## 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 | Fall 2014 | Spring 2015 | Fall 2015 | Spring 2016

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

Speaker: 郭龙（南开大学）

Title: Some combinatorial structures involving Young tableaux

Abstract: Young tableaux have numerous applications in combinatorics, representation theory, algebraic geometry. In this talk, we shall give a brief overview of some background on Young tableaux. We then discuss our two recent work on this subject. Firstly, we prove a conjecture on the generating function of a specific family of Young tableaux, proposed by Morales, Pak and Panova. Secondly, motivated by a result of Knutson, Miller and Yong for Grothendieck polynomials indexed by 2143-avoiding permutations, we use the tool of commutative algebra to give tableau interpretations for Grothendieck polynomials indexed by 1432-avoiding permutations.

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

Speaker: 夏波（中山大学）

Title: Probabilistic well-posedness of semilinear wave equations in the super-critical regime

Abstract: We know that the super-critical wave equation is ill-posed. Now a question arising from this fact is whether or not there is some "well-posedness theory" to the super-critical equation， analogous to the classical well-posedness theory? In this talk, I will introduce the concept "probabilistic well-posedness" in the sense of Burq-Tzvetkov, under which we can still expect good (local and even global) behaviours of wave equations in the super-critical regime. This is a joint work with Chenmin Sun.

Date: 12/9, 16:00-17:30

Speaker: 彦文娇（北京师范大学）

Title: Normal scalar curvature inequality on the focal submanifolds of isoparametric hypersurfaces

Abstract: . The classification of isoparametric hypersurfaces in unit spheres has been an important open problem, for which the last case was classfied very recently by [Chi16]. In this talk, I will firstly introduce some backgrounds of the isoparametric theory. An isoparametric hypersurface in unit spheres has two focal submanifolds, which are minimal submanifolds of the unit sphere, providing concrete examples to many problems. Condition A (holds only on one focal submanifold) plays a crucial role in the classification theory of isoparametric hypersurfaces in [CCJ07], [Chi16] and [Miy13]. We determine the set C_A, points with Condition A in focal submanifolds. It turns out that the points in C_A reach an upper bound of the normal scalar curvature (sharper than that in DDVV inequality [GT08], [Lu11]). We also determine the sets C_P (points with parallel secondfundamental form) and C_E (points with Einstein condition), which achieve two lower bounds of the normal scalar curvature. This talk is based on joint work with Jianquan Ge and Zizhou Tan.

Date: 12/2, 16:00-17:30

Speaker: 徐飞（首都师范大学）

Title: Brauer-Manin obstruction for rational points

Abstract: Brauer-Manin obstruction for rational points

**Abstract:** One of the fundamental problem for number theory is to decide if an algebraic variety over rational numbers has a rational point. In this talk, I will explain how Manin introduced an obstruction for existence a rational point by using Brauer group of the variety. I'll also report recent progress on this subject.

Date: 11/25, 16:00-17:30

Speaker: 夏超（厦门大学）

Title: Reilly type formula and applications

Abstract:
Bochner’s technique is one of most useful techniques in geometric analysis. In this talk, I will focus on the applications of integral version of Bochner's formula on manifolds with boundary, which is referred to Reilly’s formula, on geometric inequalities. Initiated by recent interest in studying geometric inequalities on warped product manifold, I will discuss my recent work on generalized Reilly type formula for sub-static manifolds. I will explain the formula from two points of view. One reflects a Bochner type technique on affine connection, the other reflects that in static space-time. Applications on geometric inequalities will be presented. The talk is based on joint work with Junfang Li at Birmingham.

Date: 11/11, 16:00-17:30

Speaker: 肖维（深圳大学）

Title: Dirac cohomology and Euler-Poincare pairing for weight modules

Abstract: Let g be a reductive Lie algebra over C. For any simple weight module of g, we show that its Dirac cohomology is vanished unless it is a highest weight module. This completes the calculation of Dirac cohomology for simple weight modules since the Dirac cohomology of simple highest weight modules was carried out in our previous work. We also show that the Dirac index pairing of two simple weight modules agrees with their Euler-Poincare pairing. The analogue of this result for Harish-Chandra modules is a consequence of the Kazhdan's orthogonality conjecture which was proved by Jingsong Huang and Binyong Sun.

Date: 10/28, 16:00-17:30

Speaker: 张通（Durham University）

Title: Severi inequality for varieties of maximal Albanese dimension

Abstract: The so-called Severi inequality for complex surfaces of maximal Albanese dimension dates back to a paper of Severi himself in 1932, in which a gap was found afterwards. It is Pardini who finally gave a complete proof based on the covering trick and the slope inequality of Xiao. In 2009, Mendes Lopes and Pardini proposed a question about generalizing the classical Severi inequality to higher dimensions. In this talk, I will first introduce the classical Severi inequality and explain the above two ingredients in Pardini's proof. Then I will talk about a characteristic p>0 version of the Severi inequality which, a bit unexpectedly, gives a way to the Severi inequality in arbitrary dimension.

Date: 10/21, 16:00-17:30

Speaker: 刘洋（微软亚洲研究院）

Title: Optimal Polyhedral Meshing

Abstract: Polyhedral meshes -- discretized versions of continuous shapes or domains, are the most common representation used in computer graphics and computational geometry. The generation of polyhedral meshes with desired geometric properties is essential to many applications. In the talk, I will give an introduction to Optimal Polyhedral Meshing and also report my work on this research direction. The topics include optimal isotropic/anisotropic tessellation, optimal meshes for architectural modeling and construction.

Date: 10/21, 14:00-15:30 (Different Time!)

Speaker: 冯如勇（中国科学院数学与系统科学研究院）

Title: Computation of the Galois group of linear differential equations

Abstract: Differential Galois theory is a differential analogue of ordinary Galois theory. This theory studies the Galois group of linear differential equations. In this talk, we will first give a brief introduction on this theory and then present a method to compute the Galois groups.

Date: 10/14, 16:00-17:30

Speaker: 黄勇（湖南大学）

Title: Minkowski Problems of Convex Bodies

Abstract: In this talk, the devolopment of Minkowski problems of convex bodies during the past 120 years will be reported. My talk includes two parts:
Part I: Geometric measures in the Brunn-Minkowski theory;
Part II: Solving Minkowski problems.
A recent joint work with Erwin Lutwak, Deane Yang and Gaoyong Zhang will be particularly discussed, and some open problems will be mentioned too. The main contents are based on the following references:
(1) Y. Huang, E. Lutwak, D. Yang, and G. Zhang, Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems, Acta Math, (2016), in press.
(2) R. Schneider, Convex Bodies: the BrunnCMinkowski Theory. Encyclopedia of Mathematics and its Applications, 151. Cambridge Univ. Press, Cambridge, 2014.

Date: 10/14, 9:00-10:30 (Different Time!)

Speaker: 于品（清华大学）

Title: Shock formations for 3-dimensional wave equations

Abstract: We present a mechanism of shock formation for a class of quasilinear wave equations. The proof is based on energy estimates and on the study of the differential geometry defined by the solution.

Date: 9/26, 4:00-5:30 (Different Day!)

Speaker: 张振雷（首都师范大学）

Title: Relative isoperimetric inequality

Abstract: In the talk I will introduce a notion of relative isoperimetric constant for subdomains of a manifold and explain its relation with the classical Sobolev inequality and Poincare inequality. Then I will present a geometric approach to estimate the constant, with applications to various curvature conditions.

Date: 9/18, 4:00-5:30

Speaker: 李长征（中山大学 & 韩国基础科学研究所）

Title: On the Conjecture O for G/P

Abstract: In this talk, we will discuss the Conjecture O of Galkin, Golyshev and Iritani, which‘underlies’ Gamma conjectures I and II of them. We will prove the conjecture for homogeneous varieties G/P, which is concerned with the eigenvalues of the operator on the small quantum cohomology of G/P given by the quantum multiplication of the first Chern class of G/P. This is my joint work with Daewoong Cheong.

Date: 9/18, 10:00-11:30 am (Different Time!)

Speaker: 申旭（中国科学院数学与系统研究所）

Title: Rapoport-Zink uniformization of Shimura varieties

Abstract: In this talk, we discuss some arithmetic geometrical structures of Shimura varieties. We will start with some review on Drinfeld's construction of p-adic uniformization for Shimura curves associated to an indefinite quaternion algebra over Q which is ramified at p. Then we will introduce the most general Rapoport-Zink spaces, and we will explain how to use them to uniformize certain pieces of Shimura varieties of abelian type.