Information for Graduate Students

I am looking for graduate students interested to do research in analysis and PDE. First, what do I do? My work is in the area that involves Partial Differential Equations, Fourier Analysis and Functional Analysis. Occasionally I also need to run a numerical simulation or two, mainly to gain intuition for proofs (I am not very good at numerics though). I prefer problems that originate in physics or other sciences. I am working on problems motivated by Fluid Mechanics, Quantum Mechanics, Combustion, Mathematical Biology and even Finance. I can't say I am fully applied mathematician - I work not as much on concrete and detailed problems that would have direct links to technology, but more on conceptual models of important processes. This way, you can have both beautiful, deep and rich mathematics and the feeling that you understand something new about the real world. This mix is the most delicious I have ever tasted!

PDE is a very active and vibrant field. It is certainly one of the central themes of modern mathematics, with connections to numerous other branches of mathematics. PDEs are often motivated by applications, and if you are working on PDEs you will likely find yourself interacting with scientists from other disciplines. Often, there is a certain learning curve one has to master to work on PDEs. At first, the area may seem a bit technical and too varied, even messy. However, after covering this curve the subject becomes very rewarding. Once central ideas and connections become clear, technicality and variety turn into depth and richness. Learning some new partial differential equation well becomes like making a new good friend - and the fact that there are many different PDEs becomes a big positive. Another positive aspect of working in PDE is that PDE experts are in high demand in academia. Should you decide to transition to industry, PDE background will also be helpful due to connections to applications.

If you are interested in analysis and PDE or undecided on what research you would like to do, please feel free to email me or stop by so that we can chat. In order that this conversation can have more substance, let me state some principles which I follow when mentoring research of a junior colleague. I will describe some of the directions of my research in more detail after that.

Now a bit more about my research. Partial Differential equations is an exceedingly diverse field, with very different subareas. The reason is that many processes in nature seem to be described well by PDE - a realization that probably goes back to Newton. Since nature is so rich, there are many different kinds of partial differential equations, and different tools are needed to analyze them. Calculus of variations, comparison principles for the solutions, Fourier analysis, functional analysis, stochastic calculus, ideas from geometry are just some examples of the tool sets that are used to analyze PDEs. So as someone working in PDEs, I use whatever it takes. There are two ideals that I usually try to keep in mind. I like it when mathematics is rich and beautiful, and I like when it refers to or is at least motivated by real life phenomena - problems connected to physics, biology or economics. It is not always easy to balance these two aspirations! One of my favorite tools is Fourier analysis, because it is so abundant with deep and elegant techniques and ideas.

After being pioneered by Joseph Fourier in early 1900s, Fourier analysis went on to become one of the dominant themes in mathematics and its applications for the next two centuries. It is applied widely to partial differential equations (Fourier transform turns differentiation into multiplication, and that is often beneficial - but applications get a lot richer and subtler than that). It often plays an important role in numerical simulations of PDEs, too. Fourier analysis methods are also used heavily in number theory, ergodic theory and occasionally in almost every other area of mathematics. The methods continue to develop and grow more sophisticated. Out of the four 2010 Fields medalists, for example, Smirnov is a Fourier analyst by training, and Lindenstrauss uses Fourier analysis methods extensively in his work - even though they got their awards for work done in other areas (percolation theory and statistical physics for Smirnov, ergodic theory and number theory for Lindestrauss). The set of ideas and tools that you learn in Fourier analysis just seems to be extremely adaptable and useful in all sorts of problems. Learning these techniques prepares your brain for virtually anything mathematical in a very efficient way.

In any case here are several areas that are currently central in my research.
Mathematical Fluid Mechanics
Mixing and enhancement of diffusion by fluid flow
Mathematical Biology
Quantum mechanics, Schrodinger operators and spectral theory