Mathematical Sciences: Topology, Analysis and Ergodic Theoryof Foliations

调查员将研究在调查中出现的问题。 叶状流形的分类（如 同伦类型的叶理分类空间），以及如何 解析结构（解析K理论和谱理论） 与沿着叶理的叶沿着的椭圆算子相关联 与周围流形的代数拓扑有关。 这个项目将研究全球拓扑结构在 叶理算子的光谱特性，特别是关于： 逐叶η-不变量与奇解析K-理论 横截算子的特征和渐近方法;间隙 现象的叶薛定谔算子的基础上， 固体物理布里渊区理论的推广 Fourier积分算子方法 最近的工作T。坪井介绍了一种新的方法， Haefliger分类的代数拓扑研究 叶理空间 调查人员希望延长 坪井的方法，以全面研究如何 一个分类空间的上同调与 它所分类的叶理广群， 横向可微性 他打算继续他的 横截循环上同调的应用研究 Anosov叶理不变量提供刚性现象 有负截面曲率的黎曼流形 经典的叶理问题是人们是否可以梳理 椰子上的毛（无轮）。 大量的概括， 这已经被设想和精心设计的代数和分析 机器进化来治疗它们。 仍然是潜在的想法 有一个自然的呼吁几何直觉，使一个 整个主题的特殊风味，并倾向于承担 有很多技术细节

The investigator will study problems arising in the classification of foliated manifolds (as represented by the homotopy type of foliation classifying spaces), and how the analytic structures (the analytic K-theory and spectral theory) associated to elliptic operators along the leaves of a foliation are related to the algebraic topology of the ambient manifold. This project will study the role of global topology in the spectral properties of foliation operators, especially regarding: leafwise eta-invariants and odd analytic K-theory; Chern characters and asymptotic methods for transverse operators; gap phenomenon for leafwise Schrodinger operators based on generalizing Brillouin zone theory of solid state physics via Fourier integral operator methods. Recent work of T. Tsuboi has introduced a new method for the study of the algebraic topology of the Haefliger classifying spaces of foliations. The investigator hopes to extend the approach of Tsuboi to make a comprehensive study of how the cohomology of a classifying space is related to the dynamics of the foliation groupoids that it classifies, and to their transverse differentiability. He intends to continue his research into the application of the transverse cyclic cohomology invariants of Anosov foliations to providing rigidity phenomena for Riemannian manifolds with negative sectional curvatures. The classical foliation problem is whether one can comb the hair on a cocoanut (without whorls). Vast generalizations of this have been conceived and elaborate algebraic and analytic machinery evolved for treating them. Still the underlying ideas have a natural appeal to geometric intuition which gives a special flavor to the entire subject and inclines one to bear with its many technicalities.