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Jake Solomon: Lectures

June 3, 2019
TITLE: Geodesics in the space of positive Lagrangian submanifolds

It is a problem of fundamental importance in symplectic geometry to determine when a Lagrangian submanifold of a Calabi-Yau manifold can be moved by Hamiltonian flow to a special Lagrangian. I will describe an approach to this problem based on the geometry of the space of positive Lagrangians. This space admits a Riemannian metric of non-positive curvature and a convex functional with critical points at special Lagrangians. Existence of geodesics in the space of positive Lagrangians implies uniqueness of special Lagrangians in a Hamiltonian isotopy class as well as rigidity of Lagrangian intersections. The geodesic equation is a degenerate elliptic fully non-linear PDE. I will discuss some results on the existence of solutions to this PDE.