Research Opportunities

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I am working on many research projects, almost all of which can use assistance from either graduate or undergraduate students. Since the detailed descriptions of these projects changes too frequently, it is not given here. The bigger topics are outlined on my research page . Rather, on this page, I give an idea of what kind of methods are used in these projects and what kind of background is required for them.

Undergraduate Research

Undergraduate research is probably the single-most effective way to experience what applied mathematics really is: you will use everything you have used in your courses, while learning much that is new, maybe even contributing something new of your own.

The projects I work on use many different areas of applied mathematics. They require analytical skills (pen and paper applied mathematics),and/or programming skills (applied mathematics on the computer: both numerical and symbolical computation).

What is required? As an undergraduate, you can contribute to these projects in many different ways, depending on your interests and skills. The minimal requirement is a basic understanding of ordinary differential equations ( AMATH 351 or MATH 307). A multitude of other courses are useful (but not necessary, depending on what you want to do) for the projects I work on. These might help you to narrow down what you're interested in: (order does not reflect relevance) AMATH352, 353, 383. A variety of Math, Physics and Computer Science courses can also be put to good use.

What is in it for you? Several things: (a) the fun of learning and discovering new mathematics, (b) you can do undergraduate research and receive credit, (c) what better way to convince graduate schools you can do research then to do research? (d) Depending on the results of the project, it could result in a publication, (e) for some projects, funding may be available.

Graduate Research

I expect the research you do as a graduate student to be new and original. My kind of applied mathematics consists not only of applying known mathematics, but also of contributing new mathematics, so don't be surprised if your project involves comparison with data, Riemann surfaces and stability analysis. No matter which aspect of a problem you're working on, it's almost certain that a computer will enter the picture. Symbolical and numerical computation are cornerstones of what I do.

Possible projects: (in no specific order) (a) Developing symbolic software packages (involving Riemann surfaces, solutions of integrable equations, ...); (b) Asymptotic derivation of new model equations for certain physical problems; (c) Stability analysis of known solutions of differential equations; etc.

What is required? As a graduate student, your contributions can vary greatly, depending on your interests and skills. The minimal requirement is a sound understanding of ordinary and partial differential equations, as well as the analysis of complex variables. Other areas of mathematics that I've used include (in no particular order) Riemann surfaces, algebraic geometry, Lie algebras, asymptotics and perturbation theory, Hamiltonian mechanics, numerical analysis. When working on nonlinear waves, really everything's on the table! Do you need to know all of these to be able to work on nonlinear waves? No. Can other areas of mathematics contribute? Yes!

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