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Mini-workshop on symplectic geometry

2018/06/11 11:01  点击:[]

Time: June 22, 2018

Venue: Room LA 106

 

Speaker:

  1. Reyer Sjamaar (Cornell University)

  2. Hao Ding (Southwest Jiaotong University)

  3. Farkhod Eshmatov (Sichuan University)

  4. Yi Lin (Georgia Southern University)

  5. Cheng-Yong Du (Sichuan Normal University)

     

Organized by Hongliang Shao , hongliangshao@foxmail.com

 Xiangdong Yang , yang7765330@163.com

 Hengyu Zhou ,    zhouhyu@cqu.edu.cn

 

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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