Workshop

In this talk I will explain a recent result we obtained with Iordanis Romaidis which asserts that Kauffman bracket skein modules of 3-manifolds are finitely-generated and holonomic as modules for the skein algebras of the boundary surfaces. This immediately implies a long-standing conjecture of Frohman, Gelca and Lofaro concerning the annihilator of the empty skein, and it gives a new proof (after Garoufalidis-Le) that the quantum A-polynomial of a knot is holonomic.

The approach is largely that of geometric representation theory — we develop a non-abelian analog of the holonomicity framework of Sabbah in the quantum torus, which itself echoes Bernstein and Kashiwara's approach to holonomicity for ordinary differential equations.

Higgs and Coulomb branches of quiver gauge theories form two rich families of Poisson varieties that are expected to be exchanged under so-called symplectic duality. In this talk, I will discuss the geometry and representation theory associated with these varieties and formulate a conjecture relating the two sides. The talk will be introductory and is based on arXiv:2608.16746.

Associated to a complex reductive group G is a variety originally considered by Lusztig known as the group scheme of universal centralizers. Our main result describes this variety as an affine blow up of the quotient of the cotangent bundle of a maximal torus of G by its action of the Weyl group. After making this result more precise and comparing it to previous results in the literature, we will introduce the notion of the invariant Weil restriction (also known as the transverse Hilbert scheme) and explain how our results generalize to this setting. This is joint with Victor Ginzburg.

Let k be a field. We introduce new invariants for k-schemes valued in unipotent group schemes over k. To do so, we develop the notion of "unipotent spectra" — defined to be the stabilization of Toën's category of affine stacks. 

As an application, we recover the Artin—Mazur formal groups associated to schemes. Further, we show that syntomic cohomology admits a natural refinement to a perfect unipotent spectrum. Finally, we extend Milne's work on arithmetic duality theorems to perfect unipotent spectra and apply it to refine Poincaré duality in syntomic cohomology. 

This is joint work with Shubhodip Mondal and Tasos Moulinos.