Mathematical Sciences: Metric Degeneration, Degenerating Elliptic and Parabolic Problems and Geometric Invarients of Elliptic Operators

主要研究者将用绝热极限法研究椭圆算子的几何不变量。特别是，他将尝试在一般手术过程中理解它们的行为。要研究的不变量之一是解析扭转。不变量和行列式也将被考虑。这项研究的一个动机是Bismut和Lebeau最近在算术黎曼-洛克定理上的工作。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 geometric invariants of elliptic operators via the adiabatic limit method. In particular he will try to understand their behavior under a general process of surgery. One of the invariants to be studied is the analytic torsion. The eta invariant and determinant will also be considered. One motivation for the research is the recent work of Bismut and Lebeau on the arithmetic Riemann-Roch theorem. 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.