2016 ECNU Workshop

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E-mail:fuquan_fang@yahoo.com

Capital Normal University, P.R.China

We provide an equivariant description / classification of all complete (compact or not) nonnegatively curved manifolds M together with a co-compact action by a reflection group W, and moreover, classify such W. In particular, we show that the building blocks consist of the classical constant curvature models and generalized open books with nonnegatively curved bundle pages, and derive a corresponding splitting theorem for the universal cover. This is a joint work with Karsten Grove.

E-mail:fht@nankai.edu.cn

Nankai University, P.R.China

In this talk, by applying a Mathai-Quillen type formula of Zhang and myself to the Finsler setting, we get a Finslerian Gauss-Bonnet-Chern formula for any orenited and closed Finsler manifold.

E-mail:jge@math.pku.edu.cn

Peking University, P.R.China

In this talk, we will discuss some volume comparison result for Alexandrov spaces with boundary under certain extrinsic curvature condition on the boundary.

E-mail:hong@maths.uq.edu.au

University of Queensland, Australia

We introduce a rectified $n$-harmonic map flow from an $n$-dimensional closed Riemannian manifold to another closed Riemannian manifold. We prove existence of a global solution, which is regular except for a finite number of points, of the rectified $n$-harmonic map flow and establish an energy identity for the flow at each singular time. Finally, we present two applications of the rectified $n$-harmonic map flow to minimizing the $n$-energy functional and the Dirichlet energy functional in a homotopy class.

E-mail:liu@math.ucla.edu

University of California, Los Angeles

I will present our recent results on the geometry of period maps from deformation spaces of projective manifolds into period domains, such as the algebraicity and the boundedness in complex Euclidean space of the period map lifted to the universal covers, both conjectured by Griffiths.

E-mail:jp1016@math.rutgers.edu

Rutgers University

We study the fundamental group of an n-manifold (M,p) of Ricci curvature Ric M≥-(n-1). We show that if its local universal cover (U_2 (p),p^*) of (B_2 (p),p) is not collapsed, then there exists a positive constant ϵ(n) such that for any δ>0 and any (ϵ(n),δ) regular point q^*∈B_1 (p^*), the number of short generators of π_1 (M,q) of length at most R is bounded above by some constant C(n,R,δ). In particular, this result verifies Milnor conjecture for manifolds whose universal coverings have almost maximal volume growth. This is joint work with Xiaochun Rong.

E-mail:shenshu@math.hu-berlin.de

Humboldt University, German

The relation between the spectrum of the Laplacian and the closed geodesics on a closed Riemannian manifold is one of the central themes in differential geometry. Fried conjectured that the analytic torsion, which is an alternating product of regularized determinants of the Hodge Laplacians, equals the zero value of the dynamical zeta function. In the first part of the talk, we will give a formal proof of this conjecture based on the path integral and Bismut-Goette’s V-invariants. In the second part, we will give the rigorous arguments in the case where the underlying manifold is a closed locally symmetric space. The proof relies on the Bismut’s formula for semisimple orbital integrals. This talk is based on a recent preprint arXiv:1602.00664.

E-mail:shoseto@ucsb.edu

University of California, Santa Barbara, USA

In this talk, we introduce the Laplacian eigenvalue problem and briefly go over its history. Then we will present a recent result which gives a sharp lower bound of the fundamental gap for convex domain of spheres motivated by the modulus of continuity approach introduced by Andrews-Clutterbuck. This is joint work with Lili Wang and Guofang Wei.

E-mail:zhanghc3@mail.sysu.edu.cn

Sun Yat-Sen University, P.R.China

In this talk, we will introduce a pointwise maximum principle and the (linear) geometric analysis on non-smooth metric measure spaces. In particular, we will establish a local Li-Yau’s estimate for weak solutions of the heat equation on metric measure spaces, under the Riemannian curvature-dimension condition RCD*(K,N). This talk is based on a joint work with Xi-Ping Zhu.

E-mail:weiping@nankai.edu.cn

Nankai University, P.R.China

By using the Mathai-Quillen superconnection construction of the Thom class, we show that the Euler characteristic of a compact affine manifold equals to zero. This confirms an old conjecture of S. S. Chern.