Much of the development of symplectic topology is based on studying moduli spaces of pseudo-holomorphic maps. The general scheme is to extract enumeration invariants from these moduli spaces to reveal geometric information. In this talk, I will discuss a recipe for refining the usual counting, which produces integers and bordism classes rather than rational numbers. Applications to symplectic topology and Hamiltonian dynamics will also be covered.

Gravitational instantons are non-compact Calabi--Yau metrics with L^2 bounded curvature and are categorized into six types. I will describe three projects on gravitational instantons including: (a) Fredholm theory and deformation of the ALH* type; (b) non-collapsing degeneration limits of ALH* and ALH types; (c) existence of stable non-holomorphic minimal spheres in some ALF types. These three projects utilize geometric microlocal analysis in different singular settings. Based on works joint with Rafe Mazzeo, Yu-Shen Lin and Sidharth Soundararajan.