Curves, Global Charts and Varieties.

This project lies at the intersection of two important fields of mathematics: symplectic geometry and algebraic geometry. Symplectic geometry studies spaces that arise from equations of physical motion, such as a swinging pendulum. Algebraic geometry studies geometric objects that can be described with equations built from addition and multiplication only. The structure of both types of objects can be probed by counting certain special types of curves inside them. An important conjecture in this area tells us that there are not too many of these curves. One aim of this project is to make progress towards this conjecture motivated by earlier work by the investigator describing the underlying structure of these curves. The investigator also aims to relate these counts between two geometric objects that look the same away from a smaller region. Another aim of this project is to connect certain tools in symplectic geometry, called Floer algebras, with families of smaller curves called arcs. This helps us with computations, and it also has very interesting geometric applications. The investigator will be supervising many PhD students on related projects and will mentor undergraduate and master’s students who wish to pursue a PhD in related fields. The broad aim of this project is to use earlier work by the investigator constructing global Kuranishi charts and apply it to various problems in symplectic geometry and algebraic geometry. The investigator is currently using these structural results as inspiration to give explicit upper bounds for Gromov-Witten invariants of all smooth projective varieties. This is the first result of its kind giving general bounds, rather than bounds for a specific family of varieties. In joint work with Ritter, the investigator will also make progress in proving the crepant resolution conjecture using Hamiltonian Floer cohomology, by establishing orbifold versions of quantum cohomology and (Morava) K-theoretic orbifold quantum cohomology. Another goal of this project is to describe Floer cohomology associated to any isolated hypersurface singularity in terms of jets of curves, called arcs. The investigator will also use curve counts to constrain certain symplectic hypersurfaces, develop a spectral Fukaya category as well as construct better behaved global Kuranishi charts. 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.