The project concerns symplectic topology, mirror symmetry, and their relationship. On the topological side, we study symplectic cohomology and its analogue for Lagrangian intersections (wrapped Floer cohomology). Recent advances of Bourgeois-Ekholm-Eliashberg and the author yield powerful computational tools, which we intend to exploit in order to study non-uniqueness questions for symplectic structures on open manifolds. On the mirror symmetry side, we consider decompositions of symplectic manifolds into pairs-of-pants, in the sense of Mikhalkin. One question under consideration is whether the Fukaya category can be reconstructed by gluing together pieces corresponding to each pair-of-pants. In general, this is not true and has to be modified by instanton contributions. However, there are cases where such contributions should be absent, and this would give a new viewpoint on Kontsevich's homological mirror symmetry conjecture.
From a broader perspective, a crucial issue in many current investigations in physics and mathematics is the emergence of the classical notion of space from a quantum description. In the situation under study here, the main feature is that the construction of the classical space involves a series of gradual distortions, which can be determined by complicated but explicit formulae. This is a simplified mathematical model of a general class of physical theories, and not realistic as such. However, studying such models shows us where we are facing conceptual difficulties, which is important in order to develop our understanding further.
