Stability in mirror symmetry

This workshop, sponsored by AIM and the NSF, will bring together mathematicians from a variety of backgrounds to discuss a central problem in mirror symmetry: the existence of special Lagrangian submanifolds, and their holomorphic mirrors, stable vector bundles. These two classes of objects form a set of canonical geometric objects, described by fully nonlinear partial differential equations (PDEs), which play a central role in mirror symmetry. Dating back to work of Thomas-Yau, it has long been conjectured that the existence of solutions to these nonlinear PDEs is equivalent to a purely algebraic notion of stability. This conjectural picture connects diverse fields of mathematics, ranging from stability conditions on derived categories, to symplectic geometry and the Fukaya category, to the study of the Lagrangian mean curvature flow and fully nonlinear systems of PDEs.

Topics: The central topics for the workshop are: The Lagrangian Mean Curvature Flow, singularity analysis, and connections with the Fukaya category. The deformed Hermitian-Yang-Mills equation, infinite dimensional GIT, and fully nonlinear systems. Understanding the Thomas-Yau conjecture for toric Fano manifolds and Landau-Ginzburg models.