Mathematical Sciences: Geometry and Topology of Singular Spaces

这个项目将尝试定义特征类，然后（通过热方程方法）证明黎曼曲面上二阶零度矢量束的模空间的黎曼-洛克定理。它也将试图找到一个解析的理解之间的诺伊曼和狄利克雷边界条件的微分形式在一个复杂的射影变化的光滑部分的区别。最后，我们计划将研究者和Mark Stern先前关于纯Hodge结构在投影变项的交上同调上的结果进行推广，而不依赖于d模理论。有许多定量的方法来研究弯曲的物体，其中部分不是光滑或均匀的，而是有角或尖。（想象一下一块皱巴巴的金属。）例如，有人可能会问，这些尖峰如何影响物体内部热量的扩散；或者反过来说，是否观察到的扩散特性决定了尖点的性质。对于非常复杂的光滑物体（即使是三维以上的物体），物体的曲率对其内部热扩散的影响是相当容易理解的。这个项目的一个目标是将已知的光滑物体的热扩散分析扩展到具有尖端的物体。＊＊＊

9504900 Pardon This project will attempt to define characteristic classes and then prove (by the heat equation method) a Riemann-Roch theorem for the moduli space of rank-two, degree-zero vector bundles on a Riemann surface. It will also attempt to find an analytic understanding of the difference between Neumann and Dirichlet boundary conditions for differential forms on the smooth part of a complex projective variety. Finally, it is planned that prior results of the investigator and Mark Stern concerning pure Hodge structures on the intersection cohomology of a projective variety will be extended without recourse to the theory of D-modules. There are many quantitative approaches to the study of curved objects, parts of which are not smooth or even, but instead have corners or cusps. (Imagine a crumpled piece of metal.) For instance, one may ask how these cusps affect the diffusion of heat within the object; or, conversely, whether observed properties of the diffusion determine the nature of the cusps. For very complicated smooth objects (even those of more than three dimensions), the effect of the curvature of the object on heat diffusion within it is fairly well understood. One goal of this project is to extend to objects with cusps, the known analysis of heat diffusion in smooth objects. ***