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--Lie group theory.

Algebraic structure theory, representation theory, number theory and automorphic forms, and combinatoric methods to establish the significant role of Lie groups in modern mathematics and physics, as well as the applications of the theory in analysis, differential geometry, topology and quantum mechanics in physics. 

--Modern dynamical systems.

Complex analysis and complex algebraic geometry, high-dimensional complex dynamical systems and the field theory of conformal invariance from theoretical physics and the induced interdisciplinary research on various special geometric structures. 

--Nonlinear partial differential equations and geometric analysis.

Harmonic analysis, partial differential equations especially various fluid equations. Conformal geometry, fully nonlinear PDE and its applications on Geometry and Physics, complex geometry and metric geometry, Riemann and Finsler Geometry.

--Inverse problems and imaging.

Inverse problems in partial differential equations, optics and wave propagation,reconstruction theory and applications in medical imaging. Significant contribution in applied research areas such as inverse reconstruction, compressive sensing, graphics and imaging, and industrial optimal control.

--Modern limit theorems in probability and statistics.

cutting-edge theoretical research combining Big Data, financial data modeling and analysis, stochastic matrix theory and applications, and medical trial statistics.

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