Homological and enumerative intrinsic mirror symmetry and related topics

Mirror symmetry is a deep and multi-faceted phenomenon observed by theoretical particle physicists in the context of string theory. It involves connections between different mathematical structures, such as computing volumes versus counting objects. Investigations of the phenomenon have produced a long list of fundamentally new techniques and uncovered unexpected links between hitherto largely unrelated branches of mathematics and physics. Research under this award aims to prove a part of mirror symmetry in the original setup of Calabi-Yau varieties, a distinguished class of spaces in both algebraic and differential geometry. In this setting, pairs of mirror partners have been constructed in rather complete generality, and the remaining task is to prove the two main claimed relations between the geometries: the mentioned enumerative prediction and homological mirror symmetry. This project will provide research training opportunities for students. Concerning homological mirror symmetry, the goal is to relate the intrinsic mirror ring for a degeneration of Calabi-Yau varieties constructed by the PI and Mark Gross to the symplectic monodromy ring for the degeneration, defined by Floer theory. Another related topic is the generalization of the intrinsic mirror ring to a relative quantum cohomology ring for a pair of a normal crossings divisor in a smooth projective variety. The task for enumerative mirror symmetry is to compute enumerative period integrals from the wall structure appearing in the mirror construction. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.