This project centers on a field called "symplectic topology", a part of geometry with roots in the development of celestial mechanics, whose distinctive mathematical features emerged in the 1980s. This field is today the subject of vigorous research activity. The PI, along with his graduate students and collaborators, will study connections between symplectic topology and two other parts of geometry, each superficially quite different. The first connection is to algebraic geometry, and is through the phenomenon known by the metaphorical name of "mirror symmetry". In mirror symmetry, objects from symplectic topology reappear in transformed form into algebraic form in a looking-glass world. The inter-relations between the symplectic objects and those between the algebraic objects match precisely, in an astounding way. Mirror symmetry was proposed around 1990 by physicists working in string theory, and for years, mathematicians could verify it in examples but not explain it. The PI contends that a quarter-century after its inception, the time has come to prove basic theorems conceptualizing how mirror symmetry works. The second connection is to geometry in dimensions 3 and 4. Though the two parts of the project touch on different parts of mathematics, they share common technical tools, the theories of "pseudo-holomorphic curves" and "Floer cohomology". A significant part of the proposal is to support the training of graduate students working on the two aspects of the project.
The two strands of the research proposed in this project are both based on symplectic Floer cohomology, a tool in symplectic topology that is proving as incisive and adaptable as singular cohomology is in algebraic topology. One strand explores structural aspects of the connection between symplectic topology and algebraic geometry known as mirror symmetry, in the setting of Calabi-Yau (CY) manifolds. Homological mirror symmetry, and the Fukaya category of a CY manifold, take center-stage. Key to the proposal is the investigation of logical relationships between different formulations of mirror symmetry: constructions of mirror pairs; homological mirror symmetry; Hodge-theoretic mirror symmetry; and enumeration of holomorphic curves. The other strand arises from the relationship between the symplectic and gauge-theoretic versions of Floer cohomology. The proposal is to develop a new Floer cohomology theory for 3-manifolds using the same mechanism as the hugely productive Heegaard Floer homology of Ozsvath-Szabo, but working not in a symmetric product of the Heegaard surface (made a complex curve), as one does in Heegaard Floer homology, but rather in a space of "stable pairs" on the Heegaard surface, consisting of a rank 2 holomorphic vector bundle together with a holomorphic section thereof. This theory is likely to have close relations both to Heegaard Floer homology itself, and to instanton Floer theory, and may illuminate the relations between those theories.
