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 . Analysis and its Applications

1. Functional Analysis、Noncommutative Analysis

Group people:Quanhua Xu、Xiaoping Xue

Research contents: a) Operator spaces、Quantum probability、Noncommutative/Classical matingale inequalities、Noncommutative/Classical harmonic analysis、Noncommutative Lp spaces、Banach space geometry.

b) Functional analysis、Optimal and control theory、Collective dynamics.

2. Noncommutative Analysis、Quantum Probability、Quantum Statistics and Applications in Quantum Information

Group people:Ke Li、Zhi Yin         

Research contents: Our research focuses on the mathematical aspect of quantum information. Specifically, we are  interested in quantum probability, quantum statistics, and quantum Shannon theory, which are inherently noncommutative. These research topics have been interacting fruitfully with the theories of operator space, noncommutative probability and random matrix.

3. Harmonic Analysis and its Applications

Group people:Xudong Lai

Research contents: Harmonic analysis and its applications, which include the theory of singular integral operator, elliptic equation boundary value problem on non-smooth domain, discrete Fourier restriction problem.

4. Mathematical Physics

Group people:Zhituo Wang

Research contents: a) Mathematical physics related to quantum field theory, in particular the axiomatic quantum field theory and the constructive renormalisation theory, the goal of which is to establish a mathematical theory of the quantum Yang-Mills theory, and application of renormalization theory to condensed matter physics and probability.

b) Non-commutative geometry, including deformation quantization theory and noncommutative quantum field theory. we would like to use Kontsevich's deformation quantization theory to construct more nontrivial noncommutative manifolds and to construct quantum field theory models on these manifolds.

5. Geometry Analysis and Partial Differential Equations

Group people:Chao Zhang、Zhenan Sui

Research contents:a) Priori estimates for fully nonlinear elliptic and parabolic equations on Riemannian manifolds, which include generalized Hessian type equations, prescribed curvature equations as well as equations originated in conformal geometry; Nonlinear Yamabe problem on noncompact manifolds as well as Bernstein type problems for certain fully nonlinear elliptic equations in exterior domains. 

b) Nonlinear elliptic and parabolic PDEs. The well-posedness of solutions such as weak solutions, very weak solutions, entropy solutions and renormalized solutions, etc; the regularity for nonlinear elliptic and parabolic PDEs including Holder regularity, Harnack inequalities and Calderon-Zygmund type estimates; applications of functional analysis to partial differential equations.

二、Algebras、Combinatorics and Number Theory

1. Number Theory

Group people:Yichao Zhang

Research contents:The theory of modular forms and other closely related theories, including representation theory and Kac-Moody theory. 

2. Combinatorics

Group people:Jingtao Zang

Research contents: A partition of a positive integer n, is a way of writing n as a sum of positive integers, which plays an important role in combinatorics and number theory. The spt-function counts the appearence of the smallest part for each partition. Our research are focus on the properties of the statistics on partitions and the spt-functions.

三、Applied Mathematics 

1. Differential Dynamics and its Applications

Group people:Jian Fang、Zhuchun Li

Research contents:a) Propagation dynamics of nonlinear evolution systems. Our current research focous on the influence of spatiotemproral inhomogeneous envrionment on propagation dyanmics; Modeling and analysis for species invasion.

b) Analysis and applications of collective behaviors of multi-agent systems, in particular, the flocking of interacted particles, synchronization of coupled oscillators. The aim of our study is to understand how autonomous agents organize into an ordered motion using limited environmental information and interactions. We concern the dynamical properties of multi-agent systems for flocking or synchronization, and their applications in variant subjects such as transient stability of power grids, formation control, and pattern recognition. 

2. Computational Mathematics and Optimal theory

Group people:Boying Wu、Wei Bian、Meng Xiong

Research contents:a) Theoretical and algorithmic analysis for different classes of optimization problems. Recently, our research interest mainly lies on the non-smooth, non-convex or even non-Lipschitz continuous optimization problems, which have wide applications in the sparse reconstruction. 

b) High order methods for solving convection-dominated problems using Finite difference and/orfinite volume weighted essentially non-oscillatory (WENO) methods and Discontinuous Galerkin (DG) finite element methods; Numerical analysis: superconvergence of DG and LDG methods.