Information for Prospective Graduate Students
I am always looking for graduate students interested to do research on analysis of PDE.
Firstly, I will briefly describe my research. I became interested in fluid dynamics PDE when I was exposed to the Navier-Stokes Existence and Smoothness Problem in 2008 at a summer math school in Princeton:
https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness
During my Ph.D. years at Oklahoma State University, I learned how to apply harmonic analysis tools to analysis on fluid PDEs. During my first post-doc at Washington State University, I learned how to apply functional analysis tools to populations models in mathematical biology, specifically infectious diseases. Taking advantage of my background in statistics from my Master years at Stanford University, I also learned how to apply stochastic analysis tools to fluid PDEs. During my second post-doc at the University of Rochester, I continued to learn this direction of research. Hence, currently my research can be summarized as analysis on PDEs in fluid mechanics and infectious diseases using techniques from functional, harmonic and stochastic analysis, and some quantum field theory most recently.
Before continuing any further, I would like to share a few thoughts to graduate students because I believe that a good relationship between a graduate student and an adviser relies on many things beyond just common research interests. Firstly, here is some career advice to graduate students from Terence Tao, which I initially read as a graduate student:
https://terrytao.wordpress.com/career-advice/
Besides good advice from the link above, here are a few of my thoughts.
As a graduate student, you should immediately think about what you want to do after you obtain the degree (Master or Ph.D.). In the long-run, options are primarily faculty positions at research universities or teaching universities or industry. In the short-run, for Ph.D. students, options are mainly a post-doc position (temporary but research-oriented, and necessary if you want to someday be a faculty at a research university), a tenure-track position (permanent but usually teaching-oriented), or an industry position. For master students, options are usually a Ph.D. program or an industry position.
Graduate students should give talks as much as possible, as much as time and money allow. Giving talks in the Department is great. Graduate students should also frequently go to meetings and conferences to meet new people. There are often funding opportunities for those, and they give you lots of motivation for research, as well as allow you to meet life-time colleagues and friends. A good website at which you may be able to find conferences, workshops, meetings of your interests may be
https://www.ams.org/meetings/calendar/mathcalIt may be ideal to explore various different directions of mathematics in the first 2 years as a Ph.D. student because it is difficult to switch directions or learn completely new branches later on in the career. However, after 2 or 3 years, it is important to settle, find that direction of research of your passion, and make significant progress. In my case, I was already determined to focus on fluid PDEs before I entered a Ph.D. program.
Always be conscious of your growth as a researcher. Keep important open problems in your mind all the time, e.g., be able to explain to non-experts why it is hard. However, work on challenging and yet hopefully doable problems from which you can grow by obtaining new techniques. Also, try to be a self-sufficient researcher; do not give up or ask for help after a few mathematical difficulties. Always remember that the purpose of research training during graduate school years is that you become an independent original researcher of your own afterward.
Finally, let me end by saying a few more things about my research. The direction of research on PDEs is interesting because on one hand, it requires very hard analysis from pure mathematics, and there remain many open problems that have stood firm for centuries. On the other hand, it is derived from the problems in nature (e.g., fluid mechanics, biology, physics, finance), and hence there is certainly a sense of meaningfulness in its applications. Becoming an expert on PDEs and probability may also lead to various options in both academia and industry.
If you wish to talk to me on anything above, please feel free to let me know and I will be happy to discuss. E.g., even if you may not be ready to commit to a direction of research on PDEs, I know lots of conferences and workshops which you may be interested in attending to obtain better ideas about your research interests.