UBC DG-MP-PDE Seminar: Weiyong He

We will discuss the well-known isotopy problem in symplectic geometry in dimension four. Given a compact symplectic four manifold with a fixed symplectic class, a long standing problem is to ask whether all symplectic forms in the cohomology class are path connected (isotopic to each other). This problem remains completely open and we will discuss two geometric flows related to the problem. A more special setting is to discuss a hypersymplectic structure in dimension four. A hypersymplectic structure is a triple of symplectic forms such that a linear combination of the triple is also a symplectic form. The isotopic problem in this setting asks whether a hypersymplectic structure is cohomologous isotopic to a hyperkahler structure. We will discuss the hypersymplectic flow and some recent progress of this problem.