The aim of the project is to investigate holomorphic curve invariants of symplectic manifolds, in particular the Gromov-Witten invariants, symplectic cohomology, and the Fukaya category. These invariants are of central importance in symplectic topology, and also have wider applications to low-dimensional topology (via Heegaard-Floer-type invariants), algebraic geometry (via mirror symmetry), and string theory. Following the guiding philosophy of Kontsevich's homological mirror symmetry conjecture, certain relationships between these invariants are expected, and the main aim of the project is to establish these relationships, while staying firmly rooted in explicit examples and computations.
Broadly, the project investigates symplectic manifolds, which are spaces that locally look like the phase spaces of classical mechanics. Given a symplectic manifold, one can cook up a variety of interesting structures (called 'invariants'), bearing a strong resemblance to ideas coming from string theory; in fact, in some sense these invariants describe what physics would look like for strings living on a space with a symplectic structure. Investigations by physicists have predicted surprising connections between the physics experienced by particles living on spaces with a symplectic structure, and that experienced by particles living on a completely different kind of space: a space with an algebraic structure. This deep connection between two completely different kinds of spaces is very surprising, and not at all well-understood. Even so, it is very useful for mathematicians who study invariants of symplectic manifolds, because it gives predictions for what they should be in terms of algebraic geometry, which one can then set about verifying mathematically. The project aims to follow this strategy to investigate certain kinds of symplectic invariants, which have a variety of useful applications in different areas of mathematics and physics.
