# Research Interests

Mathematical research by the Department of Mathematics and the IMS covers many branches of pure and applied mathematics. They comprise the research areas shown below.

Don't forget to check out our teachers' personal pages and archives of our seminars, conferences and journal publications.

Members of the Algebra / Number Theory group at CUHK perform world-class research at the frontier on a wide range of topics in algebraic geometry, automorphic forms, number theory and representation theory. Current research focuses of the group include:

- algebraic cobordism theory,
- finite group actions in algebraic geometry and arithmetic geometry,
- group schemes,
- Neron models,
- mass formulas,
- constructions and classification of supercuspidal representations,
- functorial lifting of automorphic representations,
- trace formulas,
- Gromov-Witten theory,
- quantum cohomology,
- homological mirror symmetry,
- toric varieties,
- ADE bundles over complex surfaces

The Algebra / Number Theory group has close connections with other institutions in Hong Kong, as well as universities and research institutes all over the world. Every semester we invite numerous world-leading experts to speak at seminars and workshops. Our group members actively collaborate with overseas scholars and receive regular invitations to speak at international conferences and workshops. We publish high-quality papers in top-notch journals such as the *Annals of Mathematics*, the *Journal of AMS* (American Mathematical Society), *Inventiones Mathematicae* and the *Duke Mathematical Journal*. Among our members are Fellows of the American Mathematics Society and Chern Prize laureates. Our group also works closely with the Geometry and Topology and PDE groups at CUHK.

Research Interests

Prof. Kwok Wai CHAN:

- SYZ mirror symmetry,
- homological mirror symmetry,
- open Gromov-Witten theory,
- (generalized) complex geometry,
- quantum field theory, and
- toric varieties.

Dr. Ping Shun CHAN:

- automorphic representations, and
- representations of
*p*-adic groups.

Prof. Conan Nai Chung LEUNG:

- Calabi-Yau geometry,
- closed and open Gromov-Witten invariants,
- SYZ conjecture for Mirror Symmetry,
- Witten-Morse theory,
- quantization,
- Rozansky-Witten invariants,
- hyperkähler geometry,
- geometry of special holonomy,
- ADE bundles over complex surfaces.

Dr. Charles Chun Che LI:

- trace formulas,
- automorphism forms, and
- representation theory.

Dr. Kelvin Chun Lung LIU:

- algebraic cobordism,
- group actions in algebraic geometry.

Prof. Jiu Kang YU: (to be confirmed)

The members of the analysis group cover a broad range of interests in mathematical analysis. Areas as diverse as real analysis, complex analysis, functional analysis and harmonic analysis are all represented. We also interact frequently with the partial differential equations group and the geometry group.

Research Interests

Prof. Dejun FENG:

- Ergodic theory,
- Dynamical systems, and
- Fractal geometry.

Prof. Ka Sing LAU:

- Functional analysis,
- Harmonic analysis,
- Wavelets,
- Analysis on fractals, and
- Applied probability.

Prof. Chi Wai LEUNG:

- Functional analysis, and
- Operator algebras.

Prof. Kung Fu NG:

- Functional analysis,
- Optimization, and
- Convex analysis.

Prof. Po Lam YUNG:

- Harmonic analysis,
- Several complex variables, and
- Partial differential equations.

The members of the computational mathematics group perform world-class research in a variety of areas including

- numerical linear algebra,
- numerical partial differential equations,
- computational differential geometry,
- imaging and image processing,
- fast solvers and domain decomposition methods,
- inverse and ill-posed problems,
- multiscale problems,
- operations research,
- financial mathematics, as well as
- bioinformatics.

The research focuses on the development of advanced numerical algorithms for a wide range of applications and their mathematical foundations.

The members publish high-quality papers in top-tier mathematical, financial, medical and physical journals, and are invited speakers in many international conferences. The group's research findings give fundamental and influential contributions to the relevant fields, and are recognized by many distinctions, honors and awards. The members also serve in the editorial boards of top international journals and panels of many funding agencies.

The group has trained outstanding graduate students who receive competitive PhD, postdoctoral and faculty positions in top institutions in Asia, Europe and USA, and are the recipients of many distinguished awards.

Research interests

Prof. Raymond Honfu CHAN:

- numerical linear algebra,
- Toeplitz solvers,
- image processing,
- queuing networks,
- financial mathematics, and
- bioinformatics.

Prof. Eric Tsz Shun CHUNG:

- discontinuous Galerkin methods,
- computational wave propagation,
- fluid flow in heterogeneous media,
- multiscale model reduction techniques,
- adaptivity for multiscale problems,
- domain decomposition methods,
- seismic imaging, and
- travel time tomography.

Prof. Ronald Lok Ming LUI:

- computational differential geometry,
- mathematical geometry processing,
- medical imaging,
- computer vision,
- 3D imaging, and
- surface registration problems.

Prof. Xiaolu TAN:

- mathematical finance,
- stochastic analysis,
- stochastic optimal control,
- numerical simulation of SDEs,
- numerical method for nonlinear PDEs,
- numerical method for optimal control problem.

Dr. Jeff Chak Fu WONG:

- computational fluid dynamics,
- inverse problems,
- mathematical biology, and
- optimal control and estimation.

Prof. Tieyong ZENG:

- image processing
- optimization
- artificial intelligence
- scientific computing
- computer vision
- machine Learning
- inverse problems

Prof. Jun ZOU:

- numerical solutions of electromagnetic Maxwell systems,
- numerical solutions of interface problems,
- ill-posed problems,
- inverse problems,
- preconditioned iterative methods, and
- domain decomposition methods.

The Geometry and Topology Group at CUHK is represented by active members who are world-class leaders in their fields of research on a wide range of topics in geometry and topology. It covers the areas of differential geometry, algebraic and complex geometry, geometric analysis, low-dimensional topology, symplectic geometry and mirror symmetry, geometric measure theory, nonlinear partial differential equations and mathematical physics. The current research focuses of the group include the following:

- ADE bundles over complex surfaces
- Geometric flows:
- (Kähler)-Ricci flow on compact and complete manifolds,
- mean curvature flow and calibrated geometry,
- curve shortening flows.

- Geometric partial differential equations:
- harmonic maps,
- Allen-Kahn equations and de Giorgi's conjecture,
- Yamabe equations,
- nonlinear Klein-Gordon equations,
- Yang-Mills fields in gauge theories,
- Minkowski problems and Monge-Ampère equations.

- Geometric topology:
- 3- and 4- dimensional manifolds,
- knot theory,
- symplectic and contact topology and geometry,
- gauge theory,
- Floer homology,
- Rozansky-Witten invariants.

- Mathematical general relativity:
- quasi-local mass,
- scalar curvature rigidity and
- positive mass theorems.

- Minimal surfaces:
- min-max constructions using geometric measure theory,
- free boundary minimal surfaces,
- classical surface theory for constant mean curvature surfaces in Euclidean spaces,
- gluing constructions of minimal surfaces.

- Mirror symmetry:
- SYZ conjecture,
- quantum cohomology,
- quantization,
- Gromov-Witten and Seiberg-Witten invariants.

- Optimal transportation:
- regularity of Monge-Ampère and Monge-Ampère type equations,
- applications from convex analysis and geometric measure theory,
- applications to geometric inequalities and PDE.

- Special geometry:
- G_2 and Spin(7)-geometries,
- Donaldson-Thomas theory,
- Topological Quantum Field Theory (TQFT).

- Sub-Riemannian geometry and integrable systems

The Geometry and Topology Group has close connections with the Institute of Mathematical Sciences (IMS) and other institutions in Hong Kong, Asia, North America and Europe. There are regular seminars by local and foreign speakers and the quarterly Geometry Colloquium co-organized with HKU and HKUST. The Group also has strong collaborative ties with many overseas scholars and is actively engaging in international conferences and workshops. Graduate students and postdocs make up a valuable component of the Group and many of them have gone on to PhD programs and academic positions in top universities, including Harvard and Stanford. Our members receive prestigious awards and honors, including the Fields Medal, Veblen Prize and Wolf Prize, and some of us are Fellows of the American Mathematics Society. Our group works closely with the Algebra and PDE Groups at CUHK.

Research interests

Prof. Thomas Kwok Keung AU:

- low-dimensional topology,
- curve shortening flows.

Dr. Kai Leung CHAN:

- symplectic geometry,
- toric geometry,
- mirror symmetry,
- SYZ mirror symmetry conjecture.

Prof. Kwok Wai CHAN:

- SYZ mirror symmetry,
- homological mirror symmetry,
- open Gromov-Witten theory,
- (generalized) complex geometry,
- quantum field theory,
- toric varieties.

Dr. Leung Fu CHEUNG:

- submanifolds,
- minimal surfaces,
- stable constant mean curvature hypersurfaces,
*p*-harmonic maps.

Dr. Chi Hin LAU:

- rational curves on projective varieties,
- variety of minimal rational tangents.

Prof. Yi-Jen LEE:

- gauge theory and symplectic topology, e.g. Seiberg-Witten theory,
- Heegaard Floer homology,
- pseudo-holomorphic curves.

Prof. Conan Nai Chung LEUNG:

- Calabi-Yau geometry,
- closed and open Gromov-Witten invariants,
- SYZ conjecture for Mirror Symmetry,
- Witten-Morse theory,
- quantization,
- Rozansky-Witten invariants,
- hyperkähler geometry,
- geometry of special holonomy,
- ADE bundles over complex surfaces.

Prof. Martin Man Chun LI:

- free boundary problems for minimal surfaces,
- Almgren-Pitts min-max theory,
- geometric measure theory,
- extremal eigenvalue problems,
- geometry of positively curved spaces,
- positive mass theorems and mathematical relativity,
- self-shrinkers in mean curvature flow,
- gluing constructions and geometric partial differential equations.

Prof. Luen Fai TAM:

- heat flow,
- Ricci flow and Kähler-Ricci flow,
- complex structure and curvature in Kähler geometry and Hermitian geometry,
- function theory,
- harmonic maps,
- mathematical general relativity,
- quasi-local mass,
- geometry of compact manifolds with boundary.

Prof. Tom Yau Heng WAN:

- geometric analysis and nonlinear partial differential equations, in particular harmonic maps between noncompact manifolds,
- mean curvature flow and other geometric flow of submanifolds,
- tropical geometry and its application to Calabi-Yau structure of general type submanifolds.

Prof. Zhongtao WU:

- knot theory and its invariants,
- 3-manifolds and Dehn surgery,
- contact 3-manifolds,
- smooth and symplectic 4-manifolds,
- Heegaard Floer homology and other gauge theoretic or holomorphic curve invariants.

Prof. Yong YU:

- geometric flows,
- fluid dynamics and liquid crystal flows.

Partial differential equations (PDEs) provide powerful tools in understanding problems stemming from geometry and various physical systems.

The principal areas involved by the PDE group at the Chinese University of Hong Kong concentrate on the study of geometric PDEs, continuum mechanics, kinetic theory and equations from biology, with the incorporation of methods from differential geometry, topology, mathematical analysis and numerical analysis. Themes are extremely varied, ranging from abstract questions (existence, uniqueness and stability of solutions) to more concrete ones (qualitative and quantitative information on behaviors of solutions, often in relation with numerical simulations).

A detailed list of domains where our faculty is engaged in is provided as follows:

1. Geometric PDEs: This area includes

- various geometric flows,
- harmonic maps between manifolds,
- equations related to geometric curvatures,
- PDE problems arising from quantum field theory,
- PDE problems arising from optimal transportation,
- minimal surface with or without free boundary,
- Minkowski problem in centroaffine geometry,
- integrable equations and mathematical general relativity.

Faculty involved in these areas include:

- Prof. Thomas Kwok Keung AU;
- Dr. Leung Fu CHEUNG;
- Prof. Kai Seng CHOU;
- Prof. Yi-Jen LEE;
- Prof. Martin Man Chun LI;
- Prof. Luen Fai TAM;
- Prof. Tom Yau Heng WAN;
- Prof. Yong YU;
- Prof. Po Lam YUNG.

2. Continuum mechanics: Most research studies in this area focus on

- compressible and incompressible Navier-Stokes equations,
- compressible and incompressible Euler equations,
- mathematical problems about transonic shocks,
- free boundary problems arising in fluid dynamics,
- liquid crystal flows,
- magnetohydrodynamic equations, and
- equations of thin film type.

Faculty involved in these areas include:

- Prof. Kai Seng CHOU;
- Prof. Renjun DUAN;
- Dr. Jeff Chak Fu WONG;
- Prof. Zhou Ping XIN;
- Prof. Yong YU.

3. Kinetic theory and equations of hyperbolic type: This area includes

- the Boltzmann equation and related kinetic equations,
- nonlinear wave equations, and
- mathematical general relativity.

Faculty in this area include:

- Prof. Renjun DUAN;
- Prof. Zhou Ping XIN;
- Prof. Po Lam YUNG.