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

Texas A&M University, College Station TX

Investigators

Abstract

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.