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). This seminar is supported by School of Mathematical Sciences and Institute of Geometry and Physics , USTC

Upcoming Talks

Date: 09/25, 16:00-17:30 管理楼1308
Speaker: 李利平(湖南师范大学数学与统计学院) 
Title:  A brief introduction to representation stability theory
Abstract: Representation stability theory, introduced by Thomas Church and Benson Farb in 2010, explores the asymptotic behavior of a sequence of representations of groups (such as symmetric groups, general linear groups, etc.), and is widely applied to investigate properties of (co)homology groups of many important examples like configuration spaces, congruence subgroups, mapping class groups in geometric group theory, algebraic topology and algebraic geometry. Recently this theory was categorified via introducing some infinite discrete categories with particular combinatorial structure and stuying their representations. In this talk I will describe the motivation, background, current developent of this theory, as well as my own contribution.
Date: 10/23, 16:00-17:30 五教5406
Speaker: 刘晔(西交利物浦大学) 
Title: Tutte多项式及其一般化 
Abstract: Tutte多项式是拟阵的一个重要不变量。在超平面配置理论中,Tutte多项式包含了超平面配置的组合和拓扑信息。本次报告我们回顾这些结果,并介绍Moci的算术Tutte多项式在环面配置理论中的应用,以及Liu-Tran-Yoshinaga的G-Tutte多项式及其应用。
Date: 11/6 ,16:00-17:30 五教5406
Speaker: Yong Wei(USTC) 
Title: Curvature flows of hypersurfaces and geometric inequalities 
Abstract:Curvature flows of hypersurfaces are characterized by a family of hypersurfaces evolving in an ambient manifold with velocity determined by their extrinsic curvatures. The equations that arise are nonlinear parabolic differential equations. The curvature flows of hypersurfaces have many applications including the proof of some sharp geometric inequalities. In this talk, I will describe some recent work on this topic, with focus on the applications of curvature flows in the proof of isoperimetric type inequalities in Euclidean space and in hyperbolic space.  
Date: 11/13 , 7:50-9:20 腾讯会议账号:950 391 9321 ; 密码112358
Speaker: Jiyuan Han(Purdue University) 
Title: Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci soliton equations 
Abstract:Let (X,D) be a log variety with an effective holomorphic torus action, and Θ be a closed positive (1,1)-current. For any smooth positive function g defined on the moment polytope of the torus action, we study the Monge-Ampere equations that correspond to generalized and twisted Kahler-Ricci g-solitons. We prove a version of Yau-Tian-Donaldson (YTD) conjecture for these general equations, showing that the existence of solutions is always equivalent to an equivariantly uniform Θ-twisted g-Ding-stability. When Θ is a current associated to a torus invariant linear system, we further show that equivariant special test configurations suffice for testing the stability. Our results allow arbitrary klt singularities and generalize most of previous results on (uniform) YTD conjecture for (twisted) Kahler-Ricci/Mabuchi solitons or Kahler-Einstein metrics. This is a joint work with Chi Li.
Date: 11/20,7:50-9:20 腾讯会议账号:950 391 9321 ; 密码112358
Speaker: Charles Cifarelli (UC Berkeley) 
Title: Shrinking gradient K\"ahler-Ricci solitons and toric geometry 
Abstract:Toric manifolds form an important class of complex manifolds with large symmetry. For compact manifolds, there is a well-known procedure which exploits this symmetry to better understand invariant K\"ahler metrics. I will give a brief survey of these results on a compact manifold $M$ and then move on to study the situation when $M$ is non-compact, with an emphasis on understanding shrinking gradient K\"ahler-Ricci solitons. There is a rich theory for such metrics in the compact setting, but the non-compact case is less well understood. In this talk, after providing some background, I will describe my recent work on the uniqueness of shrinking gradient K\"ahler-Ricci solitons on non-compact toric manifolds.  
Date: 11/27,7:50-9:20 腾讯会议账号:950 391 9321 ; 密码112358
Speaker: Weiyong He(University of Oregon)
Title: Harmonic and biharmonic almost complex structures 
Abstract:Harmonic almost complex structures was introduced by C. Wood in 1990s. We study the existence and regularity of weak harmonic almost complex structure from analytic point of view, as a tensor-valued version of harmonic maps. We introduce the notion of biharmonic almost complex structures, in particular in dimension four. We prove that biharmonic almost complex structures are smooth, and there always exist an energy-minimizing biharmonic almost complex structure. Moreover, given a homotopy class, we prove existence results which depends on whether M is spin or not. We also propose a conjecture that the homotopy class of energy-minimizing biharmonic almost complex structures do not depend on a generic background metric.
Date: 12/4,15:30-17:00 五教5406
Speaker: Mao Sheng (USTC)
Title: Motivicity vs Periodicity 
Abstract:In this talk I shall explain what the two notions in the title mean and why they should be important in algebraic geometry.
Date: 12/4,8:10-9:40 腾讯会议账号:950 391 9321 ; 密码112358
Speaker: Lu Wang (California Institute of Technology)
Title: Nonuniqueness Questions in Mean Curvature Flow 
Abstract:Mean curvature flow is the gradient flow of area functional that decreases the area in the steepest way. In general the flow will develop singularities in finite time. It is known that there may not be a unique way to continue the flow through singularities. In this talk, we will discuss some global features of the space of mean curvature flows that emerge from cone-like singularities. This is joint with Jacob Bernstein. 
Date: 12/11,8:10-9:40 腾讯会议账号:950 391 9321 ; 密码112358
Speaker: Yuanqi Wang (University of Kansas)
Title: Singular G_{2}-instantons and the associated Spectral Theory 
Abstract:We establish a relation between the spectrum of a Dirac operator on S^{5} and certain sheaf cohomologies on the complex projective plane. This operator comes from the deformation of G_{2}-instantons with 1-dim singularities.  
Date: 12/22,9:00-10:00 腾讯会议账号:950 391 9321 ; 密码112358
Speaker: Changjian Su (University of Toronto)
Title: Motivic Chern classes of Schubert cells and applications 
Abstract:The motivic Chern class in K-theory is a natural generalization of the MacPherson class in homology. In this talk, we will talk about several applications of the motivic Chern classes of the Schubert cells. These classes can be used to give a smoothness criterion for the Schubert varieties, which is used to prove several conjectures of Bump-Nakasuji-Naruse about representations of p-adic dual groups and also conjectures of Lenart-Zainoulline-Zhong about Schubert classes in hyperbolic cohomology of flag varieties. The Euler characteristics of these classes are also related to the Iwahori-Whittaker functions of the dual groups. Based on joint works with P. Aluffi, C. Lenart, L. Mihalcea, J. Schürmann, K. Zainoulline, and C. Zhong.