Department of Mathematical Sciences
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
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
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
15:40-16:10 Coffee Break
16:10-16:40 宗正宇 Geometry and Physics
16:45-17:15 陈酌 From Atiyah Classes to Homotopy Leibniz Algebras
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.
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.
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.
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.
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.
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.
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.
Title: From Atiyah Classes to Homotopy Leibniz Algebras