## 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 | Spring2014 | Fall 2014 | Spring 2015 | Fall 2015 | Spring 2016 | Fall 2016 | Spring 2017

Date: 1/26, 10:00-11:30

Speaker: 潘宣余（中科院数学与系统研究所）

Title: 1-Cycles on Fano manifolds

Abstract: In this talk, I will give a survey on the 1-cycles on Fano manifolds. It is well known that Fano manifolds have many rational curves, in particular, they are rationally connected. The geometry of Fano manifolds is governed by rational curves. So the cycles on Fano manifolds should be understood from their rational curves. Under this observation, I will also talk about a recent joint work with Cristian Minoccheri on 1-cycles on higher Fano manifolds.

Date: 12/25, 16:00-17:30 (Different Time)

Speaker: 袁原（Syracuse University）

Title: On the diameter rigidity of Kahler manifolds with positive bisectional curvature

Abstract: It follows from the comparison theorem that if the Kahler manifold has bisectional curvature at least 1, then the diameter is no greater than the diameter of the standard projective space. I will discuss a recent joint result with G. Liu regarding the rigidity when the diameter reaches the maximum.

Date: 12/22, 16:00-17:30

Speaker: 吴云辉（清华大学）

Title: The Weil-Petersson geometry of the moduli of curves for large genus

Abstract: We study the systole function along Weil-Petersson geodesics. We show that the square root of the systole function is uniform Lipschitz on the Teichmuller space endowed with the Weil-Petersson metric. As an application, we study the growth of the Weil-Petersson inradius of the moduli space of Riemann surfaces of genus g with n punctures as a function of g and n. We show that the Weil-Petersson inradius is comparable to $\sqrt{\ln{g}}$ with respect to g, and is comparable to 1 with respect to n.

Date: 12/11, 11:00-12:00 (Different Time)

Speaker: 毛井（湖北大学）

Title: The analysis of parabolic and hyperbolic inverse curvature flows

Abstract: In this talk, we would like to show some recent results on the analysis of parabolic and hyperbolic inverse curvature flows, which are based on joint-works with my collaborators.

Date: 12/11, 10:00-11:00 (Different Time)

Speaker: 陈大广（清华大学）

Title: A Penrose type inequality for graphs over Reissner-Nordstrom-anti-deSitter manifold

Abstract: We will talk about Minkowski type inquality, weighted Alexandrov Fenchel inequality for the mean convex star-shaped hypersurfaces in Reissner-Nordstrom-anti-deSitter manifold and Penrose type inequality for asymptotically locally hyperbolic manifolds which can be realized as graphs over Reissner-Nordstrom-anti-deSitter manifold. This talk is based on the joint work with Professor Haizhong Li and Dr. Tailong Zhou.

Date: 12/8, 16:00-17:30

Speaker: 熊革（同济大学）

Title: John ellipsoid, affine isoperimetric inequality and convex geometry

Abstract: It is known that there exists a unique ellipsoid of maximal volume inside a convex body (a compact convex set with non-empty interiors) in $R^n$. This ellipsoid is called John ellipsoid (named after mathematician Fritz John), and has many applications in convex geometry, functional analysis and optimizations. Particularly, that combining the isotropic characterization of John ellipsoid with the celebrated Brascamp-Lieb inequality in real analysis, plays a major role to attack reverse isoperimetric problem. In this talk, we will first briefly survey the discipline of convex geometry. Then, the classical John ellipsoid and its applications to affine isoperimetric inequalities will be addressed. With these background material and basic facts in hand, we will introduce our own work on (reverse) affine isoperimetric inequality, which include several papers already published or submitted recently.

Date: 12/8, 14:00-15:30 (Different Time)

Speaker: 乔雨（陕西师范大学）

Title: Fredholm Conditions on Singular Spaces

Abstract: In the 1980's, Alain Connes initiated his program of noncommutative differential geometry, especially the study of "bad" spaces. It turns out that Lie groupoids are an effective tool to model many analysis and index problems on singular spaces. In this talk, we first recall the notion of manifolds with corners(following the work of Melrose). Then, we review the construction of group $C^*$-algebras via the representation theory of groups, the notion of Lie groupoids, and the construction of groupoid $C^*$-algebras via the representation theory of $C^*$-algebras. Next, we present the concept of a Fredholm groupoid, which is a class of groupoids for which certain characterization of Fredholm operators is valid, and then adopt $b$-calculus, scattering calculus, and edge calculus in the frame work of Fredholm groupoids. Finally, we discuss briefly the relation between Fredholm groupoids and index theory. This is joint work with Catarina Carvalho and Victor Nistor.

Date: 12/8, 10:00-11:30 (Different Time)

Speaker: 徐泽（山东大学）

Title: On the Chow motive of generalized Kummer varieties

Abstract: It is an important problem to determine the Chow motive of a smooth projective variety. In this talk, we introduce the basic algebraic cycle theory, determine the Chow motive of generalized Kummer varieties and provide several applications to problems related to algebraic cycles.

Date: 12/1, 14:00-15:30 (Different Time)

Speaker: 徐国义（清华大学）

Title: The integral of the curvature

Abstract: The classical Gauss-Bonnet theorem in 19th century links the integral of the curvature to the topology of compact manifolds in 2-dim case. The well-known Gauss-Bonnet-Chern theorem generalize this result to higher dimension in 1940's. On the other hand, Cohn-Vossen's inequality opened the door of estimating the integral of the curvature on non-compact manifolds in 2-dim in 1930's. However, the higher dimension version of Cohn-Vossen's inequality is still missing, although Yau posed one open question along this line in 1990's. We will survey the history of the study around the integral of the curvature, from Gauss, Bonnet, Chern, Yau to the current state, my recent research result 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.

Date: 11/17, 10:00-11:30 (Different Time)

Speaker: 左康（University of
Mainz）

Title: PGL_2-crystalline local systems on the projective line minus 4 points and torsion points on the associated elliptic curve.

Abstract: In my talk I shall report my recent joint work with R.R. Sun and J.B. Yang. Given an odd prime p we take t to be a number in an unramified extensionof the p-adic number ring Z_p such that t (mod p) is not equal to 0 and 1, and C_t to be the elliptic curve defined by the affine equation y^2=x(x-1)(x-t). For q=p^n we speculate the set of points in C_t(F_q) whose order coprimes to p corresponds to the set of PGL_2(\bar F_q)-crystalline local systems on P^1- { 0, 1, infinity, t} over some unramified extension of the p-adic number field Q_p via periodic Higgs bundles and the p-adic Simpson correspondence recently established by Lan-Sheng-Zuo for GL-case and Sun-Yang-Zuo for PGL-case. In the arithmetic setting, given an algebraic number field K we introduce the notion of arithmetic local systems and arithmetic periodic Higgs bundles and speculate the set of torsion points in C_t(K) corresponds to the set of PGL_2-arithmetic local systems on P^1- { 0, 1, infinity, t} over K. It looks very mysterious. M. Kontsevich has already observed that the K3 surface as the Kummer surface of the elliptic curve C_t also appears in the construction of the Hecke operators which define the l-adic local systems on P^1- { 0, 1, infinity, t} over F_q via the GL_2 Langlands correspondence due to V. Drinfeld.

Date: 11/2, 14:00-15:30 (Different Time)

Speaker: 李平（同济大学）

Title: Alexandrov-Fenchel type inequalities, revisited

Abstract: Various Alexandrov-Fenchel type inequalites have appeared and played important roles in convex geometry, matrix theory and complex algebraic geometry. It has been noticed for some time that they share some striking analogies and have intimate relationships. In this talk we will shed new light on this by comparatively investigating them in several aspects.

Date: 10/30, 16:00-17:30 (Different Time)

Speaker: 徐佩（美国西北大学，中国科学技术大学）

Title: From Fourier Analysis to Gaussian Measures

Abstract: A Gaussian function is the exponential of a quadratic complex function whose leading coefficient is real negative. Starting from the fact that the class of Gaussian functions is invariant under Fourier transform we will discuss many well known and not so well known wonderful properties of Gaussian measures, including Stein\'92s characterization and the correlation domination inequality. We will show the special role of the Ornstein-Uhlenbeck operator (classical harmonic oscillator in quantum mechanics) and discuss its spectral property and the generalized spectral gap theorem. We will discuss the classical Poincare inequality and the logarithmic Sobolev inequality for the Gaussian measure and show that they are special cases of Beckner\'92s inequality. The prerequisite of the talk is a good knowledge of university calculus. The talk promises to be elementary and entertaining and students and mathematicians at all levels and in all fields are welcome.

Date: 10/13, 16:00-17:30

Speaker: 张伟（兰州大学）

Title: 椭圆方程凸性研究的历史与现状

Abstract: 凸性作为一个基本的几何性质，是现代几何学和分析学研究的重要论题之一。在这个报告中，我们以调和函数、torsion刚性问题和Laplace算子Dirichlet第一特征函数为例，对椭圆方程凸性研究的历史与现状做一个综述。

Date: 9/22, 16:00-17:30

Speaker: 胥世成（首都师范大学）

Title: Stability of Nilpotent Structures of
Collapsed Manifolds on the Same Scale

Abstract: We will talk about a recent work on the stability of nilpotent structures on a collapsed manifold with bounded sectional or Ricci curvature. A manifold of bounded sectional curvature is called \epsilon-collapsed, if the injectivity radius, or equivalently the volume of unit ball, at every points is less than \epsilon. The geometry/topology of a collapsed manifold can be totally described by Cheeger-Fukaya-Gromov's nilpotent structure. Similar results had been extended to manifolds of bounded Ricci curvature under some additional assumptions. The stability of locally defined nilpotent structures were essential in the work of Cheeger-Fukaya-Gromov to construct a global nilpotent structure on one fixed metric. Nilpotent structures also depend on the choice of \epsilon, the collapsed length scale one inspects. We prove that if two metrics on a manifold are L_0-bi-Lipchitz equivalent and sufficient collapsed (depending on L_0) under bounded (Ricci) curvature, then the underlying nilpotent structures are isomorphic to each other. As applications, we establish a link between the components of the moduli space of all collapsed Riemannian metrics and the set of isomorphism classes of nilpotent structures, and derive a new parametrized version of Gromov's flat manifold theorem under bounded Ricci curvature and conjugate radius.

Date: 9/15, 16:00-17:30

Speaker: 殷浩（中国科学技术大学）

Title: An introduction to neck analysis

Abstract: In this talk, we use 2-D harmonic map as an example to illustrate the basic theory of neck analysis. The proofs will be almost self-contained. We also compare with the Yang-Mills theory and outline some generalization of the argument.

Date: 9/8, 16:00-17:30

Speaker: 王芳（上海交通大学）

Title: Poincare-Einstein manifolds and conformal geometry

Abstract: This is the last part of the mini-course. In the lectures, I will introduce various topics related to the Poincare-Einstein manifolds, including existence and uniqueness theorems, topology and rigidity theorems, spectrum and resolvent for the Laplacian, scattering theory and GJMS operators, as well as fractional Yamabe problems.