Mathematical Sciences: Problems in Nonlinear Partial Differential Equations

主要研究人员继续研究非线性偏微分方程。目前的研究包括非线性波动方程解的对称性破缺，具有有限对称群的半线性方程的解，二维球面上的共形度量，Einstein-Yang-Mills方程的径向对称解，冲击波问题，以及某些拓扑问题。这些数学问题抓住了各种物理问题的特点。例如，爱因斯坦-杨-米尔斯研究中提出的问题暗示了量子效应在形成黑洞中的重要性。数学生物学、物理学和几何学中的许多问题都有表现出特定模式或对称性的解。对称破缺表示随着问题参数的变化，解的性质也发生了变化。

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.