Floer Cohomology and Birational Geometry

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Broadly speaking, this project is about the relationship between two subjects: algebraic geometry and symplectic geometry. Algebraic geometry studies geometric objects called varieties, created by equations that are built from addition and multiplication. Symplectic geometry involves geometric objects related to classical mechanics. We are interested in `topological' properties of some of these varieties such as how many `holes' they have and `enumerative' properties such as the number of curves they contain. The PI is interested in pairs of varieties which are birational to each other, which means that they become identical after removing smaller varieties. We wish to know what topological or enumerative properties they have in common. One of the main aims of this project is to see how tools from symplectic geometry can be used to investigate such issues. The PI believes these ideas are new and can be used to explore other areas of algebraic and symplectic geometry using this different perspective. The PI will help out with a program involving high school students called Seawolf math, and will also help out with a math summer camp for high school students at Stony Brook. The PI will also help out in workshops designed for graduate students learning closely related fields of study.The primary aim of this project is to understand the relationship between birational geometry and certain Floer theoretic invariants. These invariants include symplectic/contact cohomology, Floer cohomology of a symplectomorphism and the Fukaya category. The PI will use these Floer invariants to prove certain statements related to birational geometry. One of the main goals of this NSF funded project is to give a completely new approach to the crepant resolution conjecture using an extended version of symplectic cohomology. The PI will prove a weak version of this conjecture using these methods. The PI believes these methods point to certain generalizations of this conjecture. The PI will study terminal 3-fold singularities and also Newton non-degenerate singularities using Floer theory. Studying such singularities could lead to new insights just as studying quotient singularities led the PI to study the crepant resolution conjecture.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.