The study of pattern forming systems is a fundamental area of scientific investigation in which the tools of modern mathematical analysis (dynamical systems theory, the theory of partial differential equations, asymptotic analysis and probability theory) can be brought to bear on the modelling of physical systems especially near a critical transition in the behavior of the system. For the projects being studied in this proposal the PI is principally interested in patterns that arise when a continuous translational symmetry is reduced, at a critical threshold, to a discrete periodic symmetry resulting in what is often referred to as a "striped" pattern. Such patterns are for instance generic in Rayleigh-Benard convection (RBC) which is a key mechanism in the formation of many weather patterns. In RBC the striped pattern corresponds to the formation, at a critical temperature, of periodic "convection rolls" of a uniform characteristic width. A major goal of this research is to study not just the patterns that form at a critical threshold but to characterize the types of defects that arise in these patterns when one is far from threshold. The patterns studied here are stationary and, consistent with that, one may investigate the formation of defects through the minimization of a free energy which depends on a control parameter with a critical value that corresponds to the critical threshold of the physical system. The free energy studied here is known as the "regularized Cross-Newell energy" and is expressed in terms of a phase that locally defines the striped pattern away from defects. From the mathematical point of view the problem is to study the limiting behavior of minimizers to a variational problem as a critical parameter is approached. The determination of such limits is a delicate matter but the PI is developing geometric and analytical tools that should make this determination tractable.
A second part of this proposal also involves the asymptotic study of variational problems. The general context of these problems is the study of the statistics of eigenvalues of N x N Hermitean matrices as N becomes large. These statistics are taken with respect to a class of probabilistic expectations which are invariant under the symmetries that preserve the eigenvalues. This general area of study is generally referred to as random matrix theory which has become prominent because of the insights it provides into a wide range of fields including graph theory, number theory and quantum field theory. Indeed the PIand his collaborators recently extablished the existence of a large N expansion for the random matrix partition function which has played a central role over the last twenty years in discussions within the physics community concerning quantum gravity. The key to this analysis is the study of a variational problem that characterizes the asymptotic expected density of eigenvalues and the asymptotics of an associated Riemann-Hilbert problem. The current proposal will study the fine sturcture of this expansion which can provide detailed information about counting functions which enumerate graphs on topologically non-trivial surfaces. These counting functions will be determined as solutions to partial differential equations which are continuum limits of Hamiltonian (in fact completely integrable) sytems of ordinary differential equations which describe how the eigenvalue density changes as the probabilistic matrix expectations are varied. The PI will also explore the behavior of the random matrix partition function as a critical threshold in the variation of these expectations is approached. Physicists have conjectured that this phase transition can be used to determine a viable candidate for calculating field theoretic expectations in two-dimensional quantum gravity.
Both parts of this proposal have potential relevance for interesting applications. The work on pattern formation will make definite predictions on defect formation that can be tested in the laboratory. Indeed some of the predictions emerging from this work are currently being tested by experimentalists. The PI envisions applications to the stability analysis of patterns and some initial studies of the dynamics of wave patterns. In the future one may expect this work to have relevance for modelling defect structure in animal coat patterns (including fingerprints) and the evolution of plant patterns.
The methods and ideas of random matrix theory are having a broad impact; in number theory, combinatorics, random graph theory (or networks), growth processes and multivariate statistics, to name just a few areas of application. It is important that the underlying methods and results of random matrix theory be established with the highest standards ofrigor so that they can be widely used in these diverse applications. That is an overarching and fundamental goal of this part of the research proposal.
