Mini-workshop on Symplectic and Non-Commutative Geometry Talks
All talks will be held in room 211, the Integrity building (精正楼).
May 4th:
9:45-10:45:
Zhuo Chen (Qinghua University)
Title: Algebraic structures out of L-infinity pairs
(joint work with Lang Honglei, Li Yanpeng, Xiang Maosong)
Abstract: We show various structures coming from a pair (L,A), where L is an L-infinity algebra, A a subalgebra.
10:45-11:00 Break
11:00-12:00:
Hui Li (Soochow University)
Title: Hamiltonian circle actions on symplectic manifolds with certain minimal conditions
Abstract: Consider a Hamiltonian circle action on a compact symplectic manifold. Assume the even Betti numbers of the manifold are minimal or the fixed point set of the action satisfies a minimal condition, we discuss how the circle action, the integral cohomology ring and the total Chern class of the manifold determine each other, and in certain good cases, including the case when the manifold is Kaehler, we determine the manifold up to equivariant symplectomorphism or/and biholomorphism.
14:00-15:00:
Yi Lin (Georgia Southern University, USA)
Title: Hamiltonian actions on transversely symplectic manifolds
Abstract: In this talk, we introduce the notion of a Hamiltonian action on a manifold foliated by a transversely symplectic foliation. This provides a framework to study the Hamiltonian actions on the leaf spaces of such foliations, which are in general non-Hausdorff, and which include many interesting singular symplectic spaces, such as symplectic orbifolds, symplectic quasi-folds (by E. Prato), and the leaf spaces of characteristic Reeb foliations on contact manifolds, as special examples. We explain that under reasonable assumptions, the components of a moment map introduced by us are still Morse-Bott functions with even indices. This in particular leads to a foliated version of the Atiyah-Guillemin-Sternberg convexity theorem.
This talk is based on a work in progress with R. Sjamaar.
15:00-15:20 Break
15:20-16:20:
Xiaojun Chen (Sichuan University)
Title: BV algebra of noncommutative and Poisson algebras
Abstract: In this talk, we give a comparative study of the BV algebra structures that appeared in noncommutative geometry and Poisson geometry. It summarises some results of P. Xu, Ginzburg and Van den Bergh among others. The basic notion is called “calculus with duality”, which naturally gives these BV structures in a uniform way.
May 5th:
14:00-15:00:
Ping Xu (Penn State University, USA)
Title: Rozansky-Witten type invariants from symplectic Lie pairs
Abstract: In 1997,Rozansky and Witten built new finite-type invariants of 3-manifolds from hyperkaehler manifolds. It was later shown by Kontsevich and Kapranov that those invariants only depend on the holomorphic symplectic structure of the hyperkaehler manifolds. Indeed Kapranov proved that these invariants may be considered as an analogue of Chern-Simons type invariants, where the Atiyah class of the underlying complex manifold plays the role of Lie bracket. In this talk, we introduce symplectic structures on “Lie pairs” of (real or complex) algebroids, encompassing homogeneous symplectic spaces, symplectic manifolds with a $mathfrak g$-action and holomorphic symplectic manifolds. We show that to each such symplectic Lie pair are associated Rozansky-Witten type invariants of three-manifolds.
(This is a joint work with Yannick Voglaire.)
15:00-15:20 Break
15:20-16:20:
Hitoshi Moriyoshi (Nagoya University,Japan)
Title: Symplectic embeddings into complex projective space
Abstract: It is known that any Abelian variety of complex dimension n, which is also a symplectic manifold of dimension 2n, cannnot be embedded holomorphically into a complex projective space of dimension 2n. We shall prove a similar result for Kodaira-Thurston manifold in symplectic geometry context by applying the Atiyah-Singer index theorem.
16:20-16:30 Break
16:30-17:30:
Yanpeng Li (Beijing University)
Title: Double Principal Fiber Bundles
Abstract: We define the notion of double principal fiber bundles. To a given G-double principal fiber bundle and a representation of its structure group on a double vector space, we prove that the associated bundle is a double vector bundle. As an example of G-DPFB, the frame bundles of double vector bundles are studied. Also gauge transformation and connection of DPFB are studied.
（This is a joint work with Honglei Lang and Zhangju Liu.）
May 6th:
9:30-10:30:
Zuoqin Wang (University of Science and Technology of China)
Title: From symplectic reduction to equivariant spectral geometry
Abstract: Let G be a compact Lie group acting isometrically on a compact Riemannian manifold (M, g). Then each eigenspace of the Laplace-Beltrami operator is a representation of G, from which one gets a much finer structure of the Laplacian spectrum. I will explain the role of symplectic reduction in this equivariant spectral theory, and how to use the equivariant spectrum to recover Schrodinger potentials on symplectic toric manifolds. This is a joint work with V. Guillemin.
10:30-10:45 Break
10:45-11:45:
Mathieu Stienon (Penn State University, USA)
Title: Algebraic exponential maps
Abstract: Exponential maps arise naturally in the contexts of Lie theory and connections on smooth manifolds. We will explain how exponential maps can be understood algebraically, how these maps can be extended to graded manifolds and how this problem leads naturally to Dolgushev-Fedosov resolutions.