Abstract:

Representations of KLR (quiver Hecke) algebras categorify the positive part of the quantum group associated to any symmetrizable Cartan matrix. This categorical perspective makes certain symmetries more natural to study. For example, the induction and restriction functors between categories of KLR algebra modules play an important role in the theory. A closer investigation of these functors reveals surprising new symmetries. In this talk, I will explain how the induction and restriction functors for KLR algebras can be used to obtain a 2-representation of the corresponding affine positive part in type A. I will also describe a new categorification of a closely related algebra, the q-boson algebra, in all symmetrizable Kac-Moody types.

We might start 10' late.

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I will discuss various noetherianity "up to symmetry" results from the literature and highlight some of their applications. I will then describe recent non-noetherianity phenomena in positive characteristic and explain how, perhaps unexpectedly, these are also connected to uniformity results in algebra.

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I will report on ongoing joint work with Keller VandeBogert and Jerzy Weyman on the total rank of the Tor groups of modules over polynomial rings that arise from representations of Lie algebras. This work is motivated by the problem of understanding lower bounds on the total rank of free complexes with finite length homology and also the problem of computing syzygies of nilpotent orbit closures.

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The prime spectrum of a commutative ring is the underlying set of prime ideals of the ring together with the Zariski topology. A theorem proven by Reyes states that any extension of the set-valued prime spectrum functor on the category of commutative rings to the category of (not necessarily commutative) rings must assign the empty set to the n by n matrix algebra with complex entries when n is greater than 2. This suggests that sets do not serve as the underlying structure of a spectrum of a noncommutative ring. It is argued in his recent paper that coalgebras can be viewed as the underlying object of a noncommutative spectrum. In this talk, we introduce coalgebras equipped with a ringed-space-like structure, which we call ringed coalgebras. These objects arise from fully residually finite-dimensional (RFD) algebras and schemes locally of finite type over a field k. The construction uses the Heyneman--Sweedler finite dual coalgebra and the Takeuchi underlying coalgebra. We will discuss that, if k is algebraically closed, the formation of ringed coalgebras gives a fully faithful functor out of the category of fully RFD algebras, as well as a fully faithful functor out of the category of schemes locally of finite type. The restrictions of these two functors to the category of (commutative) finitely generated algebras are isomorphic. In this way, ringed coalgebras can be thought of as a generalization of RFD algebras and schemes locally of finite type.

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We show that for a reductive algebraic group \(G\) there exists an integer \(r(G)\), such that any finite set of elements in \(G\) of size more than \(r(G)\) that generates a Zariski-dense subgroup must be redundant i.e. we can remove some elements and still generate a Zariski-dense subgroup. We use this to deduce the analogous result for compact Lie groups. Thus, for example, if you have \(1000\) rotations that generate a dense subgroup of \({\rm SO}(3)\), some of them must be redundant. For non-compact Lie groups (e.g \({\rm SL}_2(\mathbb{C})\)) this fails: there are arbitrarily large irredundant topologically generating sets. The proof is mostly arithmetic: we ensure generators live in a number field in order to reduce the problem to finite groups via strong approximation and other results of this sort.

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Let \(\Sigma\) be a genus \(g\) surface with \(n\) punctures. We will define a variety that parametrizes \({\rm SL}_d\)-representations of \(\Sigma\) in which the loops around the punctures have fixed characteristic polynomial. We will discuss two properties, geometric irreducibility and smoothness. The proof of the former uses a method due to Liebeck-Shalev involving characters of the finite group \({\rm SL}_d(\mathbb{F}_q)\) and the Lang-Weil theorem from algebraic geometry. The proof of the second applies linear algebra to the differentials of the commutator and characteristic polynomial maps. Time permitting, we will mention the corresponding relative character variety, define the action of the pure mapping class group of \(\Sigma\) on that, and indicate how our two results are used in studying the orbits.

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The theory of \(F\)-singularities uses the Frobenius morphism to measure how "singular" a ring is, that is, how far it is from being a regular ring. We describe two ways to measure singularities using frobenius, respectively called the quasi-\(F\)-split height and the \(F\)-pure threshold. We describe a relationship between these two invariants, which is latent in the literature, under very specific assumptions. The proof of this statement uses sophisticated geometric machinery (including deformation theory and crystalline cohomology). We then describe a more general proof, joint with Jagathese, that uses only algebra.

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I will talk about the boundary Carrollian conformal algebra (BCCA), an infinite-dimensional Lie algebra recently discovered in the context of Carrollian physics. The BCCA is an intriguing object from both physical and mathematical perspectives, since it is a filtered but not graded Lie algebra. In this talk, I will briefly introduce the physical context in which this Lie algebra appeared, and then mention some of my recent work on the representation theory of this Lie algebra. This is joint work with Xiao He, Tuan Pham, Haijun Tan, Girish Vishwa, and Kaiming Zhao.

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