"Through my whole life my ambition as a mathematician, or rather my passion and joy, has ever been to uncover self-evident truths." Alexander Grothendieck
This webpage is primarily written for those Chinese students who might be naive enough to try to pursue his/her PhD degree under my guidance.
In general, I always discourage students to pursue an advanced degree in pure mathematics or pure physics. I have seen so many talented people who have failed in their dream on pure math or pure physics, that is one reason. The other reason is that mathematics is becoming more and more technical and specialized, so it is not as appealing as it used to be. The third reason is that it is becoming more and more political and commercialized.
However, if you are talented in mathematics, truly interested in mathematics, want to have a relatively free life, do not want to work too hard for having a reasonably comfortable life, perhaps an advanced degree in mathematics is what you need. Forget becoming a great mathematician because most famous mathematicians are not great anyway; instead, your goal is to become a math professor somewhere so that you can have a relatively free life.
Suppose that you want to get a PhD degree in mathematics. Here are some of my advices:
1. Find an advisor with powerful influence in mathematics, in general such an advisor exists only in a top university.
2. Work in a field that has better opportunity and less competition.
3. Make sure that your personality is compatible with or can be adapted to the culture in the field.
In case you cannot find a powerful advisor because you cannot go to a top university or you don't care whether your advisor is politically powerful or not, and you happen to be in HKUST and you are truly interested in mathematical physics, maybe I can assist you to some extent.
To help you decide whether I am suitable for you, let me talk a little bit about my research interest and research style. I am interested in physics and geometry, but only the simplest and cleanest part. (For example, I am very interested in the fractional quantum Hall effect, but not interested in the high temperature superconductivity.) I am more interested in discovering something unexpected rather than proving an expected statement. I like conceptually high and clean stuff, but not the technically complicated stuff. Sometime I also try to settle a fundamental problem in mathematics if I do have some ideas. In general, I am more interested in something I don't know, not something that I know, that is one reason that I rarely write two papers in the same area in the past.
Recently I have written many papers in one area for a few reasons: 1) many directions in this area are worth to explore; 2) I am probably the only current explorer in this area, so I have the burden to continue the exploration; 3) to demonstrate that I am a normal mathematician, i.e., I can be a so-called super expert in one area.
I consider myself a geometer with a keen interest in fundamental physics. The mathematics knowledge that I have used in my research so far is rather standard, so I survive by finding something hidden. Mixing two distinct objects together is the signature of my research:
Seiberg-Witten theory for 4-manifolds (1994) and Alexander theory for knots, links and 3-manifolds (1928)
Dirac Monopole (1931) and Clifford algebra (1873) (triggered from reading newspaper in the common room)
Jacobian conjecture (1939) and Legendre transformation (~1800) (triggered from a talk by Alexander Voronov)
Newton's Kepler problem (~1660) for planetary motion and Jordan algebra (~1930)
Sternberg phase space (1977) and Tulczyjew approach to classical mechanics (1973) (triggered from a talk by
Janusz Grabowski)
Assume I am suitable for you, you may proceed to read the following brief summary of my work.