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.
Spring 2025
| Date | Location | Speaker | Abstract |
|---|---|---|---|
| Jan 13 [2-3pm!] | Jennifer Brown (U Edinburgh) | Skein Categories in Non-semisimple Settings Abstract: Skein theory uses diagrammatics of braided tensor categories to build link invariants and to study moduli spaces. Many of its constructions and definitions were originally formulated for semisimple categories, but physically motivated examples are often non-semisimple. This has inspired mathematicians to update the theory to accommodate non-semisimple settings. One of the main features in non-semisimple skein theory - admissibility conditions - is not compatible with the powerful factorization homology techniques that have found purchase in skein theory. The crux of the problem is that the tensor unit is rarely admissible, complicating many constructions. In this talk we'll start by a introduction to skein theory, then explain the speaker's work with Benjamin Haïoun to reconcile admissibility conditions and factorization homology. |
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| Feb 7 [FRIDAY! 2pm] | Cailan Li (Academia Sinica) | Diagrammatics for the derived category with applications to link homology Abstract: A classical theorem of Jones says that one can obtain the HOMFLY polynomial from the Markov trace on the Hecke algebra. Khovanov categorified this construction using Ext groups between Soergel Bimodules to produce (triply graded) link homology. In this talk, we introduce a diagrammatic framework for studying (in rank 2) Ext Enhanced Soergel Bimodules, a full monoidal subcategory of Db(R-Bimodules), where R is a polynomial ring. This construction provides a novel approach to computations in the derived category, circumventing many of the usual technical challenges. As an application, we present new computations in (triply graded) link homology that hold uniformly for any positive or negative link. |
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| Feb 10 | Dylan Butson (UCLA) | Inverse Hamiltonian reduction for affine W-algebras in type A (joint with Sujay Nair) Abstract: I will outline the proof of a conjectural relation between affine W-algebras corresponding to distinct nilpotent orbits in the same Lie algebra, by first establishing a geometric proof of the finite-type, classical analogue of the statement, and then applying recently developed localization and deformation theory techniques for vertex algebras. |
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| Feb 17 | off | ||
| Feb 24 | Paolo Aluffi (FSU) | The Grothendieck class of the moduli space of pointed stable curves of genus 0 Abstract: The variety M0,n parametrizes stable genus 0 curves with n marked points. This is a central object in algebraic geometry, as the most studied and best understood moduli space of curves. Explicit constructions of this variety have been known for several decades, and recursion formulas for its Poincaré polynomial were obtained more than 30 years ago, but (to my knowledge) a more explicit expression for its Betti numbers was not available. We obtain just such an expression, in the form of an explicit generating function for the class of M0,n in the Grothendieck group of varieties, and gather more information about related generating functions. As an application, we prove an asymptotic form of log concavity for the Poincaré polynomial of M0,n. This is joint work with Matilde Marcolli, Stephanie Chen, and Eduardo Nascimento. |
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| March 3 | Gurbir Dhillon (UCLA) | On local geometric Langlands in depth zero Abstract: A fundamental theorem of Bezrukavnikov establishes local geometric Langlands with tame ramification and unipotent monodromy. We will review this result for non-specialists, and then discuss some joint works with Li--Yun--Zhu and Taylor, which extend this to the full equivalence in depth zero for restricted variation and Betti families, as conjectured by Bezrukavnikov and Ben-Zvi--Nadler, respectively. |
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| March 10 | Nathan Geer (Utah State) | Some algebra behind non-semisimple TQFTs Abstract: In this talk I will give an introductory lecture on constructing Topological Quantum Field Theories (TQFTs) from non-semisimple categories. The main goal of the talk is to give a hint of what is needed to extend the Turaev-Viro and Crane-Yetter TQFTs from the useful setting of semisimple categories to the non-semisimple world. I will do this from an algebraic and categorical point of view. In particular, I will discuss what kind of structures are needed in non-semisimple categories to give rise to (2+1)-TQFTs. Then I will remark that any spherical tensor category (in the sense of Etingof, Douglas et al.) has such structures. This work is joint with Francesco Costantino, Benjamin Haïoun, Bertrand Patureau-Mirand and Alexis Virelizier and based on arXiv:2302.04509 and arXiv:2306.03225. |
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| March 17 | spring break | ||
| March 24 | |||
| March 31 | [organizer will be absent] | ||
| April 7 | Irit Huq-Kuruvilla (Virginia Tech) | ||
| April 14 | |||
| April 21 | Kent Vashaw (UCLA) | ||
| April 28 | Justin Bloom (U Washington) |
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).