torus for arbitrary N. This is joint work with Sam Gunningham and David Jordan. |
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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).