Relative shifted symplectic structures

I will discuss the notion of relative shifted symplectic structures on sheaves of derived stacks over locally compact Hausdorff topological spaces. I will describe a general pushforward construction of relative symplectic forms and in the constructible case will
explain explicit techniques for computing such forms. As applications I will discuss a relative lift of recent results of Shende--Takeda on topological Fukaya categories, and a universal construction of symplectic structures on derived irregular character varieties. This is a joint work with Dima Arinkin and Bertrand Toen.