Title: Modularity of mirror families of log Calabi--Yau surfaces.

Abstract: In 'Mirror symmetry for log Calabi--Yau surfaces I', given a smooth log Calabi--Yau surface pair (Y,D), Gross--Hacking--Keel constructed its mirror family as the spectrum of an explicit algebra whose structure coefficients are determined by the enumerative geometry of (Y,D). As a follow-up of the work of Gross--Hacking--Keel, when (Y,D) is positive, we prove the modularity of the mirror family as the universal family of log Calabi-Yau surface pairs deformation equivalent to (Y,D) with at worst du Val singularities. As a corollary, we show that the ring of regular functions of a smooth affine log Calabi--Yau surface has a canonical basis of theta functions. The key step towards the proof of the main theorem is the application of the tropical construction of singular cycles and explicit formulas of period integrals given in the work of Helge--Siebert. This is joint work with Jonathan Lai.

Title: Applications of Higher Determinant Map

Abstract: In this talk I will explain the construction of a determinant map for Tate objects and two applications: (i) to construct central extensions of iterated loop groups and (ii) to produce a determinant theory on certain ind-schemes. For that I will introduce some aspects of the theory of Tate objects in a couple of contexts.

Title. Regularized integrals on Riemann surfaces and correlations functions in 2d chiral CFTs

Abstract. I will report a recent approach of regularizing divergent integrals on configuration spaces of Riemann surfaces, introduced by Si Li and myself in arXiv:2008.07503, with an emphasis on genus one cases where modular forms arise naturally. I will then talk about some applications in studying correlation functions in 2d chiral CFTs, holomorphic anomaly equations, etc. If time permits, I will also mention a more algebraic formulation of this notion of regularized integrals in terms of mixed Hodge structures.

Title: 2-categorical 3d mirror symmetry

Abstract: It is by now well-known that mirror symmetry may be expressed as an equivalence between categories associated to dual Kahler manifolds. Following a proposal of Teleman, we inaugurate a program to understand 3d mirror symmetry as an equivalence between 2-categories associated to dual holomorphic symplectic stacks. We consider here the abelian case, where our theorem expresses the 2-category of spherical functors as a 2-category of coherent sheaves of categories. Applications include categorifications of hypertoric category O and of many related constructions in representation theory. This is joint work with Justin Hilburn and Aaron Mazel-Gee.

CMSA Algebraic Geometry in String Theory 03.15.2022.pdf

Title:  Virtual localization for Artin stacks

Abstract:  This is a report about work in progress with: Adeel Khan, Aloysha Latyntsev, Hyeonjun Park and Charanya Ravi. We will describe a virtual Atiyah-Bott formula for Artin stacks.  In the Deligne-Mumford case our methods allow us to remove the global resolution hypothesis for the virtual normal bundle.