The Fukaya category of a log symplectic surface

Log symplectic structures constitute an important class of Poisson structures with well-behaved degeneracies: Since they are symplectic almost everywhere, many powerful techniques from symplectic geometry apply, but the degeneracy loci introduce local invariants.

As a first step in the effort to extend Floer theory, Fukaya categories and mirror symmetry to Poisson structures with degeneracies, I will present the construction of Floer cohomology and a Fukaya category for log symplectic structures on oriented surfaces, with a focus on the additional features of the theory arising from the degeneracy locus, and how this theory detects and is modified by the degeneracy loci of the structure.