Mathematical Sciences: Problems in Nonlinear Partial Differential Equations

这个项目支持研究生与乔尔·斯莫勒教授一起研究几何偏微分方程式的问题。重点研究了球对称的爱因斯坦-麦克斯韦方程和爱因斯坦-杨-米尔斯方程。更一般地，从分歧理论和对称破缺的角度研究半线性椭圆型方程和具有对称性的系统。利用关联矩阵等拓扑学技术研究了关联时变问题。正在考虑的特定方程与数学物理领域有关，如宇宙学、引力和黑洞的形成和稳定性。

This project supports graduate students working with Professor Joel Smoller on problems in geometrical partial differential equations. Particular emphasis is placed upon the study of spherically symmetric Einstein-Maxwell and Einstein- Yang-Mills equations. More generally, semilinear elliptic equations and systems with symmetry are studied from the point of view of bifurcation theory and symmetry-breaking. The associated time dependent problems are studied using topological techniques such as the connection-matrix method. The particular equations under consideration have relevance to areas of mathematical physics such as cosmology, gravitation and the formation and stability of black holes.