Mathematical Sciences: Geometry and Analysis of Foliations

小行星9400676 Heitsch教授将继续研究Bismut超连接和Chern字符的运营商定义的沿着叶的叶理。与C.Lazarov的进一步合作将在紧致流形的解析不变量扩展到叶理和开流形的不变量上，特别强调挠不变量。对拓扑空间映射的粗上同调和粗不变量作了进一步的发展。叶理的几何形状也将通过横向椭圆算子的指数来研究。 几何对象通常通过确定它们是否可以被映射或变形为标准几何对象来分类。某些几何不变量，例如，“洞”的数量在这样的映射下被保留。这个项目涉及的几何对象的不变量称为具有“叶”结构的叶状结构。上同调通过将全局对象看作是由多个部分拼接而成来产生这些不变量。 本课题的顺利完成将有助于几何分类问题的完成。 ***

9400676 Heitsch Professor Heitsch will continue research on Bismut superconnections and the Chern character for operators defined along leaves of a foliation. Further collaboration with C.Lazarov will be done on the extension of analytic invariants for compact manifolds to invariants of foliations and of open manifolds, with special emphasis on torsion invariants. A development will be made of coarse cohomology and coarse invariants for maps of topological spaces. The geometry of foliations will also be investigated through the indices of transversally ellipticoperators. Geometric objects are often classified by determining whether they can be mapped or deformed into standard geometric objects. Certain geometric invariants, for example, the number of "holes" are preserved under such maps. This project involves invariants for geometric objects called foliations that have a "leaf" structure. Cohomology produces these invariants by considering the global object as patched together from pieces. The successful completion of this project will help complete theclassification problem in geometry. ***