Matheamtical Sciences: Applications of Elliptic PDE's to Gauge Theory, Topology, and Physics

所提出的研究涉及几何偏微分方程的各个方面。线性椭圆微分算子，规范场方程，和杨-米尔斯-希格斯方程是要追求的主题之一。将使用拓扑、几何和分析方法：变分技术、模空间技术和指标理论将被应用于解决这些偏微分方程。该研究通过揭示新的流形拓扑不变量，可能对光滑流形理论产生影响。规范场在理论物理学中自然出现，试图统一自然界的基本力。杨-米尔斯-希格斯方程在早期宇宙物理学和星系形成理论中有应用。

Miller 9505170 The proposed research deals with various aspects of geometric partial differential equations. Linear elliptic differential operators, gauge field equations, and Yang-Mills-Higgs equations are among the topics to be pursued. Methods from topology, geometry and analysis are to be employed: variational techniques, moduli space techniques, and index theory are to be applied to tackle these partial differential equations. The proposed research may have consequences in smooth manifold theory by uncovering new topological invariants of manifolds. Gauge fields arise naturally in theoretical physics as attempts to unify basic forces of nature. Yang-Mills-Higgs equations have applications in early universe physics and the theory of galactic formation.