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).