2018 ECNU Winter Workshop

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University of Science and Technology of China

University of Science and Technology of China

Abstract: The relation between the spectrum of the Laplacian and the dynamical flow on a closed Riemannian manifold is one of the central themes in differential geometry. Fried conjectured a relation between the analytic torsion, which is an alternating product of regularized determinants of the Hodge Laplacians, and the Ruelle dynamical zeta function. We will formulate and show this conjecture for Morse-Smale flow. Our proof relies on Cheeger-M\"{u}ller/Bismut-Zhang theorem. This is joint work with Shu Shen.

East China Normal University

Abstract: The study of Chern number inequalities for complex projective surfaces dates back to the work of Italian school algebraic geometers. It motivates the subject of the so-called geography in algebraic geometry nowadays, which aims at finding numerical relations among intrinsic (birational) invariants of complex projective varieties. In this talk, I will focus on those projective varieties with nontrivial global one forms, which are often called irregular varieties. The goal of the talk is to give a survey of the geographical results for irregular surfaces of general type and then to introduce what we have known so far in higher dimensions.

Tongji University

Abstract：In this talk we review a notion of nonnegativity for Hermitian holomorphic vector bundles introduced by Bott and Chern and its later development. Then we discuss our recent work around it.

East China Normal University

Abstract: The famous Atiyah-Singer index theorem announced in 1963 computed the index of elliptic operators, which is defined analytically, in a topological way, by using the characteristic classes. In 1968, Atiyah and Segal established a localization formula for the equivariant index which computes the equivariant index via the contribution of the fixed point sets of the group action. It is natural to ask if the localization property holds for the more complex spectral invariants, which are not computable in a local way and not a topological invariant.
In this talk, we will establish a version of localization formula for equivariant eta-invariants，which were introduced in the 1970's as the boundary contribution of index theorem for compact manifolds with boundary and are formally equal to the number of positive eigenvalues of the Dirac operator minus the number of its negative eigenvalues, by using differential K-theory, a new research field in this century. This is a joint work with Xiaonan Ma.A.Naber.

East China Normal University

Abstract: In this talk, I will introduce of the Finsler geometry and the background of the isoperimetric problem. By using the variational theory, we give the local solution of the isoperimetric problem in 2-dimensional Finsler space forms. This is the joint work with Mengqing Zhan.

University of Science and Technology of China

Abstract:In this talk, we will recall some classical results on the differential geometry of holomorphic vector bundles, and  introduce our recent work on the existence of canonical metrics, Bogomolov type inequalities and the Hermitian Yang-Mills flow. These works are joint with Professor Li Jiayu, Zhang Pan and Professor Zhang Xi.

Northwestern Polytechnical University

Abstract: In this talk, I will present some Reilly-type inequalities for some elliptic operators on submanifolds in space forms. In the first part, we will give an upper bound for the first nonzero eigenvalue of the $p$-Laplacian with $1 < p \leq n/2 + 1$, where $n$ is the dimension of the submanifold. This part is joint work with Prof. Guofang Wei. In the second part, we will show a sharp upper bound for the second eigenvalue of a very general class of elliptic operators $L_T$ on submanifolds in space forms. This part is joint work with Xianfeng Wang. Both the results mentioned above can be regarded as natural generalizations of the famous Reilly inequality.

Tianjin University

Abstract: In this talk, we will survey some domination results in higher Teichmüller theory. Higher Teichmüller theory, initiated by Hitchin, studies the moduli space of the representations from a surface group to a semisimple Lie group, which is a generalization of classical Teichmüller theory. By the non-Abelian Hodge theory, the moduli space of the representations corresponds to the moduli space of the Higgs bundles via the equivariant harmonic maps from the universal cover of the Riemann surface to the symmetric space of the Lie group. We focus on the geometric quantities of the harmonic map, for instance the metric and the curvature. Under the background of the Hitchin fibration, we describe a conjectural picture for the behavior of these quantities and show some developments in recent years.

East China Normal University

Abstract: In this talk, we give an estimate for the volume entropy in terms of integral Ricci curvature which substantially improves an earlier estimate of Aubry and give an application on the algebraic entropy of its fundamental group. We also extend the quantitative almost maximal volume entropy rigidity of Chen-Rong-Xu and almost minimal volume rigidity of Bessieres-Besson-Courtois-Gallot to integral Ricci curvature. This is a joint work with professor Guofang Wei.

Capital Normal University

Abstract: In this talk, I will introduce an arithmetic analogue of Lefschetz-Riemann-Roch theorem in the context of Arakelov geometry which bases on Bismut and Ma's work on local index theory. If time permits, I will explain how one uses this theorem to relate some arithmetic invariants of algebraic varieties with the logarithmic derivative of L-functions.

Soochow University

Abstract:  I will talk about the parabolic frequency for solutions of the heat equation on Riemannian manifolds. We show that the parabolic frequency functional is almost  increasing on compact manifolds with nonnegative sectional curvature, which generalizes a monotonicity result proved by C. Poon and by Lei Ni.   As applications, we obtain a unique continuation result. Monotonicity of a new quantity under two-dimensional Ricci flow, closely related to the parabolic frequency functional,  is derived as well. This is a joint work with Xiaolong Li (UC Irivine).