Through this NSF Proposal, Dr. Kemp intends to engage in a detailed study of two functional inequalities, the logarithmic Sobolev inequality and the Haagerup inequality, and their descendant semigroup contraction properties, hypercontractivity and ultracontractivity. These are rich subjects, developed over the past 30 years, which have an enormous body of important applications: in heat kernel analysis, probability theory, analysis on manifolds and Lie groups, geometric group theory, and operator algebras (free probability in particular). Both functional inequalities have made significant impact both in classical and non-commutative analytic settings; each debuted in one realm and eventually found application in the other (the log Sobolev inequality moving from global analysis to non-commutative geometry, the Haagerup inequality in the other direction). Dr. Kemp's research focuses on some new and unexpected combinatorial aspects of both inequalities, particularly in the setting of analysis on holomorphic space (both classical and non-commutative). He intends to significantly extend his ideas in this subject through several projects both in global analysis (in Segal-Bargmann spaces, spaces of subharmonic functions, and spaces of sections of vector bundles) and on the non-commutative side, primarily in free probability theory. In particular, through the deep connections between free probability and random matrix theory, Dr. Kemp hopes to exploit computational matrix techniques to address some of the harder questions relating to the extremals of these functional inequalities, and extend some of his results beyond their current combinatorial limitations.
The principal intent of this NSF Proposal is to study some important aspects of the mathematical theory of heat flow and entropy. One of the most interesting ideas in recent mathematical analysis is the realization that the way in which heat flows in an object is intimately related to the global geometry of the object. For example, think of a white-hot pin being placed on a large, cool plane. The heat will diffuse quickly at first, then more slowly, over the plane, bringing the whole system eventually to the same temperature. This is not the way heat flows, for example, from a large meteorite impact on the surface of the Earth. The global geometry of the planet comes into effect; the energy from the impact travels around the world and back again, interacting with itself in a complicated fashion. Shortly after the impact the heat diffusion is similar to the pin's, but after a while, studying the heat flow gives clues to the global shape of the planet. (It's round!) One way to study heat flow, and therefore give geometric information, is through entropy the tendency of systems to disorder. One important theorem in Dr. Kemp's research, the logarithmic Sobolev inequality, asserts that, in a wide range of geometric settings, the rate of entropy is controlled by the total energy of the system, in a precise manner which is tied to global geometry.
