Tuesday, November 9, 2021
11/09/2021 - 4:15pm
The algebraic intersection form makes the homology of a closed surface into a symplectic vector space. A construction of Goldman endows spaces of representations of π_1 into certain Lie groups with a symplectic form remembering the intersection pairing on curves. A regular enough function on a symplectic manifold defines a Hamiltonian vector field—its symplectic gradient. The flow of such a vector field is called a Hamiltonian flow, and defines a family of symmetries of the symplectic structure.
We will consider (geometrically defined) functions on (geometric) loci in character varieties and study their Hamiltonian flows. For most infinite order mapping classes, we find explicit functions on the Teichmüller space whose Hamiltonian flow at time one induces the action of that mapping class. Our main tools are Thurston’s space of measured geodesic laminations and the shearing coordinates of Bonahon and Thurston for hyperbolic metrics.
11/09/2021 - 4:30pm
Shimura varieties are moduli spaces of abelian varieties with extra structures. Over the decades, various mathematicians (e.g. Rapoport, Kottwitz, etc.) have constructed integral models of Shimura varieties. In this talk, I will discuss some motivic aspects of integral models of Hodge type constructed by Kisin (resp. Kisin-Pappas). I will talk about recent work on removing the normalization step in the construction of such integral models, which gives closed embeddings of Hodge type integral models into Siegel integral models. I will also mention an application to toroidal compactifications of such integral models.