If we have are two commuting symplectomorphisms of a symplectic manifold, each one of them induces an automorphism of Floer cohomology of the other one. I will show that the supertraces of these two automorphisms are equal, developing a suggestion by Paul Seidel. As a particular case, I will explain that if a symplectomorphism f commutes with a symplectic involution, the dimension of HF(f) is bounded below by a topological quantity: the Lefschetz number of the restriction of f to the fixed locus of the involution.
In recent years, low dimensional topologists have become interested in the study of "generic" smooth maps to surfaces. The approach is similar to Morse theory, only with two dimensional target. In this talk, I will discuss a specific problem in the 4-dimensional context which is analogous to the (uniqueness of) cancellation of critical points of Morse functions. I will also indicate applications to certain pictorial descritpions of 4-manifolds in terms of curve configurations on surfaces. This is joint work with Kenta Hayano.