I will discuss the notion of shifted symplectic structures along the stalks of constructible sheaves of derived stacks on stratified spaces. I will describe a general pushforward theorem producing relative symplectic forms and will explain explicit techniques for computing such forms. As an application I will describe a universal construction of Poisson structures on derived moduli of local systems on smooth varieties and will explain how symplectic leaves arise from fixing irregular types and local formal monodromies at infinity. This is a joint work with Dima Arinkin and Bertrand Toën.
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