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破缺、有限对称群半线性方程解的对称破缺、二维球面上的共形度量、爱因斯坦-杨-米尔斯方程的径向对称解、激波问题和某些拓扑问题。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的解决方案都显示出某种模式或对称性。对称性破缺表明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.