Mathematical Sciences: Geometry and Topology of Singular Spaces

小行星9504900 这个项目将尝试定义特征类，然后（通过热方程方法）证明黎曼曲面上秩为2的零度向量丛的模空间的黎曼-罗克定理。 它还将试图找到一个分析理解之间的差异诺依曼和狄利克雷边界条件的微分形式的光滑部分的复杂的投影品种。 最后，计划将调查员和Mark Stern关于投影簇的相交上同调的纯Hodge结构的先前结果扩展到不依赖于D-模理论。 有许多定量的方法来研究弯曲的物体，其中部分不光滑或均匀，而是有角或尖点。 （想象一下一片皱巴巴的金属。） 例如，人们可能会问这些尖点如何影响物体内的热扩散;或者，相反地，观察到的扩散性质是否决定了尖点的性质。 对于非常复杂的光滑物体（甚至是三维以上的物体），物体的曲率对热扩散的影响是相当好理解的。 该项目的一个目标是扩展到具有尖点的物体，即已知的光滑物体中的热扩散分析。 ***

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. ***