UIUC-WUSTL   Joint Seminar in Symplectic and Poisson Geometry

Title: Generalized complex groupoids and their actions

Abstract: I will discuss how to use generalized complex (GC) branes to define the notion of generalized complex groupoids and their actions. Our notion of GC groupoids includes non-twisted multiplicative GC structures developed by Jotz, Stiénon and Xu. I will also explain the relation between GC actions and Hamiltonian extended actions developed by Bursztyn, Cavalcanti, Gualtieri, and Lin, Tolman. This work is in progress and joint with Rui L. Fernandes.

Title: Frames for Higher Lie Groupoids

Abstract: Symplectic and Poisson Geometry naturally lead to the consideration of higher analogues of groupoids and algebroids. Higher Lie groupoids are simplicial manifolds satisfying a horn-filling condition, while higher Lie algebroids are differential-graded manifolds concentrated in positive degrees. The Lie theory relating these structures remains elusive. In this talk, based on ongoing work with A. Cabrera, I will use the language of micro-bundles to establish a normal form for higher Lie groupoids, akin to a non-linear version of the Dold-Kan theorem, and discuss its potential implications for describing differentiation.

Title: Non-hamiltonian circle actions with a minimal number of periodic orbits.

Abstract: TBD

Title: Lie Algebroid Dirac Operators

Abstract: Dirac operators have played a vital role in much of the developments in modern differential geometry including the study positive scalar curvature and the Novikov conjecture. In this talk we will introduce Dirac operators within the setting Lie Algebroids and discuss their local index theory when the Lie algebroid is equipped with a suitable integration and trace. This talk is based on joint work with S. Liu, Y. Loizides, and A.R.H.S Sadegh.

Title: Reduction along strong Dirac maps

Abstract:  We develop a general procedure for reduction along strong Dirac maps, which are a broad generalization of Poisson moment maps. The reduction level in this setting is a submanifold of the target, and the symmetries are given by the action of a groupoid. When applied to group-valued moment maps, this framework recovers several constructions from quasi-Poisson geometry and produces new multiplicative versions of many Poisson varieties that are important to geometric representation theory. This is joint work with Maxence Mayrand.