Large scale index, positive scalar curvature and manifold topology

给出了一个没有边界的光滑紧流形M，我们研究了M适应哪种几何结构的问题。特别重要的是-受到宇宙学的启发-标量曲率，我们特别研究了M上正标量曲率的黎曼度量空间。一个长期存在的问题是：这个空间何时非空(即~M何时接纳正标量曲率的度量)？更笼统地说：这个空间的拓扑是什么？例如，它的同伦群是什么？量子物理的一个基本微分方程，旋量的狄拉克方程，正如薛定谔已经观察到的那样，与正的标量曲率密切相关。现代精化是利用算子代数及其K-理论来揭示蕴含在这种关系中的关于几何的微妙全局信息，而Dirac算子的大尺度指标理论是研究正标量曲率问题的一种相当新且非常成功的工具。它使用适合于非紧黎曼流形的算子代数，其中重点放在大尺度性质上。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Baum-Connes猜想和Novikov猜想。在这个项目中，我们将进一步发展大尺度指数理论的范式。我们将确定该范例应用的新几何情形，并且我们将建立新的几何-拓扑结构，以允许将该方法应用于封闭流形的经典情况。这将需要发展精细的局部分析方法。这些将以新的方式与来自同伦理论和同调的复杂工具相结合。

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.