I am happy to supervise students at different levels. Please read below for more descriptions of my field. If you are interested, please feel free to contact me and introduce yourself with 1) CV; 2) Transcripts; 3) English results (optional); 4) Other documents (optional), e.g., publications, certificate.
My field is the interplay between (harmonic) analysis and partial differential equations, in particular nonlinear dispersive equations. It also has many interactions with other fields such as computational mathematics, probability, fluid mechanics and signal processing etc.
The logo indicates pretty much my field.
Dispersion phenomena + Whitney decomposition of a prism
Dispersive PDE
The terminology “dispersive” is from optics. The well-known dispersion phenomenon in optics is caused by the fact that in the prism lights with different colours (frequency) have different propagation speeds. In mathematical language, the phase velocity of a wave depends on its frequency. See here for more explanation.
Dispersive PDE forms an important class of evolution equations (the unknown functions depend on time variable t and space variables x) which arise in mathematical physics, geometry and other fields. It studies the propagation of waves, but with a variable propagation speed depending on the frequency of the wave. Dispersive PDE is different from the traditional classes (e.g. elliptic, parabolic and hyperbolic types). It has some similar properties to the hyperbolic type such as wave equations which have constant propagation speed. It contains a large class of Hamiltonian PDE, such as the Schrodinger equation in quantum mechanics, and the Korteweg-de Vries equation in fluid mechanics.
For a PDE, a fundamental task is to understand its solutions. We study the qualitative and quantitative properties of the solutions through strict analysis without solving the equation explicitly.
Low regularity well-posedness
existence, uniqueness and continuous dependence
Global in time well-posedness VS blow up in finite time
Long-time dynamics for global solutions
Formation of singularities for blow-up solutions
Special solutions, e.g., solitary solutions, self-similar solutions and their properties
Global perspective as a dynamical system, instead of individual solutions
Stochastic aspects of dispersive equations
Numerical methods
Harmonic analysis
The central roles in harmonic analysis are various operators that have general backgrounds and motivations from analysis, PDE, and other fields. The purpose is to study the behaviour of these operators, e.g. boundedness, limiting behaviour. Usually, in the study one applies the "divide and conquer" strategy. First divide the operators into sub-pieces according to some patterns, then conquer each sub-piece using different strategies. A typical tool is the "Whitney decomposition" performed to the prism. Tools like this help to understand the dispersion phenomena. Harmonic analysis also provides many tools for "hard analysis" of many problems, where some subtle structures play crucial roles and need to be exploited:
Cancellation
singular integral operator, commutator, compensated compactness (NULL structure)
Orthogonality
almost orthogonality, square root cancellation, decoupling inequality (orthogonality in the non-Hilbert space, e.g. Lp space)
Oscillation
Oscillatory integral operator
Function spaces
Hardy space, BMO space, adaptive function spaces associated to the operators.
Interpolation techniques
In particular, the solution map of a nonlinear PDE is a nonlinear operator. Harmonic analysis tools naturally play crucial roles in studying it, e.g., a core task is to design some adaptive function spaces to the PDE in consideration.
Scholarships are available for competitive students
with strong mathematical background (e.g. high GPA, award, publications)
meeting the English requirements: IELTS 6.5 (each band 6.0), or equivalent
I am happy to supervise a research project for master-of-mathematcs. About the program, see here.
See here for some honours projects.
I am happy to supervise a research project for MTH3000. About the program, see here.
Summer and Winter Vacation Research (Monash only)
For Monash students, there is a Summer and Winter Vacation Research Scholarship Program. See here. I am occasionally available to supervise a research project.