The principal investigators continue studies of nonlinear partial differential equations. The present investigations include symmetry-breaking for solutions of nonlinear wave equations, solutions of semilinear equations with finite symmetry groups, conformal metrics on the two-dimensional sphere, radially symmetric solutions of Einstein-Yang-Mills equations, shock wave problems, and certain problems of topology. These mathematical questions capture features of a variety of physical problems. For instance, questions addressed in the Einstein-Yang-Mills studies have implications for the importance of quantum effects in forming black holes. Many problems in mathematical biology, physics, and geometry have solutions that show certain patterns or symmetries. Symmetry-breaking indicates changes in the character of the solutions as parameters of the problems change.
