## 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

Spring 2014: CLICK HERE

## Past Talks

Date: Jan. 6, 2014

Speaker: Chunjing XIE (上海交通大学)

Title: Steady subsonic solutions for the Euler equations

Abstract: In this talk, I will discuss the well-posedness theory for subsonic flows for steady Euler equations. The major problems are subsonic flows around a body or through a nozzle. The focus is on the flows with large vorticity.

Date: Dec. 30, 2013

Speaker: Yuan YUAN (Syracuse University)

Title: Geometry of local holomorphic maps

Abstract: Local holomorphic isometries were systematically studied in 50's by Calabi and recently number theorists applied the local holomorphic maps between bounded symmetric domains to attack the problem in number theory, which also motivates our study of such problems. I will describe the main theorems on the global extension and the rigidity for local holomorphic isometries between Hermitian symmetric spaces and explain the relation with CR geometry, minimal rational curves and etc. These are joint works with Yuan Zhang and Professor Xiaojun Huang.

Date: Dec. 23, 2013

Speaker: Min-Chun HONG (University of Queensland) at **2:00-2:50 pm **

Title: Some results on the existence of solutions to the Ericksen-Leslie system

Abstract: The Ericksen-Leslie theory describes the dynamic flow of liquid crystals. The Ericksen-Leslie system is a system of the Navier-Stokes equations coupled with the gradient flow for the Oseen-Frank free energy. In this talk, we will discuss some new results on the local existence, uniqueness and the blow up criterions of strong solutions to the Ericksen-Leslie system for the general Oseen-Frank model in 3D.

Date: Dec. 23, 2013

Speaker: Xinan MA (中国科学技术大学）at **3:00-4:30 pm **

Title: 几何非线性偏微分方程的几个问题

Abstract: 我将尽可能详细介绍我所关心的与微分几何有关的几个问题的研究历史与进展，它们往往与非线性偏微分方程的研究联系在一起。最后我也要提及偏微分方程解的几何性质的研究。
主要方法是最大值原理。

Date: Dec. 16, 2013

Speaker: Zhou ZHANG (The University of Sydney)

Title: Spatial Asymptotic Behaviors of Kahler-Ricci Flows and Kahler-Einstein Metrics

Abstract: In this talk, we mainly consider the Kahler Ricci flow over a complete non-compact manifold. The spatial asymptotic behavior is a natural and important new ingredient comparing with the closed manifold case. For the finite volume case, we have approached this problem from both geometric and analytic points of view. The Kahler-Einstein case is included, illustrating some difference. We have also studied the infinite volume case in a geometric fashion more recently. Overall, the main focus is on the quasi-projective setting. This talk is based on joint works with John Lott and Frederic Rochon.

Date: Dec. 9, 2013

Speaker: Wenxuan LU (清华大学数学科学中心)

Title: Mirror Symmetry and Wall Crossings

Abstract: We will give an overview of the study of wall crossings in the context of mirror symmetry. It is well known that the SYZ program and the homological mirror conjecture are both guiding our understandings of mirror symmetry of Calabi-Yau spaces. If one wants to combine these two pictures then the study of wall crossings is inevitable. From the perspective of differential and algebraic geometry wall crossing is a key to the investigation of the so-called instanton corrections and counting problems. We will present examples from various related directions: Calabi-Yau spaces, Moduli spaces of Calabi-Yau spaces, Hitchin's moduli spaces, gauge theory, black hole attractors, etc.

Date: Nov. 25, 2013

Speaker: Hongxin GUO (温州大学)

Title: On entropy formulas in geometric flows

Abstract: Starting from the classical Boltzmann-Shannon entropy, we construct entropy formula of Perelman's type for an abstract geometric flow. Monotonicity of the entropy holds when a technical assumption is satisfied. This provides a unified framework for entropy formulas in a variety geometric flows, which include the Ricci flow and static Riemannian manifolds with nonnegative Ricci curvature. Moreover, our technique allows us to define a new entropy formula relating Hamilton's surface entropy and Perelman's W entropy for the Ricci flow, and a new entropy for the Gauss curvature flow. This is a joint work with Robert Philipowski and Anton Thalmaier.

Date: Nov. 18, 2013 at **2:00-3:30 pm **

Speaker: Andrey LOSEV (ITEP, Russia)

Title: Segal axioms of Quantume Field Theory and Renormalization

Abstract: I will discuss the Dirac-Segal axioms of QFT as a proper language in modern description of QFT and compare it to functional integral approach. I explain the notion of figure observables and local observables as a particalar case.I will show that local observables are the tangent space to the space of QFT, their transport under deformations of QFT and absence of ultraviolet divergenses.Then I comment the issue of renormalization from this point of view.

Date: Nov. 18, 2013 at **4:00-5:30 pm **

Speaker: Yingyi WU (中国科学院大学)

Title: A non-CSC HCMU metric with cusp singularities and conical singularities

Abstract: An HCMU metric is a conformal metric which has a finite number of singularities on a
compact Riemann surface and satisfies the equation of the extremal Kaehler metric. In this paper we consider a kind of non-CSC HCMU metrics which has two kinds of singularities: cusp singularities and conical singularities. This is a joint work with Prof. Qing Chen and Bin Xu at USTC.

Date: Nov. 11, 2013

Speaker: Xiaojun CHEN (四川大学)

Title: Derived noncommutative Poisson structures and their applications

Abstract: Recently, Crawley-Boevey introduced a notion of noncommutative Poisson structure for an associative algebra $A$ (viewed as a noncommutative space). It has a very good property that it naturally induces a Poisson structure on the moduli space of representations of $A$. Crawley-Boevey called such noncommutative Poisson structure an $H_0$-Poisson structure. In this talk we generalize Crawley-Boevey's result to all higher degrees. It is represented by a Lie bracket on the cyclic homology $\mathrm{HC}_\bullet(A)$ of $A$, which then induces a (graded) commutative Poisson structure on the derived moduli space of representations of $A$. After that, we give several interesting examples and applications in/to algebra and geometry, such as Calabi-Yau algebras and Fukaya category. Joint work with Berest, Eshmatov and Ramadoss.

Date: Nov. 4, 2013 at **2:00-4:00 pm **

Speaker: Huijun FAN (北京大学)

Title: On tt* geometry (II)

Abstract: Fourth talk of A short course in quantum Landau-Ginzburg theory

Date: Oct. 28, 2013 at **4:00-5:30 pm **

Speaker: Huijun FAN (北京大学)

Title: Spectrum theory of Schroedinger operators associated to singularities

Abstract: First talk of A short course in quantum Landau-Ginzburg theory

Date: Oct. 28, 2013 at **2:00-3:30 pm **

Speaker: Fang WANG (上海交通大学)

Title: On the Radiation Field for Wave Equations

Abstract: Definitions for the radiation field by Lax-Phillips and Friedlander are introduced for standard wave equation on Minkowski space-time, which can be generalized to curved space-time (such as Schiwarzschild space-time) and nonlinear wave equations (such as Einstein vacuum equations). Regularities of the radiation field are studied. Mapping property for Moller wave operator, which maps initial data to the radiation field, are investigated. In the case that the Molloer wave operator is an (or a local) isomorphism, the characteristic initial problem is considered.

Date: Oct. 21, 2013

Speaker: Minxin HUANG (中国科学技术大学交叉科学中心)

Title: Refined stable pair invariants for E-, M- and [p,q]-strings

Abstract: We use mirror symmetry, the refined holomorphic anomaly equation and modularity properties of elliptic singularities to calculate the refined BPS invariants of stable pairs on non-compact Calabi-Yau manifolds, based on del Pezzo surfaces and elliptic surfaces, in particular the half K3. The BPS numbers contribute naturally to the five-dimensional N=1 supersymmetric index of M-theory, but they can be also interpreted in terms of the superconformal index in six dimensions and upon dimensional reduction the generating functions count N=2 Seiberg-Witten gauge theory instantons in four dimensions. Using the M/F-theory uplift the additional information encoded in the spin content can be used in an essential way to obtain information about BPS states in physical systems associated to small instantons, tensionless strings, gauge symmetry enhancement in F-theory by [p,q]-strings as well as M-strings.

Date: Oct. 14, 2013

Speaker: Wen HUANG (中国科学技术大学)

Title: Furstenberg Conjecture and related problems

Abstract: We will review some results on Furstenberg Conjecture and discuss some related problems.

Date: Sep. 23, 2013 at **2:00-3:30 pm (Different Time!)**

Speaker: Jiajing ZHANG (四川大学)

Title: Bundles over rational surfaces and Weierstrass cubics

Abstract: This work in progress is joint with Naichung Conan Leung. In this talk, we shall study principal G-bundles over certain rational surfaces S, and describe the relation between the moduli space of the surfaces S, containing D as a possibly singular anti-canonical curve, and the moduli space of certain principal G-bundles over D. We show that these two moduli spaces are isomorphic to each other. When D is singular (nodal or cuspidal Weierstrass cubics), moduli spaces considered here are in fact affine spaces, whose compactifications are very interesting.

Date: Sep. 16, 2013

Speaker: Xiaoguang MA (清华大学数学科学中心)

Title: Dunkl Operators

Abstract: Dunkl operators were introduce by Dunkl in 1989. They generalized the definition of differential operators. In this talk, I will begin with the definition of Dunkl operators and will tell you some applications of these operators.

Date: Sep. 9, 2013

Speaker: Minxian ZHU (清华大学数学科学中心)

Title: Regular representations of quantum groups at roots of unity

Abstract: I will discuss how the quantum function algebra decomposes as a bimodule of the quantized enveloping algebra at a root of unity. If time permits, I will explain how this relates to the structure of vertex algebras of differential operators on a complex semisimple Lie group.