Abstract. Moduli spaces often have interesting enumerative properties. The goal of this talk is to introduce some enumerative results on solutions of matrix equations and zero-dimensional sheaves over singular curves. To motivate them, I first discuss several moduli spaces in general, which I put onto the "unframed" side and the "framed" side. The unframed side includes the commuting variety AB=BA of n x n matrices, the variety of commuting matrices satisfying polynomial equations (the titular "matrix points on varieties"), and the moduli stack of zero-dimensional coherent sheaves on a variety. The framed side includes the Hilbert scheme of points on a variety, or more generally, the Quot scheme of zero-dimensional quotients of a vector bundle on a variety. The enumerative properties to be considered are point counts over finite fields, the Euler characteristics of complex varieties, and the motive in the Grothendieck ring of varieties. I will explain some general connections within and between the two sides, and known results for smooth curves and smooth surfaces. Finally, I will discuss recent results on singular curves. This talk is based on arxiv: 2110.15566 and ongoing joint work with Ruofan Jiang.

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Abstract. One could say that an ulterior motive for this talk is to understand, through an example, how to construct derived moduli spaces out of L-infinity algebras (which are homotopical generalizations of Lie algebras). L-infinity algebras produce derived moduli spaces or stacks, and we will see how one can change (up to homotopy) the underlying L-infinity algebra and compare their corresponding derived spaces/stacks. In this talk, we will use a discretization of matrix valued differential forms to produce an ind-finite (i.e., inductive limit of finite dimensional pieces) model of a derived moduli space whose classical locus is the moduli space of complex vector bundles with flat connections on a (closed, oriented) topological 3-manifold M with a triangulation K_M.

It is known that a natural derived enhancement of the moduli space of flat bundles of a closed, oriented 3-manifold should carry a -1 shifted symplectic structure. We will see how such a -1 shifted symplectic structure can be constructed on our ind-finite model out of L-infinity data. In fact, this -1 shifted symplectic structure can be generated out of the Chern–Simons action functional and some homotopy data associated with it. One can think of this lift of the -1 shifted symplectic structure as an example of -1 shifted prequantization. The existence of this associated data implies that a certain homotopy version of the Chern–Simons functional induces a d-critical structure on the classical moduli space of vector bundles with flat connection on M.

This talk is based on ongoing joint work with Kai Behrend.

Abstract. An involution on a ring A is a self-map of order 2 that is additive and reverses the order of multiplication. I will motivate the work by explaining the classification of involutions of central simple algebras over fields. In this context, the involution acts either trivially or as a Galois action on the centre.

I will explain what an Azumaya algebra A on a scheme X is, where the action on X can be considerbly more complicated, and give a definition of "involution" that fits this context. I will then give a classification of involutions into types, based on the local classification of the involution at the fixed points of the action of the involution on X. I will then present work in progress dedicated to determining which types actually arise. If time permits, I will make some remarks about extending the definition of "Brauer group" to this setting. The entire talk is based on joint work with Uriya First.

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Abstract. Papadakis and Petrotou showed that the anisotropy of a quadratic form implies the Hard Lefschetz theorem for simplicial homology spheres in characteristic 2. These spheres are a special case of the broader family of pseudo-manifolds. In this talk, I will explain how we define a mixed volume on pseudo-manifolds using an integration map, which is linear over a connected sum decomposition. Using this decomposition, we explicitly describe the quadratic form and prove an additional combinatorial conjecture of Papadakis and Petrotou, which turns out to imply all characteristic 2 anisotropy results for pseudo-manifolds and spheres. This is joint work with Kalle Karu.

Abstract. Fano schemes are moduli spaces that parameterize linear spaces contained in an embedded projective variety. In this talk, I investigate the Fano schemes parameterizing linear subspaces of symmetric matrices whose elements are all singular. I discuss their irreducibility and smoothness and characterize when they are connected. As an application, I outline how to use the geometry of these schemes to give alternative proofs for several results on subspaces of singular symmetric matrices.

Abstract. The Weyl discriminant is a polynomial function on a Lie algebra (which generalizes the discriminant of the characteristic polynomial of a matrix), which arises in many questions in representation theory. We compute the log canonical threshold of this polynomial, and from it, make a more precise qualitative statement about local integrability of Fourier transforms of orbital integrals that appear in Harish-Chandra's work and are ubiquitous in harmonic analysis on reductive p-adic groups. This is joint work in progress with Itay Glazer and Yotam Hendel.

Abstract. This is a joint work with Alexander Braverman and David Kazhdan. The goal is to understand Hecke operators on the Schwartz space of half-densities over the stack of G-bundles on a curve over a local non-archimedean field. Assuming that the curve has good reduction, we reduce the problem to studying functions on G-bundles over the reductions of the curve modulo powers of the maximal ideal. We introduce an analogue of the subspace of cuspidal functions and study it using tools from representation theory. It contains the usual space of cuspidal functions for the reduction of the curve over the finite field. We identify some subspaces of cuspidal functions, which show up modulo the square of the maximal ideal, with functions on certain Hitchin fibers.

Abstract. We can study how a given scheme X fits into a family using the tools from the deformation theory. One begins by using infinitesimal methods, studying possible obstructions, and attempting to construct a family called a versal deformation, which collects all possible deformations. If X is a smooth complete toric variety, combinatorial descriptions of the space of first-order deformations and the obstruction to second-order deformation given by the cup product have been studied. In this talk, I will present these descriptions with an example of a smooth complete toric threefold with a quadratic obstruction. In addition, I will discuss my current research, which provides a combinatorial iterative procedure for finding higher-order obstructions.

Abstract. Work of Jacques Dixmier and Colette Moeglin from the 1970's and 1980's established a beautiful characterization in representation theory of Lie algebras; if L is a finite dimensional complex Lie algebra, the annihilators of irreducible representations of the enveloping algebra U(L) are precisely locally closed points in Spec(U(L)), and further, those are the prime ideals P where the center of Frac(U(L)/P) is equal to C.

A similar correspondence has since been studied for many other classes of associative algebras, beyond enveloping algebras, under the name of the Dixmier-Moeglin equivalence. In this talk, based on joint work with Jason Bell and L\'eon Burkhardt, I will introduce and motivate the Dixmier-Moeglin equivalence, discuss in what generality it is expected to hold, and present a recent result, proving the Dixmier-Moeglin equivalence for certain skew polynomial rings of Gelfand-Kirillov (GK) dimension less than 4. Time permitting, I will also present a counterexample in larger GK Dimension.