LEAPS-MPS: Noncommutative Geometry and Topology of Quantum Metrics

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This project in noncommutative metric geometry focuses on the theme of approximation, a fundamental notion in mathematics and the physical sciences, in which one seeks to understand a complex object by studying its simpler relatives which are “nearby” according to some measure of distance. The Gromov-Hausdorff distance -- an important tool from metric geometry which measures the distance between subsets of a space, rather than just individual elements -- has seen many applications across the mathematical and physical sciences. This project will employ generalizations of the Gromov-Hausdorff distance to develop new approaches to the approximation of infinite-dimensional algebras arising from quantum mechanics (known as C*-algebras) by their finite-dimensional counterparts. These new methods will advance the state of the art in noncommutative metric geometry and build connections with the structure theory of C*-algebras as well as quantum information theory. The PI will offer research opportunities for undergraduate and high school students, who will play a central role in the project. A diverse group of students will be recruited, with a focus on professional development and a view toward broadening participation in mathematics and science in the next generation. Noncommutative metric geometry (NCMG) was largely motivated and built by structures arising from noncommutative topology (NCT) and noncommutative geometry (NCG), and this project aims to provide further connections between these three areas. The importance of this pursuit lies in the fact that the classical counterparts of metric geometry, topology, and differential geometry have many connections, and finding analogous results in the noncommutative realm to results in the classical realm has proven to advance mathematics overall. In particular, NCT and the classification of C*-algebras rely heavily on C*-algebras arising as inductive/direct limits of C*-algebras, and due to quantum versions of the Gromov-Hausdorff distance provided by NCMG, one can show that these inductive limits are limits in a metric sense and even establish metric convergence of sequences of inductive limits. However, in various cases including the case of approximately finite-dimensional (AF) algebras, convergence of certain sequences of inductive limits has only been attained using purely NCMG and NCT structure without using natural structures arising from NCG like spectral triples. A primary goal of this project is to enrich these results with spectral triple data and thus produce a missing connection among these areas in the noncommutative realm. This project will also investigate other classes of inductive limits and their finite-dimensional approximations as well as further associated structures such as Hilbert C*-modules. A focused look at some of the individual quantum metric spaces arising from these investigations yields connections to metric geometry and quantum information theory.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.