Structural Results in Floer Theory and Mirror Symmetry

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Symplectic geometry is an area broadly concerned with the global shape of physical systems, such as those tracking the possible positions and momenta of particles in a constrained configuration. Insights from mathematical physics have led to powerful tools, collectively called Floer theory, for extracting properties of such systems (for instance the number of periodic orbits associated to a system with given kinetic and potential energy); unfortunately the appearance of difficult differential equations makes these tools challenging to apply. This project seeks to simplify the study of Floer theory of a large class of spaces by developing systematic rules for computation (via e.g., cut and paste) and by establishing new relationships between different types of Floer theory. Using these, the project aims to solve open problems about the structure of Floer theory and find new applications to mirror symmetry (a remarkable geometric duality first arising in string theory) and the study of singularities (abrupt changes) in symplectic geometry. The PI will also train and encourage mathematics students through workshops, new course and seminar content, and judging of K-12 science fairs.This project aims to develop new structural results in Floer theory and mirror symmetry using input from (and with applications to) the study of singularities of various forms in symplectic geometry. In one direction, the project aims to show that wrapped Fukaya categories satisfy expected van-Kampen style locality properties, and deduce as a consequence axiomatic and sheaf-theoretic characterizions of Fukaya categories of Stein manifolds (using in part the singular structure of their Lagrangian skeleta). In a related direction, the project aims to geometrically calculate the Hochschild invariants of Fukaya categories associated to certain Landau-Ginzburg models (i.e., holomorphic functions), and deduce new applications to the study of singularities of such functions. The third and final direction is to understand the effect of hidden singularities (in the sense of having a singular mirror) on Fukaya categories, with computational and structural consequences.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.