|
My research is in Analysis. Below I list some of the
research projects I can suggest if you
choose to be my PhD student:
- The
singularity formation and instability in the two-dimensional
incompressible equations of fluid dynamics (2-d Euler and SQG
equations). The goal here is to obtain mathematically rigorous
justification of some numerical and experimental evidence. These
problems are of ``Math Olympiad type``, brainteasers that do not require
vast math background so if you want to work on them you can start right
away [ps].
- The generic
behavior of evolution equations. Many processes in nature are
governed by evolution equations (loosely speaking, the Cauchy problem in
the infinite dimensional space). One of the standard problems is to
understand the smoothness of the solutions as time evolves. Using
various tools of Harmonic Analysis one can try to study the
``generic” behavior of the Sobolev norms [ps].
- The wave
propagation in multidimensional medium. This project will focus on
scattering theory of multidimensional Schrodinger equation. The goal is
to understand the scattering mechanism when the potential is rough:
decays slowly and has little symmetry. The methods of the classical
Potential Theory, Probability, and Complex Analysis turn out to be
applicable here [ps].
- Approximation
theory. The general theory of orthogonal polynomials is a classical
subject. It became quite popular recently due, in particular, to some
connections with math physics. In spite of recent progress in this area,
there are many interesting problems left wide open [ps].
The students making significant progress in their research
might be supported through NSF funds.
|