Simons Collaboration Workshop, Jan. 10-13, 2018

Video

Abstract: I will discuss the general mirror symmetry question for
symplectic del Pezzo surfaces in a setup that goes beyond the
Hori-Vafa construction. I will explain how homological mirror symmetry
considerations lead to an explicit description of the mirror map and
will discuss some consequences that can be checked directly. This is a
joint work with Auroux, Katzarkov and Orlov.

Video

Abstract: The derived Knorrer periodicity compares the derived category of coherent sheaves on a projective hypersurface with that of matrix factorizations of its defining equation. I’d like to talk about a parallel development in curve counting, including Chang-Li’s p-field invariant, Chang-Li-Li’s algebraic theory of (narrow) FJRW invariant and Polishchuk-Vaintrob’s cohomological field theory, from the viewpoint of cosection localization.

Video

Abstract:  After explaining some results in Lagrangian Floer theory in the presence of an anti-symplectic involution, I will present a definition of twisted sectors, which is suitable for  Lagrangian Floer theory in orbifold setting.  The first part is based on joint works with K. Fukaya, Y.-G. Oh and H. Ohta and the second is based on a joint work (in progress) with B. Chen and B.-L. Wang.

Video

Abstract A fundamental result for K3 surfaces is the Kulikov-Persson-Pinkham theorem on degenerations of K3 surfaces. In this talk, I will explore higher di mensional analogues of it and potential applications. Specifically, as a consequence of the minimal model program, Fujino has a obtained a weak analogue of the KPP Theorem for K-trivial varieties. I will then discuss some relationships between the dual complex of the central fiber and the monodromy of the degenerations. I will then explain some consequences of this for Hyperkaehler manifolds and Calabi-Yau 3-folds.

Video

Video

Abstract: Given a Lagrangian submanifold L, we can consider a formal deformation theory of $L$ which is developed by Fukaya-Oh-Ohta-Ono. This provides a local mirror (with respect to L), given by the Lagrangian Floer potential function on the formal Maurer-Cartan space of L. Then, we can canonically construct a localized mirror functor  from Fukaya category to the matrix factorization category. Given two different Lagrangian submanifolds, we explain how to glue these local mirrors to obtain a global mirror model, and also how to glue their localized mirror functors to obtain a global version of homological mirror functor.

This is a joint work in progress with Hansol Hong and Siu-Cheong Lau.

Video

Video

Abstract: We find the Floer-theoretical gluing between local moduli of Lagrangian immersions, and use it to study wall-crossing for local Calabi-Yau manifolds.  It is a joint work with Cho and Hong.  In a joint work with Hong and Kim, we apply the technique to recover the Lie theoretical mirror of Gr(2,n).

Video

Notes

Abstract: A symmetric quiver with g nodes is described by a symmetric adjacency matrix of size g.  The same data defines a “framing” of a certain genus-g Legendrian surface in the five-sphere, and the invariants of the quiver conjecturally relate to the open Gromov-Witten (GW) invariants of a non-exact Lagrangian filling of the surface.  (Physically, both data count the same BPS states but from different perspectives.)  Further, cluster theory can be exploited to conjecturally obtain all open GW invariants of Lagrangian fillings of a wider class of Legendrian surfaces described by cubic planar graphs.

In this talk, I will describe these observations, which build on prior work of others and are​ explored in joint works with David Treumann and Linhui Shen.

Video

Abstract: We discuss some physical and geometric aspects of Kodaira-Spencer gravity (BCOV theory) on Calabi-Yau geometry and explain how quantum master equation leads to integrable hierarchies

Video

The construction of cohomology classes in the compactified moduli spaces of curves based on the quantum master equation on cyclic cochains will be reviewed. For the simplest category consisting of one object with only the identity morphism it produces the generating function for products of the psi-classes. The talk is based on the speaker’s works “Modular operads and Batalin-Vilkovisky geometry” (MPIM Bonn preprint 2006-48 (04/2006)) and “Noncommutative Batalin–Vilkovisky geometry and matrix integrals” (preprint Hal-00102085 (09/2006)).

Video

Abstract: I will talk about a mirror theorem for minuscule flag
varieties.  The mirror theorem asserts that the Dubrovin quantum connection of a minuscule flag variety coincides with the character D-module of a geometric crystal (serving as a Landau-Ginzburg model). The idea of the proof is to recognize the former as a Galois object and the latter as an automorphic object, and apply the (ramified) geometric Langlands correspondence. Some surprising connections to Kloosterman sums and sheaves will appear.This is joint work with Nicolas Templier.

Abstract:  The Satake-Baily-Borel (SBB) compactification is an projective algebraic completion of a locally Hermitian symmetric space. This construction, along with Borel’s Extension Theorem, provides the conduit to apply Hodge theory to study the moduli spaces (and their compactifications) of principally polarized abelian varieties and K3 surfaces.

Most period domains are not Hermitian, and so one would like to generalize SBB in the hopes of similarly applying Hodge theory to study the moduli spaces (and their compactifications) of more general classes of algebraic varieties.  In this talk I will present one such generalization.  This work joint work with M. Green, P. Griffiths and R. Laza.

Video

Abstract: I shall explain a program to relate the arithmetic of Calabi-Yau hypersurfaces in toric varieties or flag varieties, to their period integrals at the large complex structure limit. In particular, we prove a recent conjecture of Vlasenko regarding higher Hasse-Witt matrices. This work follows Katz’s description of Frobenius action in terms of local expansions. It is joint work with Huang, Lian and Yau.

Video

Abstract: We give a description of the moduli space of K3 surfaces polarized
by the even unimodular lattice of signature (1,9)
as the moduli space of A-infinity structures on a fixed graded ring,
and discuss its application to homological mirror symmetry.
This is a joint work with Yanki Lekili.