LEAPS-MPS: Noncommutative Geometry and Topology of Quantum Metrics

这个非对易度量几何的项目集中在近似的主题上，这是数学和物理科学中的一个基本概念，其中一个试图通过研究它的简单亲戚来理解一个复杂的对象，这些亲戚根据某种距离测量“附近”。 Gromov-Hausdorff距离是度量几何的一个重要工具，它测量空间子集之间的距离，而不仅仅是单个元素之间的距离，在数学和物理科学中有许多应用。 该项目将采用Gromov-Hausdorff距离的推广来开发新的方法，以通过有限维对应物来近似量子力学中产生的无限维代数（称为C*-代数）。 这些新方法将推进非对易度量几何的发展，并与C*-代数的结构理论以及量子信息理论建立联系。 PI将为本科生和高中生提供研究机会，他们将在项目中发挥核心作用。 将招募一批多样化的学生，重点是专业发展，并着眼于扩大下一代对数学和科学的参与。 非对易度量几何（NCMG）主要是由非对易拓扑（NCT）和非对易几何（NCG）产生的结构激发和建立的，本项目旨在提供这三个领域之间的进一步联系。这种追求的重要性在于这样一个事实，即度量几何，拓扑学和微分几何的经典对应物有许多联系，并且在非交换领域中找到与经典领域中的结果类似的结果 已经被证明能促进数学的整体发展特别是，NCT和C*-代数的分类严重依赖于C *-代数作为归纳/直接极限而产生的C *-代数，并且由于NCMG提供的量子形式的Gromov-Hausdorff距离，人们可以证明这些归纳极限是度量意义上的极限，甚至建立归纳极限序列的度量收敛。然而，在各种情况下，包括近似有限维（AF）代数的情况下，某些序列的归纳极限的收敛只达到使用纯粹的NCMG和NCT结构，而不使用自然结构所产生的NCG像谱三元组。这个项目的主要目标是丰富这些结果的光谱三重数据，从而产生一个失踪的连接这些地区之间的非对易领域。本项目还将研究其他类别的归纳极限及其有限维近似，以及进一步相关的结构，如希尔伯特C*-模。通过对这些研究中产生的一些单独的量子度量空间进行重点关注，可以将其与度量几何和量子信息理论联系起来。该奖项反映了NSF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。

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.