Through the framework of tropical geometry, we analyze the critical behavior of the Landau-Ginzburg mirror of toric/non-toric blowups of possibly non-Fano toric surfaces. After a brief review on the mirror construction for log Calabi-Yau surfaces, we introduce a method for identifying the precise location of critical points of the superpotential (or equivalently, non-displaceable fibers). We further show their non-degeneracy for generic parameters, proving closed and open string mirror symmetry.

In this talk, I’ll explain a definition of local curve-counting theory, specifically local Gromov-Witten (GW) theory, of some simple normal crossing surfaces (namely, shrinkable surfaces), generalizing local mirror symmetry of a smooth del Pezzo surface. Such generalization is motivated by physics of M-theory on a Calabi-Yau 3-fold containing a simple normal crossing surface and also by 3-fold canonical singularities. I’ll first review the background, describe the embeddability problem of shrinkable surfaces, and then discuss how we can apply GW theory to singular surfaces. No knowledge on physics is assumed. This talk is partly based on joint work with S. Katz.

On toric Fano fibrations; 

We discuss the classification of germs of toric Fano fibrations, extending work of A. Borisov in the case of Q-factorial toric singularities. As an application, we verify in the toric case a conjecture of V. Shokurov on the existence of complements with bounded index and prescribed singularities.

On Seshadri constants; 

The Seshadri constant of a polarized variety (X,L) at a point x measures how positive is the polarization L at x. If x is very general, the Seshadri constant does not depend on x, and captures global information on X. Inspired by ideas from the Geometry of Numbers, we introduce in this talk successive Seshadri minima, such that the first one is the Seshadri constant at a point, and the last one is the width of the polarization at the point. Assuming the point is very general, we obtain two results: a)  the product of the successive Seshadri minima is proportional to the volume of the polarization; b) if X is toric, the i-th successive Seshadri constant is proportional to the i-th successive minima of a suitable 0-symmetric convex body. Based on joint work with Atsushi Ito.

In this talk, I will discuss my recent research that establishes the minimal model program for Q-factorial foliated dlt algebraically integrable foliations and log canonical generalized pairs. This is joint work with G. Chen, J. Han, and L. Xie.

 In this talk, we will introduce the concept of  K-semistable domains and compute them for various examples. We will also present some important properties of these domains and reveal the connection with wall crossing property for K-stability.

Spectral networks and non-abelianization were introduced by Gaiotto-Moore-Neitzke and they have many applications in mathematics and physics. In a recent work by Nho, he proved that the non-abelianization of an almost flat local system over the spectral curve of a meromorphic quadratic differential is actually the same as the family Floer. Based on the mirror symmetry philosophy, it is then natural to ask how holomorphic vector bundles arise from spectral networks and non-abelianization. In this paper, we construct toric vector bundles on toric surfaces via spectral networks and non-abelianization.

For a compact Lie group G, its massless Coulomb branch algebra is the G-equivariant Borel-Moore homology of its based loop space. This algebra is the same as the algebra of regular functions on the BFM space. In this lecture series, we will explain how this algebra acts on the equivariant symplectic cohomology of Hamiltonian G-manifolds when the symplectic manifolds are open and convex. This is joint work with Eduardo González and Dan Pomerleano.