Mini-workshop on Geometry, Topology and their Applications
Student No.:50
Time:08:40-17:00, 2017-4-8/4-9
Instructor:Hai Lin, Yunhui Wu, Guoyi Xu  
Place:Lecture Hall, Floor 3, Jin Chun Yuan West Building
Starting Date:2017-4-8
Ending Date:2017-4-9






April 8, 2017 Saturday

08:40-09:10    Registration 


09:10-10:00    Zuoqin Wang (University of Science and Technology of China)


10:00-10:15    Tea Break 

10:15-11:05    Weixu Su (Fudan University)  

11:10-12:00    Wenyuan Yang (Peking University)   

12:00-14:00    Lunch 

14:00-14:50    Ming Fang (Academy of Mathematics and Systems Science, CAS) 

14:50-15:05    Tea Break

15:05-15:55    Hao Zheng (Peking University)

16:00-16:50    Gang Li (Shandong Universaity) 

16:50-17:00    Group Photos 

17:00-17:30    Tea Break & Free Discussion 


17:30         Conference Dinner in Jin Chun Yuan Hotel 


April 9, 2017 Sunday

08:40-09:10    Registration 

09:10-10:00    Bobo Hua (Fudan University) 

10:00-10:15    Tea Break 

10:15-11:05    Renjie Feng (Peking University) 

11:10-12:00    Yang Su (Academy of Mathematics and Systems Science, CAS)

12:00          Lunch




Titles and Abstracts:



1.      Ming Fang, Academy of Mathematics and Systems Science, CAS

Title: Characteristic-Free GL (n) Homomorphisms between Symmetric, Exterior and Divided Powers
Abstract: Symmetric, exterior and divided powers are fundamental objects in polynomial representation theory of general linear groups. They are defined over any communicative rings. In this talk, I will talk about their homomorphisms as GL(n)-modules, and show in particular that some of these are also defined over any commutative rings.


2.      Renjie Feng, Peking University, China

Title: Critical Radius and Supremum of Random Spherical Harmonics
Abstract: We first consider deterministic immersions of the d-dimensional sphere into high dimensional Euclidean spaces, where the immersion is via spherical harmonics of level n. The main result of the article is the, a priori unexpected, fact that there is a uniform lower bound to the critical radius of the immersions as n→∞. This fact has immediate implications for random spherical harmonics with fixed L2-norm. In particular, it leads to an exact and explicit formulae for the tail probability of their suprema by Weyl's tube formula, and also relates this to the expected Euler characteristic of their upper level sets. We will mainly concentrate on the geometry aspect of the program, such as Weyl's tube formula in the integral geometry. The talk is very elementary and accessible to all students. This is the joint work with R. Adler.


3.      Bobo Hua, Fudan University, China

Title: Total Curvature of a Planar Graph with Nonnegative Curvature
Abstract: For a planar graph with nonnegative combinatorial curvature, we show that its total curvature is an integral multiple of 1/12. This is a joint work with Yanhui Su.


4.      Gang Li, Shandong Universaity, China

Title: On uniqueness of Conformally Compact Einstein Metrics with Berger Sphere Infinity
Abstract: We show that for a Berger metric $\hat{g}$ on $S^3$, the non-positively curved conformally compact Einstein metric on the 4-ball with $(S^3, [ \hat{g}])$ as its conformal infinity is unique up to isometries and it is the metric constructed by Pedersen [1].
Reference: [1] H. Pedersen, Einstein Metrics, Spinning Top Motions and Monopoles,Math. Ann. 274 (1986) 35 - 59.


5.      Weixu Su, Fudan University, China

Title: Variation of Extremal Length Functions on Teichmuller Space
Abstract: Extremal length is an important conformal invariant on Riemann surface. It is closely related to the geometry of Teichmuller metric on Teichmuller space. By identifying extremal length functions with energy of harmonic maps from Riemann surfaces to R-trees, we study the second variation of extremal length functions along Weil-Petersson geodesics. We show that the extremal length of any measured foliation is a pluri-subharmonic function on Teichmuller space.


6.      Yang Su, Academy of Mathematics and Systems Science, CAS

Title: On the Mapping Class Group of Spin Homology CP^3
Abstract: The mapping class group of a smooth manifold is an important algebraic object associated to the manifold, which reflects the symmetry of the geometric object. The mapping class groups of surfaces are in the central place of low dimensional geometry and topology. The computation of the mapping class group is in general difficult, in high dimensions, only a few examples are known. In this talk I will present a recent work, joint with M.Kreck, on the mapping class group of a class of 6-manifolds --- those having the same homology as that of CP^3 and being spin. This is the first step towards the understanding of the mapping class group of complex 3-dimensional hypersurfaces.


7.      Zuoqin Wang, University of Science and Technology of China

Title: Improved Error Term in Weyl Asymptotic Formula for Planar Disk
Abstract: In this talk I will discuss some recent results regarding the remainder term in the Weyl asymptotic formula for Laplacian eigenvalues of the planar disk. The main idea is to convert the eigenvalue counting problem into a lattice point counting problem, and then to use methods from harmonic analysis and number theory. This is a work in progress, joint with Jingwei Guo and Weiwei Wang.


8.      Wenyuan Yang, Peking University, China

Title: Statistically Convex-Cocompact Actions
Abstract: In this talk, I will describe a class of statistically convex-cocompact actions of groups with a contracting element. This could be thought of as a statistical version of convex-cocompact Kleinian groups in a general setting. In particular, it includes relatively hyperbolic groups, CAT (0) groups with rank-1 elements, mapping class groups etc. Our main result shows that this class of groups have purely exponential growth, and generic elements are contracting elements. This produce several new results in the above class of groups and recovers some known results via different methods. For instance, one corollary is a strengthening of Maher's result that pseudo-Anosov elements are generic in mapping class groups. Another is Knieper's result non-rank 1 geodesics are exponentially small in rank-1 compact manifolds.


9.      Hao Zheng, Peking University, China

Title: Factorization Homology and Topological Order
Abstract: I will review of the theory of factorization homology emphasizing on dimension one and two by giving some elementary examples. Then I will show that the factorization homology of a surface with an anomaly-free coefficient system gives the ground state degeneracy of the associated 2+1D topological order. This is a joint work with Ai Yinghua and Kong Liang.