Floer Theory, Arc Spaces, and Singularities

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

This project is concerned with the development of certain tools for use in two areas of mathematics: algebraic geometry and symplectic geometry. Algebraic geometry studies mathematical objects called varieties that are solutions of equations built from addition and multiplication. Symplectic geometry is the natural geometry that emerges when one studies Hamilton's equations, which are equations describing the motion of classical physical systems. A very important collection of tools, called Floer theory, has been utilized to solve many problems in both of the subject areas above. However, these tools are hard to use since the computations involved can be quite difficult. A part of this project involves finding better computational techniques via an object called the arc space. Another part of this project is concerned with finding more refined ways of counting curves inside varieties as well as establishing efficient foundations for such counts using ideas from symplectic geometry. Additionally, the PI will help mentor a summer workshop for graduate students as well as engage with the Stony Brook math summer camp for high school students. In his role as the graduate director at Stony Brook, the PI will be involved in many student-centered activities. The broad aim of this project is to better understand Floer theory and its relationship with symplectic and algebraic geometry. To this end, the PI will utilize arc spaces to compute various Floer algebras. The arc space of a variety is the space of holomorphic maps from a disk to that variety. The PI will show that Floer cohomology of iterates of the monodromy of an isolated hypersurface singularity is compactly supported cohomology of jets of certain arcs. Another project is to compute the full contact homology of any isolated singularity in terms of its arc space. The PI aims to prove that there is a spectral sequence computing symplectic cohomology of affine varieties whose pages are also compactly supported cohomology groups of jets of certain arcs at infinity. The PI, along with collaborators Abouzaid and Smith, has a project defining Morava K-theoretic Gromov-Witten invariants in a more efficient way as well as over some other generalized cohomology theories. This project will also use the new idea of constructing a global Kuranishi chart from an enlarged moduli space of curves. Finally in joint work with Ritter, the PI will investigate the crepant resolution conjecture using Hamiltonian Floer cohomology. 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.