Invariants from Equivariant Transversality in Symplectic Topology and Some Results on the Rouquier Dimension of Wrapped Fukaya Categories

Abstract

In this thesis, we construct several invariants in low-dimensional topology and symplectic topology, including a symplectic definition of generalized Casson invariants, an extension of Taubes' Gromov invariants to symplectic Calabi-Yau threefolds, and integer-valued genus 0 Gromov--Witten type invariants for general compact symplectic manifolds, based on solutions to various equivariant transversality problems in symplectic topology. In a different direction, we also study the Rouquier dimension of wrapped Fukaya categories of Liouville manifolds to obtain applications in algebraic geometry and symplectic geometry.