"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 Chinese students who may be interested in pursuing a PhD under my guidance.
In general, I discourage students from pursuing an advanced degree in pure mathematics or pure physics. I have seen many talented people give up their dreams in these fields after obtaining their advanced degrees. Mathematics has become more technical and specialized, and is not as appealing as it used to be. Additionally, it has become more political and commercialized.
However, if you are talented in mathematics, truly interested in the subject, want a relatively free life, and do not want to work too hard for a comfortable life, an advanced degree in mathematics may be suitable for you. Forget about becoming a great mathematician, as most famous mathematicians are not necessarily great. Instead, aim to become a math professor somewhere so that you can have a relatively free life.
If you wish to pursue a PhD in mathematics, here is some advice:
1. Find an advisor with significant influence in mathematics. Such an advisor typically exists only at a top university.
2. Work in a field that has better opportunities and less competition.
3. Ensure that your personality is compatible with the culture in the field or can be adapted to fit.
In case you cannot find a powerful advisor because you cannot attend a top university or you do not care about your advisor's political power, and you happen to be at HKUST and are truly interested in mathematical physics, perhaps I can assist you to some extent.
To help you decide whether I am suitable for you, let me briefly describe my research interests and style. I am interested in physics and geometry, but only in the simplest and cleanest parts. For example, I am very interested in the fractional quantum Hall effect, but not interested in high temperature superconductivity. I am more interested in discovering something unexpected rather than proving an expected statement. I like conceptually high and clean work, but not technically complicated work. Sometimes I also try to solve fundamental problems in mathematics if I have some ideas. In general, I am more interested in something I do not know rather than something I do know. That is why I rarely write two papers in the same area. However, I have been focusing my research efforts in one particular area for several reasons. Firstly, there are many interesting and promising directions within this area that are worth exploring. Secondly, as I am currently the only researcher exploring this area, I feel a sense of responsibility to continue the exploration and make progress. Finally, by becoming a "super expert" in this area, I aim to demonstrate my proficiency as a mathematician and showcase my ability to make significant contributions to a specific field.
I am a geometer with a keen interest in fundamental physics. In my research, I enjoy finding something hidden between two subjects, and here are some examples:
These are just a few examples of the types of research projects I enjoy working on. I am always on the lookout for new and interesting ways to combine disparate areas of mathematics and physics to uncover new insights and make progress on long-standing problems.
If you believe I am suitable for you, you may proceed to read a brief summary of my work.