The proposed research is in the area of nonlinear hyperbolic systems of partial differential equations arising in mathematical physics and deals specifically with regularity, break-down, and large-time behavior of solutions to the initial value problem for wave maps. While the ultimate goal of any study in this area is the understanding of the dynamics of physically relevant field theories such as General Relativity and Yang-Mills, valuable insight can be gained by first focusing on wave maps, which is a simpler geometric field theory exhibiting many similarities with the above, and in which many of the difficulties in dealing with those physical theories are present in a more transparent way. Wave maps (known to physicists as sigma-models) are the hyperbolic analogue of harmonic maps between manifolds, where the domain manifold instead of being Riemannian is Lorentzian. They thus satisfy a system of semilinear wave equations with a nonlinearity which is quadratic in the gradient. Some of the problems to be studied are (1) existence of smooth stationary solutions to the equations and their stability under small perturbations, (2) constructing all possible self-similar wave maps of the Minkowski space, (3) investigating the genericity of singularities arising from self-similar solutions, and (4) finding examples of blowup in two space dimensions. Hyperbolic partial differential equations lie at the heart of mankind's efforts to quantify and understand the evolutionary phenomena in nature. From subatomic particles to galactic clusters, every phenomenon known to Man which involves the propagation of signals and disturbances at finite speeds is modeled by a hyperbolic differential equation. Examples of such phenomena are the production and propagation of sound waves (acoustics), water waves (hydrodynamics), seismic body and surface waves (elastodynamics), electromagnetic waves (electrodynamics), and gravitational waves (general relativity). Despite considerable progress in recent years in the study of nonlinear hyperbolic systems, the basic questions regarding regularity, break-down and large time behavior of their solutions in more than one space dimension remain largely unanswered. Progress in this area requires a good understanding of continuum physics and is only possible through rigorous mathematical analysis of a host of simpler problems, each one modeling only a few of the many difficulties present in the actual physical problem. With such a long-term plan, it is equally important to educate the next generation of scientists who will have the physical and mathematical ability, as well as the courage and enthusiasm, to continue the work being done today. This requires a serious re-evaluation of the existing mathematics curriculum and the development of new courses dealing specifically with mathematics as it relates to continuum physics at all levels of college and graduate education.
