Abstracts Spring 2024

Speaker:  Mina Aganagic

Title:  Homological mirror symmetry and link  invariants

Abstract: There is a new family of homological mirror pairs, rooted in representation theory of simple Lie (super)algebras. A virtue of this family is that homological mirror symmetry can be understood explicitly.  One application of the theory is to categorification of quantum link invariants.

Speaker:  Hülya Argüz

Title:  The KSBA moduli space of log Calabi-Yau surfaces

Abstract: The KSBA moduli space, introduced by Kollár-Shepherd-Barron, and Alexeev, is a natural generalization of "the moduli space of stable curves" to higher dimensions. It parametrizes stable pairs (X,B), where X  is a projective algebraic variety satisfying certain conditions and B is a divisor such that K_X+B is ample. This moduli space is described concretely only in a handful of situations: for instance, if X is a toric variety and B=D+\epsilon C, where D is the toric boundary divisor and C is an ample divisor, it is shown by Alexeev that the KSBA moduli space is a toric variety. Generally, for a log Calabi-Yau variety (X,D) consisting of a projective variety X and an anticanonical divisor D, with B=D+\epsilon C where C is an ample divisor, it was conjectured by Hacking-Keel-Yu that  the KSBA moduli space is still toric (up to passing to a finite cover). In joint work with Alexeev and Bousseau, we prove this conjecture for all log Calabi-Yau surfaces. This uses tools from the minimal model program, log  smooth deformation theory and mirror symmetry.

Speaker:  Alexander Polishchuk

Title: Moduli of supercurves and the superstring measure

Abstract: I’ll discuss some aspects of the geometry of supercurves related to the study of the superstring measure. These will include: stable supercurves, analog of Mumford isomorphism and superperiods.

Speaker:  Constantin Teleman

Title: Towards Coulomb branches for disconnected groups

Abstract:  Coulomb branches for (the "N=4, 3D SUSY gauge theory” of) compact connected Lie groups control the topological boundary conditions of the theory, namely equivariant Gromov-Witten theory. After a quick review of the base (connected) case I will explain the problem arising, and proposed solution in special cases, when disconnected groups are introduced. This involves the theory of matrix factorization, the (partly conjectural) Kapustin-Rozanski-Saulina 2-category, and the notion of ‘condensation of topological defects’. The examples I will discuss are joint work in progress with John Nolan and Colleen Delaney.