Mathematical Sciences: Singularities in Determined AnalyticPDE and Associated Qualitative Properties of Solutions

首席研究员将研究低维非线性实解析偏微分方程的奇异性。他的方法的一个关键特征将是使用多值解和几何方法来研究奇点。本研究的一般目标是找出控制解奇异性的偏微分方程的几何不变量。该奖项将支持微分几何和全局分析领域的研究。微分几何是研究空间几何与空间上定义的解析概念之间关系的学科。全局分析是通过拼凑局部信息来研究空间的整体几何和拓扑特性。这些数学领域在其他科学中的应用包括复杂分子的结构、液气边界和宇宙的大尺度结构。

The principal investigator will study the singularities in low dimensional non-linear real analytic partial differential equations. A key feature of his approach will be the use of multivalued solutions and a geometric approach to the study of singularities. A general objective of the research will be to identify geometric invariants of partial differential equations which control singularities of solutions. This award will support research in the general area of differential geometry and global analysis. Differential geometry is the study of the relationship between the geometry of a space and analytic concepts defined on the space. Global analysis is the study of the overall geometric and topological properties of a space by piecing together local information. Applications of these areas of mathematics in other sciences include the structure of complicated molecules, liquid-gas boundaries, and the large scale structure of the universe.