Mathematical Sciences: Geometry and Analysis of Foliations

[400676] Heitsch教授将继续研究Bismut超连接和沿叶理的叶定义算子的chen性质。进一步与C.Lazarov合作，将紧流形的解析不变量推广到叶形和开流形的不变量，特别强调扭转不变量。对拓扑空间映射的粗糙上同调和粗糙不变量作了进一步的发展。叶理的几何也将通过横向椭圆算子的指标来研究。几何对象通常通过确定它们是否可以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”的数量在这样的地图下是保留的。该项目涉及具有“叶子”结构的称为叶状的几何对象的不变量。上同调通过将全局对象看作是一块块拼接在一起而产生这些不变量。本课题的成功完成将有助于完成几何学中的分类问题。＊＊＊

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. ***