Possible Student Research Projects
Analysis is the area of mathematics which studies the rigorous ideas and proofs behind calculus. It is a rich and beautiful area of study, with surprising constructions and bizarre counterexamples. If you like working with inequalities and estimates, studying limits, or analyzing functions, then analysis is the topic for you.
Analysis is not just theory; it has many important applications to all areas of science. My particular research is in the field of partial differential equations. Many scientific rules, when expressed mathematically, relate a function to its derivatives. This is called a differential equation. Some of the most important differential equations come from physics, which was my first major in college. There are also important differential equations studied in biology, economics, and chemistry, as well as other parts of mathematics, like geometry.
I focus on the qualitative behavior of solutions to partial differential equations. This includes determining for how much time solutions exist, whether or not they have asymptotes or otherwise blow up, how big the solutions are, and other important information which may be useful to laboratory scientist and engineers. We usually can't solve the equations exactly, so our job is to provide as much information as possible from limited mathematical clues.
I reseach two different types of equations. First, I study dispersive wave equations such as the Korteweg-deVries Equation and the Schrödinger Equation. These equations model the behavior of all kinds of waves in nature, including water, light, and quantum-mechanical particles. It is a challenging but exciting area that has seen a great deal of activity in recent years. Professor Terence Tao of UCLA recently won the Fields' Medal, the highest prize in mathematics, in part for his work on dispersive partial differential equations. Second, I study nonlinear elliptic equations. These equations determine the equilibrium solutions of all sorts of situations, from the temperature distribution in an air-conditioned room to electrical charge. They can even help determine your probability of winning a poker game. This is a long-established area of research that many of the most famous mathematicians have worked on--people like Laplace, Green, and Dirichlet.
I have supervised several student projects, including masters theses, undergraduate theses, undergraduate summer projects, and informal work with several students. Topics have included ranking systems such as the BCS, general relativity, traffic modeling, fractals, wave equations, DNA modeling, and free boundary problems. I am always happy to work with a student on a project idea they have, even if I am not an expert in the area.
If you are interested in doing a research project in analysis or differential equations at any level, I would be happy to work with you. This could include a theoretical project in real analysis, or a project based on a specific application in differential equations or partial differential equations. If you have a particular area of interest, either theoretical or applied, please come talk to me about it and we can find a problem that suits you. I have also listed several projects below that come from my own interests.
Analysis is can be an abstract and complex field of study, but it doesn't have to be. The field of ordinary differential equations has lots of fun problems that can be looked at with only a background in linear algebra and ordinary differential equations. If you need to learn more to work on your chosen project, I can give you reading material on any extra topics that come up.
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