Introduction to the Workshop： 

Unlike most of the math workshops, this one consists of the survey talks tailored for young mathematicians, graduate students and advanced undergraduate students. The talks will mainly focus on picturing the (local or global) landscape of the areas that the speakers are interested in, with emphasis on open problems which might act like 'singularities' of such landscape. We hope the talks might give attendees the impetus to 'remove the singularity' and initiate discussions.

Confirmed speakers: 

Bobo Hua(Fudan)

Mijia Lai(SJTU)

Yi Li (SJTU)

Yi Liu (PKU)

Jing Mao (WHUT)

Fang Wang (SJTU)

Zuoqin Wang (USTC)

Chao Xia (XMU)

Guoyi Xu (THU)

Shicheng Xu (CNU)

Bin Zhou (PKU)

Schedule

November 5th, 2016 (Saturday)

9:00-9:50 Zuoqin Wang

Title: Sounds in the appearance of symmetry -- An introduction to equivariant spectrum

Abstract: Let $(M, g)$ be a Riemannian manifold endowed with an isometric action of a Lie group $G$. Then there is a natural structure inside the Laplacian spectrum of $M$. I will give a brief introduction to this structure: what is it? what do we know about it? how to use it?

10:00-10:50 Bin Zhou

Title: Geometric variational problems of Monge-Ampere type

Abstract: We discuss the recent development of the variational problems of a class of Monge-Ampere type functionals. Among these functionals, there are two important ones from geometry. One is associated with the affine mean curvature equation and the other one arises in the study of Calabi's extremal metrics on toric Kahler manifolds. The Euler equations of both functionals are fourth order partial differential equations, which are elliptic when the solution is a convex function. Both equations can be written as a system of a Monge-Ampere equation and a linearized Monge-Ampere equation. We introduce the progress on the existence and regularity of maximizers (or minimizers) of these functionals.

10:50-11:10 break

11:10-12:00 Guoyi Xu

Title: When the fundamental group of a Riemannian manifold is finitely generated?

Abstract: For every compact Riemannian manifold, it is well known that the fundamental group is finitely generated. For complete non-compact Riemannian manifolds, the fundamental group possibly is not finitely generated. A natural question is: which complete Riemannian manifolds have finitely generated fundamental group? We will survey the progress in this question from Bieberbach, Cheeger-Gromoll, Gromov to more recent work by Kapovitch and Wilking, and my recent work will also be presented. No technical proofs in the talk, some elementary topology and Riemannian geometry knowledge is enough to understand most of the talk.

12:00-14:00 Lunch

14:00-14:50 Chao Xia

Title: Geometric inequalities for hypersurfaces via Elliptic and Parabolic PDE approaches

Abstract: In this talk, I wil start from some elliptic and parabolic PDE approaches to isoperimetric inequality in Euclidean space by Gromov, Trudinger, Gage-Hamilton, etc.. Then I will collect several extensions of such approaches for general Alexandrov-Fenchel type inequalities for hypersurfaces in space forms and warped product manifolds by Reilly, Trudinger, Chang-Wang, Huisken-Ilmanen, Guan-Li,  Brendle-Hung-Wang  etc. and some of my recent works. Compared with the Euclidean space, the AF inequalities for hypersurfaces in space forms and warped product manifolds are far from complete. Finally, I would like to point out the weighted AF inequalities in space forms and warped product manifolds and its relationship with Penrose type conjecture in static space-time proposed by Brendle-Wang.

14:50-15:10 break

15:10-16:00 Mijia Lai

Title: Some equations arising from conformal geometry

Abstract: In this talk, I will briefly survey some equations in conformal geometry that have been studied intensively since the resolution of the famous Yamabe problem.

16:10-17:10 Bobo Hua

Title: Discrete curvature on graphs

Abstract: We will survey some definitions of discrete curvature on graphs and related problems in discrete geometric analysis.

November 6th, 2016 (Sunday)

9:00-9:50 Fang Wang

Title: Poincare-Einstein Manifolds and Fractional Yamabe problem.

Abstract: In this talk I will mainly give a survey about the Poincare-Einstein manifolds, as well as the Fractional Yamabe problem on their conformal infinity.

10:00-10:50 Yi Liu

Title: Virtual 1-domination of 3-manifolds

Abstract: Given an oriented closed hyperbolic 3-manifold, we show that every closed oriented 3-manifold can be mapped onto by a finite cover of that hyperbolic 3-manifold, with the mapping degree equal to 1. When the target is equipped with a Riemannian metric, the Lipschitz constant of the map can be controlled as small as wanted. This is joint work with Hongbin Sun.

10:50-11:10 break

11:10-12:00 Jing Mao

Title: The overdetermined elliptic problem and some recent progress

Abstract: In this talk, first, we would like to give an introduction to the overdetermined elliptic problem together with some classical conclusions in the Euclidean space. Then recent progress including a partial answer to the Berestycki-Caffarelli-Nirenberg conjecture on hyperbolic surfaces will be mentioned.

12:00-14:00 Lunch

14:00-14:50 Shicheng Xu

Title: Quantitative rigidity of space forms and almost flat manifolds under lower Ricci curvature bounds

Abstract: This is a survey on rigidity and stability theorems under lower Ricci curvature bounds for space forms and almost flat manifolds. We will start from the well-known results in positive Ricci curvature including round sphere’s pinching volume rigidity/stability by [Perelman, Colding, Cheeger-Colding] and nth/(n+1)th eigenvalue pinching rigidity/stability by [Petersen, Bertrand, Honda, Aubry]. Secondly we will talk about the world of negative lower Ricci curvature bound, including volume pinching rigidity/stability under 1-degree map by [Bessières-Besson-Courtois-Gallot] and volume entropy/1st eigenvalue pinching rigidity/stability by [Ledrappier-Wang, Chen-Rong-Xu] for closed hyperbolic manifolds.  Compared to spherical and hyperbolic case, there are only rare results for flat manifolds. But we will also talk about more recently rigidity/stability results for almost flat manifolds under lower Ricci curvature bounds. If there is time, we will talk about a conjecture on the local rewinding volume pinching stability of space form, which is raised in a series of joint papers with Lina Chen and Xiaochun Rong.

14:50-15:10 Break

15:10-16:00 Yi Li

Title: Long time existence on Ricci-harmonic flow

Abstract: We give a survey on recent results on Ricci-harmonic flow.

16:10-17:00 Discussion