# Joint Symplectic Seminar — October 28, 2017

Penn State, Department of Mathematics
114 McAllister Building , University Park, PA

09:30–10:00 Coffee & Snacks
10:00–11:00 Markus Pflaum
11:30–12:30 Liat Kessler
15:00–15:30 Benjamin Hoffman
16:00–17:00 Xiang Tang

Markus Pflaum (University of Colorado Boulder)

Inertia Groupoids and their singularity structure or why we need a concept of stratified groupoids

The inertia space of a compact Lie group action or more generally of a proper Lie groupoid has an interesting singularity structure. Unlike the quotient space of the group action, respectively the groupoid, the inertia space can not be stratified by orbit types, in general. In the talk we explain this phenomenon and provide a stratification and local description of the inertia space. Moreover, we show that that leads naturally to the concept of a stratified groupoid which lies in between the one of a Lie groupoid and the one of a topological groupoid. Finally we show that a de Rham theorem holds for inertia spaces and explain the connection of the inertia space with the non-commutative geometry of the underlying groupoid. The talk is based on joint work with Farsi and Seaton and with Posthuma and Tang.

Liat Kessler (University of Haifa at Oranim)

Extending Homologically trivial symplectic cyclic actions to Hamiltonian circle actions

We ask whether every homologically trivial cyclic action on a symplectic four-manifold extend to a Hamiltonian circle action. By a cyclic action we mean an action of a cyclic group of finite order; it is homologically trivial if it induces the identity map on homology. We assume that the manifold is closed and connected. In the talk, I will give an example of a homologically trivial symplectic cyclic action on a four-manifold that admits Hamiltonian circle actions, and show that it does not extend to a Hamiltonian circle action. I will also discuss symplectic four-manifolds on which every homologically trivial cyclic action extends to a Hamiltonian circle action. I will deduce corollaries on the existence of homologically trivial cyclic actions and on embedding finite-order cyclic subgroups of the group of Hamiltonian symplectomorphisms in circle subgroups. This work applies holomorphic methods to extend combinatorial tools developed for circle actions to study cyclic actions.

Benjamin Hoffman (Cornell University)

Positivity and Potentials in Poisson Geometry

The set of Hermitian matrices has a natural Poisson structure and is a completely integrable system, with action variables given by the Gelfand-Zeitlin cone. The Gelfand-Zeitlin cone also parametrizes crystal bases of irreducible highest weight modules of  $U_q(\mathfrak{gl}_n)$ . Our goal is to construct completely integrable systems on  $\mathfrak{g}^*$ , for compact simple  $\mathfrak{g}$ of any type, using ideas from the study of crystal bases. I will speak about the first step toward that goal. This is joint with A. Alekseev, A. Berenstein, and Y. Li.

Xiang Tang (Washington University in St Louis)

Symplectic 2-groupoids

Motivated by the problem of integrating Courant algebroids, we will introduce the concept of symplectic 2-groupoid. As the first step toward the more general study, we will discuss constant symplectic 2-groupoids and the associated Courant algebroids.