Variational Problems and Dynamics

The research is aimed at solving mathematical problems that are not only significant as mathematics, but that have been suggested by problems arising in physics and biology. The analysis of the equations governing physical and biological processes often requires a precise quantitative understanding of the relative sizes of various quantities involved in these processes, and this is provided by mathematical inequalities, often of a geometric nature. The quest for a better understanding of these processes is in part quest for new and more precise mathematical inequalities. One way of discovering and proving such mathematical inequalities, which is central to the project, is through the consideration of auxiliary dynamical processes that evolve the state of a system into a form that is amenable to analysis. This area of research has been fruitful not only in producing results that are of interest to a wider scientific community, but also in engaging the interest of Ph.D. students. The intellectual merit of the research is that it will produce not only significant new mathematics, but results that are relevant to physical and biological sciences as well. These applications in other fields guarantee a broad impact of the work, which is further enhanced by the involvement of students, contributing to training of the next generation of researchers.

Among the many nonlinear evolution equations that arise in the description of physical and biological systems are the Boltzmann equation and the Keller-Segel equations for chemotaxis. For both of these, an essential source of information on the behavior of solutions is a priori functional inequalities. For example, solutions of the Boltzmann equation tend towards equilibrium solutions, and the rate at which this happens is governed by an inequality relating relative entropy and the entropy production forced by the evolution. Such functional inequalities are established by completely solving a variational problem: finding the minimum value of some functional, determining the full set of minimizing functions, and finally, proving results that assert that if the value of a functional is close to the optimal value, then its argument must be close to an optimizer. Such complete solutions of variational problems are not only of interest for studying the evolution of physical systems, but also, variational problems can sometimes be best solved by studying an appropriate dynamics associated with them. This interplay between nonlinear dynamics and variational problems has been the source of much recent progress. This project focuses on variational problems and on nonlinear evolution equations, with emphasis on those problems in which the investigator expects a particularly fruitful interplay. A second focus is on operator and trace inequalities for quantum systems. Investigation of these is motivated by problems in quantum statistical mechanics and quantum information theory, and again there is close interplay between quantum dynamics and the inequalities to be investigated, except that here functions are replaced by operators and non-commutativity issues arise.