Analysis and Differential Equations

• Matthew Rudd
Professor Rudd studies nonlinear differential equations, concentrating primarily on nonlinear elliptic and parabolic partial differential equations. He is currently preoccupied with two kinds of problems: those with variational structure and those in which geometry plays a significant role.

Thanks to their structure, many variational problems are amenable to a variety of nonlinear analysis techniques, typified (in the elliptic case) by minimization or min-max methods for functionals defined on appropriate Banach spaces. These problems provide lots of beautiful examples of the interplay between abstract analysis and concrete models of real-world phenomena, and the underlying structure leads naturally to computational paradigms like the finite element method. Some specific areas of interest include multiplicity results for quasilinear boundary-value problems, variational problems with very fast or very slow growth conditions, and sub/supersolution methods for elliptic and parabolic problems with constraints.

In many variational problems, one looks for a weak solution, e.g., an element of some Sobolev space that satisfies an integrated form of the equation. In other problems, one looks for a viscosity solution, a continuous function that satisfies certain inequalities when approximated from above or below by smooth functions. This is the case for many level-set equations, including those that model the curvature-dependent motion of curves and hypersurfaces. These geometric initial-value problems are intimately related to two-player discrete-time games, and one of Dr. Rudd's projects is the development of algorithms based on these connections. He is also working on some problems related to k-Hessian equations, which arise naturally in geometric problems and involve a fascinating blend of various PDE and analysis tools.

Studying PDEs provides plenty of opportunities to prove theorems, develop models of things that actually happen, and/or write code to implement algorithms. What more could you want?

• Frank Gao
Professor Gao studies geometric aspects of functional analysis. This includes, in particular, studies of metric entropy and properties of norms of function spaces.

Consider the following example: How many probability distribution functions one can find on a d-dimensional unit cube, so that the mutual distance between any two is at least epsilon? Of course, the answer depends on the distance one uses. However, even in the simplest cases, only partial results are available. Many similar questions can be asked, and the answers are likely to be unknown. Note that problems like this are related to geometric functional analysis, approximation theory, small deviations in probability theory, etc., and have vast applications in empirical processes in statistics. There are some analytic and probabilistic tools available; yet more need to be discovered -- a very interesting research area.

• Steve Krone
Professor Krone uses differential equations in the study of biological populations. These come about in two ways. First of all, ordinary and partial differential equations are often used to directly create deterministic models in biology. These types of models also arise as scaling limits of interacting particle systembsmean field and hydrodynamic limits corresponding to ODE and PDE, respectively. In this sense, differential equations can be used to obtain information about the more complex particle systems. Conversely, a particle system can be thought of as providing a microscopic picture of the dynamics that are modeled by a differential equation. Understanding the connections and differences between these types of models provides much insight into the systems they model. More information on these applications to biology can be found on the Bioinformatics and Mathematical Biology page.