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

几何的数学领域探索了空间中点，线和形状的属性，关系和测量，为我们的物理世界的空间和结构方面提供了见解。 其实际应用涵盖建筑、工程和空间理解，实现安全设计、高效结构和有效导航。 刚性结果在现代几何中起着举足轻重的作用，它决定了几何对象在特定变换下的稳定性和保持性。 其中，曲率约束下的刚度研究具有特殊的意义。 标量曲率在这种情况下是主要的兴趣，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。 该项目的主要目标是开发新的方法来解决长期存在的问题和开放的标量曲率相关的问题。 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。 指数理论的最新进展导致了从分析的角度理解曲率和刚性之间的相互作用的重大突破，并引发了对标量曲率的兴趣和活动的激增，开辟了令人兴奋的新方向。 除了探索这一新的景观，该项目还为本科生和研究生提供培训和指导的机会，专注于K理论，指数理论和非交换几何领域的研究。该项目的主要目标是推进“奇异”空间（如奇异空间或不完整度量空间）上的指数理论的发展。除了其内在的数学兴趣，奇异空间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更高签名问题（如诺维科夫猜想）。主要研究者（PI）与合作者一起，为具有奇点的流形开发了一种新的指数理论。值得注意的是，当应用到标量曲率问题，这个新的指标理论允许比较标量曲率，平均曲率和二面角的黎曼度量流形上的奇点。这个理论的应用已经产生了有趣的结果，解决了Gromov提出的重要命题的标量曲率，包括Gromov的立方不等式猜想和Gromov的二面角极值和刚性猜想。与紧致光滑流形上的经典指标理论相反，奇点的存在对在具有这种奇点的空间上建立一个连贯的指标理论提出了重大挑战。类似地，不完备流形上的许多几何问题由于度量的不完备性而遇到类似的挑战。该项目的一个主要组成部分是进一步发展索引理论技术，以有效地解决奇异性和度量不完整性带来的挑战。作为应用，这些技术将导致一些重要的Gromov标量曲率的正决议，诺维科夫猜想和粗糙的鲍姆-康纳斯猜想的新类groups.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.