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.

  Abstract:

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.

  Abstract:

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.

  Abstract:

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.

  Abstract:

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.

  Abstract:

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.

  Abstract:

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.

  Abstract:

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.

  Abstract:

The exchange property, introduced by Crawley and Jonsson in 1964 in the study of direct decompositions of algebraic systems and later extended to modules and rings by Warfield, plays a central role in modern decomposition theory. One of the main open problems in the area is whether the finite exchange property implies the full exchange property. This talk surveys the development of exchange theory from its module-theoretic origins to its ring-theoretic formulation via exchange rings. The last part of the talk is based on joint work with A. Cigdem Ozcan and focuses on lifting theory, including idempotent, regular, and unit lifting ideals and morphisms, and their interaction with local morphisms.

  Abstract:

The border rank of tensors is a widely studied topic with practical applications to theoretical computer science and algebraic statistics. New lower bounds were obtained for the matrix multiplication tensor using techniques from representation theory and algebraic geometry. In this talk, we will prove non-trivial border rank lower bounds for a class of \({\rm GL}(V)\)-invariant tensors using Young flattenings. We will see how this comes down to proving results on ranks of certain maps between schur functors, the proofs of which surprisingly uses deep results in representation theory and commutative algebra.

  Abstract:

I will discuss the rational cohomology of \({\rm SL}_n(R), {\rm Sp}_{2n}(R)\), and their principal congruence subgroups for \(R\) a number ring. Borel-Serre showed that these groups satisfy a (co)homological duality that lets us study their cohomology groups via certain representations called the Steinberg modules, which have a beautiful combinatorial description in terms of Tits buildings. I will describe a conjecture of Lee-Szczarba on the top cohomology of principal congruence subgroups of \({\rm SL}_n(\mathbb{Z})\), and its resolution due to Miller-Patzt-Putman. I will then discuss forthcoming work on generalizations of this to other Euclidean rings, and also to symplectic groups.

  Abstract:

Consider a finite modular tensor category \(\mathcal A\). In [DGGPR] the authors exhibit a 3-dimensional topological field theory \(Z_{\mathcal A}: \operatorname{Bord}_{\mathcal A} \to \operatorname{Vect}\), which, in the case where \(\mathcal A\) is semisimple, recovers the usual Reshetikhin-Turaev TQFT. In the present work we show that this extends naturally to a TQFT \(Z_{\operatorname{Ch}(\mathcal A)}\), which takes values in the symmetric tensor category \(\operatorname{Ch(Vect)}\) of linear cochains. This cochain valued theory furthermore respects (certain classes of) homotopies.

[DGGPR] M. De Renzi, A. M. Gainutdinov, N. Geer, B. Patureau-Mirand, and I. Runkel. 3-dimensional TQFTs from non-semisimple modular categories. Sel. Math. New Ser., 28(2):42, 2022.

  Abstract:

Let \(R\) be a polynomial ring, \(J\subseteq R\) an ideal, and \(\mathfrak{m}\) a maximal ideal containing \(J\). We consider invariants of the pair \((R, J)\) which measure the singularities of the embedding \({\rm Spec}(R/J) \subseteq {\rm Spec}(R)\) at \(\mathfrak{m}\): the log canonical threshold (lct) in characteristic zero and the \(F\)-pure threshold (fpt) in positive characteristic. A smaller value of the lct/fpt means that the embedding is "more singular;" we seek to classify pairs which are as singular as possible. In 1972, Skoda showed that \({\rm lct}_{\mathfrak{m}}(R, J) \geq 1/{\rm ord}_{\mathfrak{m}}(J)\), where \({\rm ord}_{\mathfrak{m}}\) denotes the order of vanishing at \(\mathfrak{m}\). Skoda's bound has been generalized and refined many times since; among these improvements is a 2014 result by Demailly and Pham using mixed multiplicities of \(J\) and \(\mathfrak{m}\). We extend Demailly and Pham's lower bound to positive characteristic and study the pairs \((R, J)\) for which \({\rm lct}_{\mathfrak{m}}(R, J)\) (or \({\rm fpt}_{\mathfrak{m}}(R, J)\)) equals the lower bound. We classify these "extremal pairs" in the standard graded case, the codimension 1 case, and the dimension 2 case, confirming special cases of a conjecture by Bivia-Ausina.

  Abstract:

I will review recent constructions of oligomorphic tensor categories generalizing Deligne's \({\rm Rep}(S_t)\). Then, I will lean into the model theoretic part of the question. Specifically, I will explain where there are no continuous families like the original \({\rm Rep}(S_t)\) and where you should look for \(n\)-parameter families, i.e., depending on \(n\) free variables. Ultimately, these questions are closely related to classes of structures in model theory.

  Abstract:

Recently, Balmer-Gallauer deduced the tensor-triangular geometry of the so-called "derived category of permutation modules," which controls both the usual modular representation theory of a finite group as well as that of its "\(p\)-local" subgroups. Their construction of "permutation twisted cohomology" plays a key role in their deduction in the case of elementary abelian \(p\)-groups; here the authors deduce far stronger geometric results. In this talk, after reviewing some basics about tensor-triangular geometry and permutation modules, we'll describe how one can utilize endotrivial complexes, the invertible objects of this category, to extend Balmer-Gallauer's results for elementary abelian \(p\)-groups to all \(p\)-groups.

  Abstract:

In a classic paper from 1990, Beilison, Lusztig and MacPherson gave a geometric realization of the quantized enveloping algebra of \({\rm gl}_n\) by defining a convolution product on the space of invariant functions over the variety of pairs of n-step partial flags over a finite field. This construction was generalized by Rosso to the mirabolic setting by modifying the points on the variety to include the additional data of a vector. A presentation for this "mirabolic quantum group" in terms of generators and relations was recently given by Fan, Zhang and Ma. I will describe this construction of the mirabolic quantum group and discuss its representation theory. Time permitting, I will also discuss a mirabolic quantum Schur-Weyl duality that this algebra satisfies with a mirabolic version of the Hecke algebra of Type A.

  Abstract:

For \(G\) an abelian group and \(A\) a \(k\)-algebra, it is well-known that a \(G\)-graded, \(k\)-central \((A, A)\)-bimodule is equivalent to a \(G\)-graded left \(A^{\text{e}}\)-module. This characterization is what allows us to update the results of Reyes and Rogalski regarding \(\mathbb{N}\)-graded twisted Calabi-Yau (tCY) algebras to the abelian \(G\)-graded case. In particular, we prove that an algebra is \(G\)-graded tCY if and only if it is \(G\)-graded and tCY. In the nonabelian case, however, we cannot assume there is such a bimodule correspondence. In this case, questions about non-abelian \(G\)-graded tCY algebras become homological questions about \(G\)-graded \((A, A)\)-bimodules and the category thereof. In this talk, we motivate the study of nonabelian group-graded tCY algebras by examining a result of Crawford which gives an example of a 2-dimensional AS regular algebra graded by a finite nonabelian group. We discuss progress and major obstructions found by working in the category of graded bimodules, and present potential solutions found by translating the problem to the category of relative \(((A,A),H)\) bimodule-comodules.

  Abstract:

Symplectic resolutions arise in representation theory (Springer resolution), algebraic geometry (Hilbert--Chow morphism), and mathematical physics (3D mirror symmetry). There is a program to classify all possible symplectic resolutions of a given singularity. This classification simplifies when the singularity is conical, as it suffices to resolve any neighborhood of the cone point. In ongoing work with Travis Schedler, we extend the perspective beyond conical singularities. Surprisingly, local resolutions of conical neighborhoods extend and glue uniquely to a global resolution, provided they are monodromy-free and chosen compatibly.