Graduate study at the Lefschetz Center

Applied analysis

Applied mathematics involves modeling, analysis and computation. Analysis has been a traditional strength of the Lefschetz Center at Brown. This involves both mathematical rigor and an appreciation of scientific problems where mathematics can contribute. This is a good area of research if you are interested in physical problems, but like to prove theorems. It usually involves trading scientific breadth for mathematical depth in the sense that we often study `ideal' models stripped of messy `real' features. This is simply because it takes a long time, and much hard work, to make progress in mathematics. However, the rigorous analysis of a few key problems often yields broad scientific insights.
I work on mathematical problems arising in fluid mechanics, materials science and physical chemistry. Fluid mechanics is an old subject that continues to challenge us. It is not hard to see why: a quick glance at van Dyke's beautiful album of fluid motion reveals a fascinating array of flow patterns that stubbornly resist explanation. Mathematical analysis has an important role to play in these problems: with a few exceptions, there is little ambiguity about the basic equations. But they are nonlinear and hard, and every picture in the album seems to require new ideas and methods. For example, an outstanding problem is to understand the hierarchical organization of vortices in turbulent flows. This fascinated Leonardo da Vinci five hundred years ago, and continues to fascinate us today.

Mathematical interest in materials science and physical chemistry is more recent and has been driven by spectacular experimental advances (in short, the ability to see more and more, at smaller and smaller scales). Modeling such phenomena is a challenge: we no longer have a single master equation, but instead a proliferation of models with different regimes of validity. However, a closer inspection reveals a wealth of unexpored connections to analysis, combinatorics, differential equations, geometry and probability. It is both strange and profound, that we seem to require sophisticated `pure' mathematical tools to understand `real' microscopic features in these areas.

Preparation and admission

An undergraduate program in mathematics, or the sciences or engineering with a significant mathematical component is a good basis for graduate study in the Division. Specific mathematical preparation is less important at this stage than general ability and motivation. The formal application procedure may be found in the graduate handbook. The entering class usually has about ten students, and two or three choose to work with faculty in the Lefschetz Center. Most students find Brown a pleasant environment.
If you are an undergraduate student at a US university contemplating graduate school, please consider one of the many REU programs supported by the National Science Foundation . At this time, the Division does not fund undergraduate summer internships for students at non-US universities. However, we certainly welcome international applicants for the Ph.D program.

Coursework and prelims

Once at Brown, it is essential to take the graduate courses in analysis, differential equations, and probability. It is also important to take a sequence of classes in engineering or the sciences to `see how they do it'. Rule of thumb: if you studied science as an undergrad, take more math, if you studied math, take more science.
The first couple of years are a good time to explore your options in the division. Please attend seminars even if they seem bewildering. They do become more comprehensible with time (well, at least many of them). It is very important to speak to a few faculty members and find an advisor by the end of the second year. This is followed by an unpopular rite of passage - the prelims. Prelim preparation is a good way to take stock, but it is not an end in itself. The year after your prelims is the best time to come to grips with the core technical issues in some field and a realistic understanding of research in applied mathematics.


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On 27 Jul 2006, 15:06.