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

算子代数的K理论与奇点空间索引论

• 批准号: 2247322
• 负责人: Zhizhang Xie
• 金额: $ 24.58万
• 依托单位: Texas A&M University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247322&HistoricalAwards=false
• 关键词: theory; Operator; Algebras; Index; Theory

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.

几何学的数学领域探讨了空间中点，线和形状的属性，关系和测量，从而提供了对我们物理世界的空间和结构方面的见解。它的实际应用涵盖了体系结构，工程和空间理解，实现了安全的设计，有效的结构和有效的导航。刚性结果决定了在特定变换下的几何对象的稳定性和保存，在现代几何形状中起了关键作用。其中，曲率约束下的刚性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，因为与其他曲率笔记相比，它在适当的情况下表现出柔韧性和刚性。该项目的一个主要目的是开发新的方法来解决与标态曲率有关的长期猜想和开放问题。各种分析方法，包括索引理论的技术，将有助于实现项目的目标。索引理论提供了一组强大的工具，用于研究几何结构的刚度，通过研究差异操作员及其相关索引的特性。索引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显着突破，并引发了标态曲率的兴趣和活动的激增，打开了令人兴奋的几何学新方向。除了探索这一新景观外，该项目还为本科和研究生提供了培训和心理机会，重点是在K理论，索引理论和非交通性几何学领域进行研究。该项目的主要目的是促进索引理论在“单一”空间上的发展（例如与奇异的空间一样，或与奇异的空间相同或与necomplete noselete noctlede norderte normerics necomplete noclede noclete normits”。除了其内在的数学兴趣外，关于奇异空间的索引理论还具有两个重要的应用：在几何形状中的标态曲率问题和拓扑中的较高签名问题（例如Novikov猜想）。首席研究员（PI）与合作者一起为具有奇异性的流形开发了一种新颖的索引理论。值得注意的是，当应用于标量曲率问题时，这种新的索引理论可以比较标态曲率，平均曲率和二面角的二面角，这是在具有奇异性的流形上的。通过解决格罗莫夫对标量曲率提出的重要猜想，该理论的应用已经产生了有趣的结果，包括格罗莫夫的立方体不平等猜想和格罗莫夫的二面体极端和僵化的猜想。与紧凑的平滑流形的经典索引理论相反，奇点的存在在制定这种奇异性空间上的连贯索引理论方面是一个重大挑战。同样，由于度量的不完整，许多不完全流形的几何问题遇到了类似的挑战。该项目的一个主要组成部分是进一步开发索引理论技术，以有效解决奇异性和度量不完整所带来的挑战。作为应用，这些技术将导致对格罗莫夫在标量弯曲方面的一些重要猜想以及对新组的诺维科夫猜想和粗糙的鲍姆 - 康涅斯猜想的积极解决方案。这奖反映了NSF的法定任务，并通过基金会的知识优点和广泛的criperia criperia criperia criperia cripteria criperia crighitia criperia criperia criperia criperia criperia cripteria secalluation the诚实。