What analysis, combinatorics, and quilted spheres can tell us about symplectic geometry

Over the past three decades, symplectic geometers have constructed powerful curve-counting invariants of symplectic manifolds. The chief example is the Fukaya category, which revealed a deep connection with algebraic geometry via Kontsevich's Homological Mirror Symmetry conjecture. In this talk, I will describe my program to relate the Fukaya categories of different symplectic manifolds. The key objects are "witch balls" (coupled systems of PDEs whose domain is the Riemann sphere decorated with circles and points), as well as the configuration spaces of these domains, which are posets called "2-associahedra". I will describe applications to symplectic geometry, algebraic geometry, and topology.