## Welcome to GAP seminar

This seminar aims at filling up the gap between graduate level math and research math, and enlarging the scope of graduate students as well as more advanced researchers. In each talk, the speaker will focus on a currently active research topic. He/she will spend one hour or so to introduce the background of the topic, including basic conceptions/examples, important known results and major problems etc. The last part of the talk will usually be more technical and is related to the speaker's own work. The subjects of future talks are mainly chosen from geometry(G), algebra, analysis(A), and mathematical physics(P).

## Upcoming Talks

## Past Talks

Archive of GAP Seminar: Fall 2013 | Spring 2014

Date: January 6 (Different Time: Tuesday!), 2015

Speaker: Chenxu HE (University of Oklahoma)

Title: Geometry of gradient Ricci solitons

Abstract: Ricci solitons are self-similar solutions to Hamilton's Ricci flow and they play a particular role in the singularity analysis of Ricci flow. They are also natural generalization of Einstein manifolds. We present a few results in the study of geometry of gradient Ricci solitons, including the stability problem for the shrinking Ricci solitons with respect to Perelman's shrinker entropy and show the classification on symmetric spaces of compact type, and the deformation of steady gradient Ricci solitons and show that the infinitesimal deformation is trivial in low dimension.

Date: December 29, 2014

Speaker: Rongjie Lai (Rensselaer Polytechnic Institute, USA)

Title: Geometric Methods for Analyzing Unorganized Data in 3D and Higher

Abstract: Point clouds are the simplest and most basic forms for data representation in 3D and higher. Examples include those used in 3D modeling, image science, the Internet and many others. Analyzing and inferring the underlying structure from the point clouds are crucial in many applications. However, these raw data or primitive representations in Rn are lack of global parameterization and far from intrinsic, which obstruct further intrinsic and global analysis for given data. On the other hand, point clouds from practical problems are usually with certain coherent structures which allow us to model them as points sampled from lower dimensional Riemmannian manifolds in a high dimensional space. In this talk, I will discuss our work on solving PDEs on point clouds. It can be applied to manifolds with arbitrary dimensions and co-dimensions. This approach enables us to propose geometric methods for analyzing and understanding point clouds through solutions of PDEs on point clouds. As applications, I will demonstrate our methods to intrinsic geometric quantities extraction, comparison and registration for point clouds based on solutions of certain geometric PDEs. (Joint work with Hongkai Zhao)

Date: December 22, 2014

Speaker: 何跃（南京师范大学）

Title: On sharp lower bound of the spectral gap for a Schr\"odinger operator and some related results

Abstract: In this paper, we give an easy proof of the main results of Andrews and Clutterbuck's paper [Proof of the fundamental gap conjecture, J. Amer. Math. Soc. 24 (2011), no. 3, 899--916] which gives both a sharp lower bound for the spectral gap of a Schr\"odinger operator and a sharp modulus of concavity for the logarithm of the corresponding first eigenfunction. We arrive directly at same estimates by the `double coordinate' approach and asymptotic behavior of parabolic flows. Although using the techniques appeared in the above paper, we partly simplify the method and argument. This maybe help to provide an easy way for estimating spectral gap. Besides, we also get a new lower bound of spectral gap for a class of Sch\"odinger operator.

Date: December 15, 2014

Speaker: 魏达盛（中国科学院数学与系统科学研究院）

Title: Strong approximation with Brauer-Manin obstruction

Abstract: For an algebraic variety, strong approximation may imply the existence of integral points of its integral model. In this talk, I will introduce some varieties which satisfy strong approximation.

Date: December 8, 2014

Speaker: 张红莲 （上海大学）

Title: On the structures and representations of quantum affine algebras

Abstract: It is well known that quantum affine algebras admit two realizations because they are both quantum Kac-Moody algebras and quantum affinizations. In general, a quantum affinization is not a quantum Kac-Moody algebra. The representation theory of quantum affine algebras is very rich and interesting. In this talk, I will review some results on the structures and representations of quantum affine algebras and discuss some related problems.

Date: December 1, 2014

Speaker: Xiaohua YAO (华中师范大学)

Title: Uniform Sobolev estimates for Laplace-Beltrami operators on compact manifolds

Abstract: This talk is concerned with the uniform Sobolev estimates of the Laplace-Beltrami operator on a compact manifolds M. Specifically, if the pair (1/p,1/q) is on the Sobolev line 1/p−1/q=2/n and p<2(n+1)/(n+3) and q>2(n+1)/(n−1), then we will prove that the resolvent is uniformly Lp-Lq-bounded outside a parabola opening to the right and a small disk centered at the origin, which is optimal on Sphere (or Zoll manifolds ). Using the shrinking spectral project estimates, we further obtain a logarithmic improvement over the parabolic region for resolvent estimates on manifolds with non-positive sectional curvature and a power improvement for flat torus.

Date: November 24, 2014

Speaker: 张晓（中国科学院数学与系统科学研究院）

Title: Some Geometrical Problems in General Relativity

Abstract: There has been a natural connection between geometry and gravity since Einstein. In this talk, we will discuss some geometrical problems relating to the positivity of the total energy-momentum in general relativity . The unsolved problems on the collision and the nonlinear stability of black holes will also be discussed.

Date: November 10, 2014

Speaker: 王成波 (浙江大学)

Title: Strauss猜测介绍

Abstract: 在本报告中，我们将介绍Strauss猜测的历史、概貌以及近年来该领域的一些最新进展。Strauss猜测是关于形如\partial_t^2 u-\Delta u =|u|^p的典型半线性波动方程何时具有小初值整体解的一个猜测。该猜测经过Georgiev、Glassey、John、Lindblad、Sogge、Tataru、周忆等数学家的努力已经得到解决。近年来，在各种时空背景下，该问题的研究又得到重视，取得了很多有意义的进展。最近，我们与Lindblad、Metcalfe、Sogge和Tohaneanu合作证实了Schwarzschild/Kerr黑洞时空背景下的Strauss猜测。

Date: November 3, 2014

Speaker: 王庆雪 (复旦大学)

Title: Introduction to dilogarithm function and Bloch group

Abstract: In this talk, we will start with the basic definitions of dilogarithm and Bloch group. We will explain the motivations for studying them. In particular we will discuss the close relation between the Bloch group and the Quillen's K-group K_3(F) for a field F, and a regulator map connecting the dilogarithm and Bloch group. We will give a few examples which use them to construct some invariants for geometric structures on 3-dimensional manifold.

Date: October 20, 2014

Speaker: 刘党政 (中国科学技术大学)

Title: 随机矩阵与黎曼Zeta函数

Abstract: 传说Hilbert和Polya曾猜想黎曼Zeta函数的非平凡零点与某个Hermite算子的特征值相对应. 从70年代以来, Montgomery, Katz与Sarnak及Keating与Snaith等人的工作表明随机矩阵与Zeta函数有着密切的关联：随机矩阵特征值间隔对应临界线上零点间隔，而特征多项式的矩对应Zeta函数在临界线上的矩.这些联系大多是猜测性的，但仍为Hilbert-Polya猜想提供了更坚实的证据.本报告介绍随机矩阵理论中两个重要的例子：圆酉系综(Circular Unitary Ensemble, CUE)和高斯酉系综(Gaussian Unitary Ensemble, GUE),及其同黎曼Zeta函数的关联.

Date: October 14, (Different Time!) 2014

Speaker: Kang ZUO (University of Mainz and 中国科学技术大学)

Title: Algebraic Geometry and Hodge Theory

Abstract: Hodge theory plays an important role in Algebraic Geometry. In my lecture I shall overview Hodge structures on curves and abelian varieties, Higgs bundles and some basic questions in Algebraic Geometry.

Date: September 22, 2014

Speaker: Zhihua CHANG (Bar-Ilan University)

Title: Descent theory and its application in Lie theory

Abstract: Given a ring extension $S/R$, a basic question is whether an $S$-module $M$ is of the form $N \otimes_R S$ for some $R$-module $N$. The descent theory gives a solution to this question in the case where $S/R$ is a faithfully flat extension.
My talk will begin with an introduction on faithfully flat ring extensions. Then I will explain how the descent theory answers the question mentioned above. As an application, I will present some Lie-theoretical results obtained by using the descent theory. Such an application is based on the key observation that every twisted loop Lie algebra can be obtained by applying a descent construction to a untwisted loop Lie algebra.

Date: September 15, 2014

Speaker: Zhou ZHANG (Sydney University)

Title: The General Kahler-Ricci Flow

Abstract: In the past ten years or so, the study of Kahler-Ricci flow has been reformed quite essentially. By allowing the Kahler class to evolve with time, we reveal its natural relation with the Minimal Model Program in algebraic geometry as outlined in Tian's Program. This also brings the subject a lot closer to the study of Ricci flow, and tightens the relation with pluripotential theory in several complex variables. With more tools and concepts introduced into this research field from other major branches of pure mathematics, we have got many answers with even more problems for future work. In this talk, we aim at a description of the whole story, accessible for the general audience.