Large scale index, positive scalar curvature and manifold topology

给定一个无边界的光滑紧致流形M，研究了M容纳何种几何结构的问题。特别重要的是-由宇宙学的动机-标量曲率，我们特别研究了M上的正标量曲率的黎曼度量空间。一个长期存在的问题是：什么时候这个空间是非空的（即~什么时候M允许一个度量的正标量曲率）？更一般地说：这个空间的拓扑结构是什么？ 例如，它的同伦群是什么？量子物理学的一个基本微分方程，旋量的狄拉克方程，与正标量曲率密切相关，正如薛定谔已经观察到的那样。现代的改进，其中的研究计划是一部分，使用算子代数和他们的K-理论，揭示了微妙的全球信息的几何编码在这个关系。更具体地说，大规模指数理论的狄拉克运营商是一个相当新的和非常成功的工具，研究正标量曲率问题。它使用适应于非紧黎曼流形的算子代数，重点是大尺度性质。近年来，这方面有了巨大的发展。该理论与算子代数和拓扑学中长期存在的深层次问题有关，如鲍姆-康纳斯猜想和诺维科夫猜想。在这个项目中，我们将进一步发展大规模指数理论的范式。我们将确定新的几何情况下，这种范式适用，我们将使新的几何拓扑结构，允许应用的方法为经典的情况下，一个封闭的流形。这将需要发展精细的局部分析方法。这些将以新的方式与同伦理论和同源性的复杂工具相结合。

Given a smooth compact manifold M without boundary we study the question which kinds of geometric structure M accommodates. Of special importance is ---motivated by cosmology--- the scalar curvature, and we study in particular the space of Riemannian metrics of positive scalar curvature on M. A long-standing question is: when is this space non-empty (i.e.~when does M admit a metric of positive scalar curvature)? More generally: what is the topology of this space? For example, what are its homotopy groups?A fundamental differential equation of quantum physics, the Dirac equation for spinors, is closely connected to positive scalarcurvature, as already observed by Schr\"odinger. Modern refinements, of which the research program is a part, use operator algebras and their K-theory to unveil the subtle global information about geometry which is encoded in this relation.More specifically, large scale index theory of the Dirac operator is a rather new and very successful tool to study positive scalar curvature questions. It uses operator algebras adapted to non-compact Riemannian manifolds, where the focus is on thelarge-scale properties. This has seen tremendous development in recent years. The theory connects to long-standing and deep questions in operator algebras and topology, like the Baum-Connes conjecture and the Novikov conjecture. In the project, we will develop further the paradigm of large scale index theory. We will identify new geometric situations where this paradigm applies, and we will make new geometric-topological constructions which allow to apply the method for the classical case of a closed manifold. This will require the development of fine local analytic methods. These will be combined in new ways with sophisticated tools from homotopy theory and homology.