Gromov-Witten Theory, Non-Archimedean Geometry, and Mirror Symmetry

This thesis consists of three projects related to enumerative geometry and mirror symmetry, with an eye towards birational geometry. The first project studies how certain non-archimedean Gromov-Witten invariants of log Calabi-Yau surfaces, called infinitesimal cylinder counts, behave under blowup. We discuss the case of primitive cylinders, and establish a formula that expresses cylinder counts on a blow up of a toric surface in terms of counts in a simpler surface. The proof of the formula uses non-archimedean geometry techniques in an essential way to produce suitable degenerations of the geometric objects enumerated by the counts. The next two projects introduce and study the notion of F-bundle, a structure which can be used to formulate mirror symmetry type results using the language of differential geometry. Our spectral decomposition theorem provides a canonical decomposition for F-bundles satisfying a condition called maximality. We develop the theory of framing, and use it to obtain reconstruction theorems for isomorphisms between maximal F-bundles. As an application of this theory, we prove the uniqueness of certain decompositions of quantum cohomology related to birational geometry, complementing the existence results found in the literature. We also extend the framework of F-bundles to the setting of equivariant mirror symmetry, and prove an unfolding result which can be used to strengthen mirror symmetry statements from the small quantum cohomology to the big quantum cohomology. We apply this unfolding theorem to the equivariant mirror symmetry of general flag varieties, for which only the small quantum cohomology mirror symmetry was known until now.