Complex Analysis

Complex analysis is an important research direction in modern mathematics. Its main areas of study include the theory of value distribution of meromorphic functions and the theory of complex analytic dynamical systems. The theory of value distribution, based on the fundamental theory by Nevanlinna, investigates the distribution of function values and their related properties. Complex analytic dynamical systems study the dynamical properties of function self-iterations, primarily focusing on the dynamical properties of rational functions, polynomials, and transcendental functions.

Harmonic Analysis

Harmonic analysis, also called Fourier analysis, originated from the study of convergence problem of Fourier series. Modern theory of harmonic analysis starts from the Calderon-Zygmund singular integral operator theory in the 1950s, and is closely related to mathematical branch such as partial differential equations. Its core content mainly includes two components: 1. theory of various types of function spaces; 2. properties and applications of singular integral and related operators on function spaces.

Analytic Number Theory

Analytic number theory is a branch of mathematics which mainly uses analytical methods to investigate number theory problems. It originated from the investigation of the distribution of prime numbers, Goldbach's conjecture, Waring's problem and lattice problem. The main research methods include complex variable integral method, circle method, sieve method, exponential sum method, characteristic method, etc. Analytic number theory is related to many important branches of mathematics, such as differential equations, elliptic curves, etc.

Ordinary Differential Equations of Fractional Order

Fractional order operators are nonlocal operators that can effectively characterize the memory and global correlation of physical processes. Ordinary differential equations of fractional order are an extension of ordinary differential equations of integer order, and have wide applications in many fields such as physics, chemistry, mechanics, economics, and biology. This research field mainly studies the existence, uniqueness, stability, and related properties of solutions to ordinary differential problems of fractional order.

Nonlinear Partial Differential Equations

Partial differential equations, as one of an important branch of modern mathematics, play an important role in many disciplines, such as physics, biology, and engineering. Our research mainly concern the well posedness, long-time behaviour, and singular limit of several types of nonlinear partial differential equations which are mostly derived from physics. The main research methods include energy methods, Sobolev space theory, and asymptotic analysis.

Integrable Systems and Their Applications

As an important branch in the realm of mathematical physics, the theory of integrable systems concerns nonlinear differential/difference equations with abundant features. In classical mechanics, there are typical finite-dimensional integrable systems such as the Newton's two-body problem，three integrable spinning tops (the Euler top, the Lagrange top and the Kovalevskaya top) and the geodesic flow of an ellipsoid. Prototypes of infinite-dimensional integrable systems include the Korteweg-de Vries, nonlinear Schrödinger, sine-Gordon and Kadomtsev-Petviashvili equations. The research object of this field is integrable systems in mathematics and physics, with the aims of investigating their algebraic and geometric structures, constructing their solutions and exploring their applications in various fields.

Fractal Geometry and Its Applications

Fractal geometry studies the non-smooth geometric objects, which appears in physical world, the field of natural science, areas of mathematics and engineering problems, such as winding coastline, transforming endless Brownian motion trajectory, and so on.  Fractals are too irregular to be analysed using the traditional calculus or geometry of smooth objects, and we need alternative techniques. Since this field has achieved tremendous applications, it develops rapidly and becomes an active cross discipline research.  

Representation theory of algebras

The current representation theory is often referred to as the representation theory of quivers or as the Auslander-Reiten theory. The final aim of the representation theory is to get the complete description of all indecomposable modules of an algebra up to isomorphism.

Mathematical Physics and Algebraic Geometry

Mathematics and Physics are tightly linked subjects. Besides differential equations, mathematical physics is related to geometric ideas. For example, the applications of algebraic geometry in string theory.