Refined counts of holomorphic curves and applications

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.