MIT Infinite Dimensional Algebra Seminar (Spring 2025)

A periodic pencil of flat connections on a smooth algebraic variety $X$ is a linear family of flat connections $\nabla(s_1,...,s_n)=d-\sum_{i=1}^r\sum_{j=1}^ns_jB_{ij}dx_i$, where $\lbrace x_i\rbrace$ are local coordinates on $X$ and $B_{ij}: X\to {\rm Mat}_N$ are matrix-valued regular functions. A pencil is periodic if it is generically invariant under the shifts $s_j\mapsto s_j+1$ up to isomorphism. I will explain that periodic pencils have many remarkable properties, and there are many interesting examples of them, e.g. Knizhnik-Zamolodchikov, Dunkl, Casimir connections and equivariant quantum connections for conical symplectic resolutions with finitely many torus fixed points. I will also explain that in characteristic $p$, the $p$-curvature operators $\lbrace C_i,1\le i\le r\rbrace$ of a periodic pencil $\nabla$ are isospectral to the commuting endomorphisms $C_i^*:=\sum_{j=1}^n (s_j-s_j^p)B_{ij}^{(1)}$, where $B_{ij}^{(1)}$ is the Frobenius twist of $B_{ij}$. This allows us to compute the eigenvalues of the $p$-curvature for the above examples, and also to show that a periodic pencil of connections always has regular singularites. This is joint work with Alexander Varchenko.

We will discuss the functor of geometric Whittaker coefficients in the context of quantum geometric Langlands program. Concretely, we will prove (modulo the spectral decomposition conjecture) that the functor of quantum geometric Whittaker coefficients is conservative on the category of cuspidal automorphic D-modules. The proof will combine generalizations of representation-theoretic and microlocal methods from the preceding works of Faergeman-Raskin and Nadler-Taylor respectively.

This talk is based on joint work with Terry Gannon. We will discuss the interplay of tensor categories $\mathcal{C}$ with some group action $G$ and orbifolds $\mathcal{V}^G$ of vertex operator algebras $\mathcal{V}$ (VOAs for short). More specifically, we will show how the categorical structure of $\mathrm{TwMod}_G \mathcal{V}$ allows one to not only simplify previous results done purely through VOA techniques but vastly extend them. One such example is the Dijkgraaf-Witten conjecture, now a theorem, which describes how the category of modules of a holomorphic orbifold should look like. Additionally, our techniques also allow us to expand the modular fusion categories known to arise from VOAs, we show that every group-theoretical fusion category comes from a VOA orbifold. Lastly, we will discuss how VOAs with group actions give concrete realizations of $G$-Tambara Yamagami categories for nilpotent $G$.

Working over a field of positive characteristic, the higher Verlinde categories are obtained by taking the abelian envelope of quotients of the category of tilting modules for the algebraic group $\mathrm{SL}_2$. These finite symmetric tensor categories have been introduced by Benson-Etingof-Ostrik and, independently, Coulembier. This construction has been generalized by Sutton-Tubbenhauer-Wedrich-Zhu to Lusztig’s quantum group for $\mathfrak{sl}_2$ at an arbitrary root of unity, thereby yielding the mixed higher Verlinde categories. I will discuss the properties of these finite braided tensor categories. In particular, I will explain how to construct an analogue of the quantum Frobenius-Lusztig functor, derive a Steinberg tensor product formula for the simple objects, and identify the symmetric center of the mixed Verlinde categories.

We will describe a bridge between the theories of quantum projective spaces of Artin, Schelter, Tate and Van den Bergh and the theory of big quantum groups at roots of unity of De Concini, Kac and Procesi. The most interesting examples of quantum projective spaces of dimensions 3 and 4 are provided by the families of 3 and 4 dim Skyanin algebras. They are associated to an elliptic curve, an invertible sheaf on it and an automorphism of the curve. The most interesting case from representation theoretic point of view is when the automorphism has finite order, in which case the algebra is finite over its center. By combining techniques from the above two sets of works (Poisson geometry and noncommutative projective geometry), we will give a classification of the irreducible representations of the 3 and 4 dim Sklyanin algebras in the finite order case and a computation of their dimensions. This is a joint work with Xingting Wang (LSU) and Chelsea Walton (Rice Univ).

The volume of a locally symmetric space is essentially a product of special values of zeta functions. More generally, Hirzebruch's proportionality theorem (extended by Mumford) tells us how to integrate any Chern class polynomial on a locally symmetric space.

We give an analog and extension of these results in the function field case where new phenomena show up. Locally symmetric spaces will be replaced by the moduli space of Drinfeld Shtukas with multiple legs, and special values of zeta functions will be replaced by a linear combination of their derivatives of various order. This is joint work with Tony Feng and Wei Zhang.

K-theoretical Coulomb branches are expected to have cluster structure. Cautis and Williams categorified this expectation. In particular, they conjecture (and prove in type A) that the category of perverse coherent sheaves on the affine Grassmannian is a cluster monoidal categorification. We will discuss recent progress on this conjecture. In particular, we construct cluster short exact sequences of certain perverse coherent sheaves. We do that by constructing a bridge, relating this (geometric) category to the (algebraic) category of finite dimensional modules over the affine quantum group. This is done by relating both categories to the notion of Feigin--Loktev fusion product.

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Exclusion processes are stochastic particle lattice systems modeling fluctuations in classical diffusive systems. Their extensive study over the past decades led to the formulation of the now called macroscopic fluctuation theory - a framework extending Bolztman's theory to non-equilibrium diffusive systems. In the first part of the talk, I will describe the newly proposed quantum extension of exclusion processes, aiming at formulating a quantum version of the macroscopic fluctuation theory. The focus will be on the interplay between group theory and probability theory at play in solving these models. In the second part, I will describe elements of a theory of structured random matrices which has emerged from studying these models.

In this talk, we first give a gentle introduction to cluster algebras.
Next, we construct common triangular bases for almost all the known (quantum) cluster algebras from Lie theory. These bases provide analogs of the dual canonical bases, long anticipated in cluster theory. In cases where the generalized Cartan matrices are symmetric, we show that these cluster algebras and their bases are quasi-categorified.
Finally, we extend cluster algebras from finite ranks to infinite ranks. This extension allows us to show that, for cluster algebras associated with double Bott-Samelson cells, the standard bases can be computed via a braid group action when the Cartan matrix is of finite type.

We will present new insights in alterfold theory to study modular tensor categories. We provide streamlined quick proofs and broad generalizations of a wide range of results on modular invariance of modular tensor categories. It is based on joint work with M. Shuang, Y. Wang and J. Wu, arXiv:2412.12702.

The representation theory of quantum affine algebras has been the subject of intense study for almost 30 years, an important aspect of which are the q-characters introduced by Frenkel-Reshetikhin. I will survey recent developments that generalize this framework (specifically that of the Hernandez-Jimbo category O) to quantum loop groups associated to any Kac-Moody Lie algebra, and introduce new shuffle algebra tools for the computation of q-characters.