数学科学系

Department of Mathematical Sciences

活动简介： 清华大学数学科学学术节为数学系和丘成桐数学科学中心联合举办的学术活动，由部分科研前线的教师介绍相关领域的研究现状及其科研工作，旨在增进师生之间的了解以及教师之间的交流。届时在报告中将安排茶歇及自由讨论，希望广大师生踊跃参加，在活动中有所收获。

时间： 12月10日全天

地点： 郑裕彤讲堂

**Schedule**

** 2016年12月10日**

**Session 1**

8:30-9:00 于品 从波方程的观点看引力波与黑洞

9:05-9:35 吴昊 地震定位问题的数学理论

9:40-10:10 Babak Haghighat 6d SCFTs and their tensionless strings

10:10-10:40 **Coffee Break**

**Session 2 **

10:40-11:10 何凌冰 Sharp bounds for Boltzmann and Landau collision operators

11:15-11:45 陈宗彬 Modular forms, modular curves and Ramanujan conjecture

12:00-13:30 **Lunch**

**Session 3**

14:00-14:30 刘思齐 Matrix Integral, Hodge Integral, and Integrable system

14:35-15:05 荆文甲 Homogenization theory, inverse problems and their interactions

15:10-15:40 陈大广 The estimates of the gaps of consecutive eigenvalue of

Lapalcian

15:40-16:10 **Coffee Break**

**Session 4**

16:10-16:40 宗正宇 Geometry and Physics

16:45-17:15 陈酌 From Atiyah Classes to Homotopy Leibniz Algebras

** Abstract**

于品

题目：从波方程的观点看引力波与黑洞

摘要：Einstein 的场方程是广义相对论中描述时空中物质分布的最重要的方程。从数学的角度来看，可以将它视为一个有十个变量十个方程的双曲型偏微分方程组。我们将讲述如何将场方程理解为波动方程（尝试从数学上“定义”“引力波”）并且应用这样的理解来阐述黑洞形成的机制。

吴昊

题目：地震定位问题的数学理论

摘要：我们将介绍地震定位问题基本背景和优化模型。以此为基础，我们给出了三种全新的地震定位方法。以上介绍方法比传统方法具有巨大优势，并且有望用于地震的实时定位和全球地震重定位。这一系列工作是与杨顶辉教授合作的。

Babak Haghighat

Title: 6d SCFTs and their tensionless strings

Abstract:Six-dimensional SCFTs (superconformal field theories) are the quantum field theories with the maximal amount of supersymmetry in the highest dimension which at the same time admit conformal symmetry. Despite the unique status they enjoy, and despite the fact that they have been instrumental in construction lower dimensional theories, they remain among the least understood theories. This is mainly related to the fact that we do not have a Lagrangian description of these theories. Moreover, if we go slightly away from the conformal point we get a theory of interacting almost tensionless strings. This talk aims at illuminating the role these strings play in understanding the quantum Hilbert space of these 6d theories.

何凌冰

Title: Sharp bounds for Boltzmann and Landau collision operators

Abstract: In this talk, we will provide a stable method to get sharp bounds for Boltzmann and Landau operators in weighted Sobolev spaces and in anisotropic spaces. The main ingredients are two types of dyadic decompositions performed in both phase and frequency spaces and also the geometric decomposition to catch the main structure of the operator. As applications, we will show that our results are related closely to the Cauchy problem and asymptotic problem for the Boltzmann equation.

陈宗彬

Title: Modular forms, modular curves and Ramanujan conjecture

Abstract: Firstly, I will give a sketch of the work of Serre, Kuga-Shimura and Deligne on the Ramanujan conjecture. This is a typical example of how to associate a Galois representation to an automorphic form. Then I will try to explain how to generalize this construction to higher dimension using Shimura variety and the trace formula.

刘思齐

Title:Matrix Integral, Hodge Integral, and Integrable system

Abstract:Matrix integral is a classical topic in mathematics. It is introduced by physicist E. Wigner, and has many interesting applications in physics, probability theory, mathematical statistics, numerical analysis, and number theory. It is revealed by the celebrated Witten conjecture that matrix integral is also the bridge among two-dimensional quantum gravity, the moduli space of stable curves, and the Korteweg-de Vries (KdV) hierarchy. Hodge integrals are the integrals of certain natural cohomological classes on the moduli space of stable curves, which are very important in modern mathematical physics. In our previous work, we showed that Hodge integral is also related to a certain generalization of the KdV hierarchy. We also conjectured a mysterious relation between matrix Integral and Hodge Integral. Recently, we proved this conjecture. This is a joint work with Boris Dubrovin, Di Yang, and Youjin Zhang.

荆文甲

Title: Homogenization theory, inverse problems and their interactions

Abstract: Partial differential equations with highly oscillatory coefficients arise in many applications, and the fine-scale oscillations of the physical media are often unknown and, hence, modeled as random. The theory of homogenization and its quantitative versions amount to finding large scale approximations of such equations, and to characterizing the fine-scale, often random, fluctuations around these mean approximations. PDE based inverse problems, loosely speaking, aim at reconstructing the coefficients of PDEs from knowledge of their solutions. They are ill-posed in general, but in some practical settings and equipped with sufficient prior information, they can be solved by exploring the structures of the solutions to the forward problems. I will introduce some research topics that I have been working on in these fields, and point to some interesting connections between them.

陈大广

Title: The estimates of the gaps ofconsecutive eigenvalue of Lapalcian

Abstract: In this talk, we will talk about the estimates of the gaps ofconsecutive eigenvalue of Lapalcian. For the eigenvalue problem of the Dirichlet Laplacian on the bounded domain in Euclidean space $\mathbb R^n$, we obtain the estimates for the upper bounds of the gap of consecutive eigenvalues, which are the best possible in the meaning of the orders of eigenvalues.

This is the joint work with Professor Hongcang Yangand Dr. Tao Zheng.

宗正宇

Title: Geometry and Physics

Abstract: In this talk, I will explain the beautiful connection between modern high energy physics and geometry. One motivation for mathematians to become interested in physics phenomena and build the corresponding mathematics theories comes from the various dualities in physics. These physcis dualities give highly nontrivial predictions in mathematics such as mirror symmetry, Gromov-Witten/Donaldson-Thomas correspondence, and so on. I will introduce several modern enumerative geometry including Gromov-Witten theory, Donaldson-Thomas theory, stable pairs, and so on, which play central roles in modern mathematical physics. Then I will discuss the mathematical aspect of mirror symmetry, Gromov-Witten/Donaldson-Thomas correspondence and other interesting dualities. These beautiful areas involve many subjects in mathematics including algebraic geometry, symplectic geometry, topology, categorification, representation theory, homological algebra, and so on.

陈酌

Title: From Atiyah Classes to Homotopy Leibniz Algebras