Abstract

In the late 1980’s, Floer developed generalized Morse theories, where the generators of the complex are critical points of action functionals on various moduli spaces, solutions of certain ODEs, and the gradient flow lines are solutions of certain elliptic PDEs. Floer’s breakthrough construction is based on that era’s other breakthrough works done by many people. In this talk, we will learn one of his theories, called Hamiltonian Floer (co)homology and Symplectic (co)homology, which is a Morse (co)homology on the loop space of a symplectic manifold with Hamiltonian function on it. After reviewing definitions and some properties, we introduce criteria for affine varieties to admit uniruled subvarieties of certain dimensions. The measurements are from long exact sequences of versions of symplectic cohomology. We provide applications of the criteria in birational geometry of log pairs in the direction of the Minimal Model Program.