Clark-Atlanta University

Reeder-Yu introduced certain low positive depth supercuspidal representations of p-adic groups called epipelagic representations. These representations generalize the simple supercuspidal representations of Gross-Reeder, which have the lowest possible depth. Epipelagic representations also arise in recent work on the Langlands correspondence; for example, simple supercuspidals appear in the automorphic data corresponding to the Kloosterman l-adic sheaf. In this talk, we report on a project whose goal is the construction of "mesopelagic representation of Iwahori type", higher depth analogues of simple supercuspidal  representations.

Symplectic geometry is a relatively new branch of geometry. However, a string theory-inspired duality known as "mirror symmetry" reveals more about symplectic geometry from its mirror counterparts in complex geometry. M. Kontsevich conjectured an algebraic version of mirror symmetry called "homological mirror symmetry" (HMS) in his 1994 ICM address. HMS results were then proved for symplectic mirrors to Calabi-Yau and Fano manifolds. Those mirror to general type manifolds have been studied in more recent years, including my research. In this talk, we will introduce HMS through the example of the 2-torus T^2. We will then outline how it relates to HMS for a hypersurface of a 4-torus T^4, in joint work with Haniya Azam, Heather Lee, and Chiu-Chu Melissa Liu. From there, we generalize to hypersurfaces of higher dimensional tori, otherwise known as "theta divisors." This is also joint with Azam, Lee, and Liu.

Parity sheaves are some constructible complexes defined on some stratified space where the strata satisfies some parity vanishing conditions. They are introduced by Carl Mautner, Daniel Juteau and Geordie Williamson in 2014. In characteristic 0 they coincide with the intersection cohomology complexes but in positive characteristic they are new and important objects. On flag variety they can be used as the "p-canonical basis" for Hecke algebras. It was noticed that on finite flag variety as well as affine Grassmannian the parity sheaves correspond to the tilting sheaves. One expectation was to find similar relation on nilpotent cone, which Achar and Mautner started exploring in 2012. Another expectation was to understand the modular Springer correspondence in terms of parity sheaves. To study that one imporant conjecture has to be solved was made by Mautner. It says the parabolic induction functor defined on the nilpotent cones must preserves parity complexes. We break down this conjecture in two pieces and first try to prove that the parabolic) induction functor sends parity sheaves associated to a cuspidal pair to a parity complex for classical groups. This is a ongoing project with Pramod N. Achar.

There is a spectral sequence converging to symplectic cohomology of an affine variety whose E1-page consists of symplectic cohomology of the complement of a hypersurface in the affine variety. We will talk about a contact (as well as S1-equivariant) version of the spectral sequence and provide applications including the invariants of fibered knots and the invariants of isolated hypersurface singularities. The data of each page of the spectral sequence consist of the invariants. If time permits, we will talk about properties of the relative contact homology and relations to other Floer-theoretic invariants.

Under the Gordon-Stafford functor, every filtered modules over the type A rational Cherednik algebra corresponds to an equivariant coherent sheaf on the Hilbert scheme of points on the plane. As conjectured by Gorsky-Negut, under the decategorification of this functor, the images of the finite-dimensional representations are closely related to the knot superpolynomials. I'll talk about work in progress towards this conjecture using mirabolic D-modules equipped with the Hodge filtration.

Donaldson and Uhlenbeck-Yau established the classical result that on a compact Kahler manifold, an irreducible holomorphic vector bundle admits a Hermitian metric solving the Hermitian-Yang-Mills equation if and only if the vector bundle is Mumford-Takemoto stable. Motivated by the characterization of supersymmetric B-branes in string theory and mirror symmetry, Collins-Yau asked if a line bundle admits a solution of the deformed Hermitian-Yang-Mills (dHYM) equation is equivalent to it is stable with respect to certain Bridgeland stability conditions. In this talk, we will discuss a partial answer to this question for a set of line bundles on a Weierstrass elliptic K3 surface. This is joint work with Tristan Collins, Jason Lo, and Shing-Tung Yau.

We classify all additive invariants of open Petri nets: these are N-valued invariants which are additive with respect to sequential and parallel composition of open Petri nets. In particular, we prove two classification theorems: one for open Petri nets and one for monically open Petri nets (i.e., open Petri nets whose interfaces are specified by monic maps). Our results can be summarized as follows. The additive invariants of open Petri nets are completely determined by their values on a particular class of single-transition Petri nets. However, for monically open Petri nets, the additive invariants are determined by their values on transitionless Petri nets and all single-transition Petri nets. Our results confirm a conjecture of John Baez (stated during the AMS' 2022 Mathematical Research Communities workshop).