UC Berkeley

Polyfold theory, as developed by Hofer, Wysocki, and Zehnder, is a relatively new approach to resolving transversality issues that arise in the study of J-holomorphic curves in symplectic geometry. This approach has recently led to a well-defined Gromov--Witten invariant for J-holomorphic curves of arbitrary genus, and for all closed symplectic manifolds. In this thesis we prove the Effective, Grading, Homology, Zero, Symmetry, Fundamental Class, and Divisor Gromov--Witten axioms for $J$-holomorphic curves of arbitrary genus, and for all closed symplectic manifolds.