Relative symplectic cohomology and quantitative deformation theory

Abstract: Consider a Liouville domain D embedded in a closed symplectic manifold M. To D one can associate two types of Floer theoretic invariants: intrinsic ones like the wrapped Fukaya category which depend on D only, and relative ones which involve both D and M. It is often the case that the intrinsic invariant is amenable to computation. On the other hand, the relative invariants are important, at least in SYZ mirror symmetry, as one can reconstruct the global Floer theory by a local to global principle. Thus it is a fundamental question if the relative invariants can be understood as  a deformation of the intrinsic invariant. It turns out the question need to be approached quantitatively. By shrinking the Liouville domain, the answer is often positive. This circle of ideas is at the heart of a program joint with Mohammed Abouzaid and Umut Varolgunes for a general approach to homological mirror symmetry. I will discuss a work in progress on the application of this circle of ideas to the reconstruction problem in mirror symmetry via relative symplectic cohomology.