K-theory of Operator Algebras and Index Theory on Spaces of Singularities

几何学的数学领域探索空间中点、线和形状的性质、关系和测量，为我们的物理世界的空间和结构方面提供见解。它的实际应用跨越建筑、工程和空间理解，实现安全的设计、高效的结构和有效的导航。刚性结果在现代几何中起着举足轻重的作用，它决定了几何物体在特定变换下的稳定性和保存性。其中，曲率约束下的刚度研究具有特殊的意义。在这种情况下，标量曲率是主要的兴趣，因为与其他曲率概念相比，它在适当的情况下表现出灵活性和刚性。该项目的一个主要目标是开发新的方法来解决与标量曲率相关的长期猜想和开放问题。各种分析方法，包括指数理论的技术，将有助于实现项目的目标。指标理论通过研究微分算子及其相关指标的性质，为研究几何结构的刚性提供了一套强有力的工具。指标理论的最新进展在从分析的角度理解曲率和刚性之间的相互作用方面取得了重大突破，并引发了对标量曲率的兴趣和活动的激增，为几何开辟了令人兴奋的新方向。除了探索这一新领域外，该项目还为本科生和研究生提供培训和指导机会，重点研究k理论、指标理论和非交换几何领域。本项目的主要目标是推动“奇异”空间（如奇异空间或不完全度量空间）上指标理论的发展。除了其固有的数学兴趣之外，奇异空间上的指标理论有两个重要的应用：几何中的标量曲率问题和拓扑中的高特征问题（如诺维科夫猜想）。主要研究者（PI）与合作者一起，发展了一种新的具有奇点的流形指标理论。值得注意的是，当应用于标量曲率问题时，这个新的指标理论允许在具有奇点的流形上比较标量曲率、平均曲率和黎曼度量的二面角。该理论的应用已经产生了有趣的结果，解决了Gromov关于标量曲率的重要猜想，包括Gromov的立方不等式猜想和Gromov的二面体极值和刚性猜想。与紧光滑流形上的经典指标理论相比，奇点的存在对在具有此类奇点的空间上建立相干指标理论提出了重大挑战。同样，由于度量的不完备性，许多不完备流形上的几何问题也遇到了类似的挑战。该项目的一个主要组成部分是进一步发展指标理论技术，以有效地解决奇点和度量不完备所带来的挑战。作为应用，这些技术将导致Gromov关于标量曲率的一些重要猜想的正解，以及Novikov猜想和粗糙Baum-Connes猜想的新类群。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The mathematical field of geometry explores the properties, relationships, and measurements of points, lines, and shapes in space, providing insights into the spatial and structural aspects of our physical world. Its practical applications span architecture, engineering, and spatial understanding, enabling secure designs, efficient structures, and effective navigation. Rigidity results, which determine the stability and preservation of geometric objects under specific transformations, have played a pivotal role in modern geometry. Among them, the study of rigidity under curvature constraints holds particular significance. The scalar curvature is of primary interest in this setting because, in contrast to other notions of curvature, it exhibits both flexibility and rigidity under suitable circumstances. A main objective of this project is to develop new approaches to address long-standing conjectures and open questions related to scalar curvature. Various analytical methods, including techniques from index theory, will be instrumental in achieving the project’s goals. Index theory provides a powerful set of tools for studying the rigidity of geometric structures by investigating the properties of differential operators and their associated indices. Recent advances in index theory have led to significant breakthroughs in understanding the interplay between curvature and rigidity from an analytical point of view and have sparked a surge of interest and activity in scalar curvature, opening exciting new directions in geometry. In addition to exploring this new landscape, this project offers training and mentoring opportunities for undergraduate and graduate students, focusing on research in the fields of K-theory, index theory, and noncommutative geometry.The primary objective of this project is to advance the development of index theory on “singular” spaces (such as spaces with singularities, or spaces with incomplete metrics). In addition to its intrinsic mathematical interest, index theory on singular spaces has two significant applications: scalar curvature problems in geometry and higher signature problems in topology (such as the Novikov conjecture). The principal investigator (PI), together with collaborators, has developed a novel index theory for manifolds with singularities. Notably, when applied to scalar curvature problems, this new index theory allows for comparisons of scalar curvature, mean curvature, and dihedral angles of Riemannian metrics on manifolds with singularities. The application of this theory has already yielded interesting results by solving important conjectures posed by Gromov on scalar curvature, including Gromov's cube inequality conjecture and Gromov's dihedral extremality and rigidity conjecture. In contrast to the classical index theory on compact smooth manifolds, the presence of singularities poses a significant challenge in formulating a coherent index theory on spaces with such singularities. Similarly, many geometric problems on incomplete manifolds encounter similar challenges due to the incompleteness of the metric. A major component of this project is to further develop index theory techniques to effectively address the challenges posed by both singularity and metric incompleteness. As applications, these techniques will lead to positive resolutions of some important conjectures of Gromov on scalar curvature, and the Novikov conjecture and the coarse Baum-Connes conjecture for new classes of groups.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.