This Presidential Young Investigator Award provides support for research in Applied Mathematics. The general areas of work will be partial differential equations, differential geometry and Hamiltonian mechanics. Emphasis will be on studies of nonlinear waves, Ginsburg-Landau theory of magnetic vortices, sigma-models, wave guided and conformally invariant equations (strings). Specific objectives will be to analyze the stability of infinite dimensional Hamiltonian systems with symmetry groups. Related work will be done on conjugating flows of general nonlinear evolution equations with corresponding linear flows. This theory is well understood in finite dimensions but in the infinite dimensional setting (i.e. partial differential equations) new difficulties present themselves which are only partially understood. Work is also planned on problems involving complicated phenomena of continuum mechanics. Of particular interest are questions related to superconductivity and gauge-invariant theories such as understanding magnetic vortices in superconductors. Efforts will be made to understand the dynamics of the breakup of strong vortices of high vortex number to weak ones. Also, the asymptotic behavior of generalized minimal surface equations on Minkowski space will be studied to obtain a better picture of closed strings in the conformally invariant field equation of string theory.
