Investigations in Calabi-Yau modularity and mirror symmetry

This is the author’s PhD thesis. Two main sections address various aspects of mirror symmetry for compact Calabi-Yau threefolds and the roles that classically modular varieties play in string theory compactifications. The main results include a study, and finding an application to the higher genus problem, of infinite Coxeter symmetries in the sets of Gopakumar-Vafa invariants; provision of a new class of solutions to the supersymmetric flux vacuum equations which have elsewhere been conjectured to give weight-two modular manifolds; provision of two new conjectural examples of weight-four modular varieties (rank-two attractors); and discussion of a set of numerical relations between infinite sums of Gromov-Witten invariants and critical L-values.