Joint Topology Seminar

Abstract. I will explain how the pseudo-holomorphic quilts of Wehrheim-Woodward can be used to understand the Fukaya category of a product of symplectic manifolds (under the usual assumptions which prevent bubbling). Applying this to the product of the 2-torus with itself, we obtain a correspondence between Lagrangians in the 4-torus of vanishing Maslov index and coherent sheaves on a certain abelian variety. In the case of genus 2 surfaces, we use some sheaf theory on the complex side to obtain numerical restrictions, in the spirit of the Arnol'd conjecture, on the number of intersections of such a surface with the fibres of a Lagrangians torus fibration. This is work in progress, joint with Ivan Smith.