Time: June 22, 2018
Venue: Room LA 106
Speaker:
Reyer Sjamaar (Cornell University)
Hao Ding (Southwest Jiaotong University)
Farkhod Eshmatov (Sichuan University)
Yi Lin (Georgia Southern University)
Cheng-Yong Du (Sichuan Normal University)
Organized by Hongliang Shao , [email protected]
Xiangdong Yang , [email protected]
Hengyu Zhou , [email protected]
Supported by National Natural Science Foundation of China;
College of Mathematics and Statistics
Convexity properties of presymplectic Hamiltonian actions
(9:30-10:20)
Abstract: A celebrated theorem in symplectic geometry, due to Atiyah, Guillemin-Sternberg, and Kirwan, states that the image of the moment map of a Hamiltonian compact Lie group action is a convex polytope. I will present a generalization of this result, obtained in joint work with Yi Lin, to compact Lie group actions on presymplectic manifolds.
; Some properties of quasi-symplectic groupoid reduction
; (10:30-11:20)
Abstract: Quasi-symplectic groupoids and their moment map theory are introduced by Ping Xu to unify into a single framework various reduction theories in the literature. In this talk, I will first consider Hamiltonian dynamics on Hamiltonian $\mathcal{G}$-spaces and on their reduced spaces, and then I will prove a commuting reduction theorem which is a quasi-symplectic groupoid version of Hamiltonian reduction by stages.
TBA (14:30-15:20)
Localization formula for Riemannian foliations
(15:30-16:20)
Abstract: A Riemannian foliation is a foliation on a smooth manifold that comes equipped with a transverse Riemannian metric: a fiberwise Riemannian metric $g$ on the normal bundle of the foliation, such that for any vector field $X$ tangent to the leaves, the Lie derivative $L(X)g=0$. ;In this talk, we would discuss the notion of transverse Lie algebra actions on Riemannian foliations, which is used as a model for Lie algebra actions on the leave space of a foliation. Using an equivariant version of the basic cohomology theory on Riemannian foliations, we explain that when the action preserves the transverse Riemannian metric, there is a foliated version of the classical Borel-Atiyah-Segal localization theorem. Using the transverse integration theory for basic forms on Riemannian foliations, we would also explain how to establish a foliated version of the Atiyah-Bott-Berline-Vergne integration formula, which reduce the integral of an equivariant basic cohomology class to an integral over the set of invariant leaves. This talk is based on a very recent joint work with Reyer Sjamaar.
; Groupoid of morphisms of groupoids and groupoid extension
; (16:30-17:20)
Abstract: In this talk we introduce various constructions on (Lie) groupoids originated from orbifold Gromov-Witten theory. We first consider the construction of groupoid of morphisms of groupoids, this would be the foundation for studying moduli spaces in orbifold Gromov-Witten theory. As an application we give a definition of group action on groupoids. Then we consider general groupoid extension/groupoid fibration with fiber being general groupoids. A gerbe is a special case of groupoid extension. We classify groupoid extensions. Finally, we discuss the morphism groupoids of fiber class morphism to a groupoid extension. This is the first step to study the orbifold Gromov-Witten theory of groupoid extensions. This talk is based on recent joint works with Pro. B. Chen, Pro. R. Wang and Pro. Y. Wan.
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