LEAPS-MPS: Noncommutative Geometry and Topology of Quantum Metrics

LEAPS-MPS：量子度量的非交换几何和拓扑

• 批准号: 2316892
• 负责人: Konrad Aguilar
• 金额: $ 18.97万
• 依托单位: Pomona College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2316892&HistoricalAwards=false
• 关键词: LEAPS; MPS; Noncommutative; Geometry; Topology

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.

该项目在非交通性度量的几何形状中着重于近似的主题，近似值，数学和物理科学的基本概念，其中人们试图通过研究其更简单的亲戚来理解复杂的对象，这些亲属根据距离的一些测量而“附近”。 Gromov-Hausdorff距离 - 度量几何的重要工具，它测量了空间的子集之间的距离，而不仅仅是单个元素 - 在整个数学和物理科学中都看到了许多应用。该项目将采用Gromov-Hausdorff距离的概括，以开发新的方法，以通过其有限维度对应物引起的量子力学（称为C*-Elgebras）产生的无限维代数近似。这些新方法将在非共同度量的几何形状中推进艺术的状态，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，他们将在该项目中发挥核心作用。将招募一群潜水员的学生，重点是专业发展，并希望扩大对下一代数学和科学的参与。非交通性度量几何形状（NCMG）在很大程度上是由非交通性拓扑（NCT）和非交通性几何形状（NCG）产生的结构来建立的，该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一个事实，即公制几何，拓扑和差异几何形状的经典对应物具有许多联系，并且在非共同领域中找到类似的结果以在古典领域的结果，已证明可以促进数学的整体发展。特别是，NCT和C* - 代数的分类在很大程度上取决于C* - 代数作为C* - 代数的电感/直接限制，并且由于NCMG提供的Gromov-Hausdorff距离的量子版本，因此可以表明，这些归纳性限制在限制中限制了序列的限制和序列的限制。但是，在各种情况下，包括大约有限维（AF）代数的情况，仅使用纯NCMG和NCT结构来实现某些电感限制序列的收敛，而无需使用来自NCG的天然结构，例如光谱三元组。该项目的一个主要目标是通过光谱三数据丰富这些结果，因此在非交通域中的这些领域之间存在缺失的联系。该项目还将研究其他类别的归纳限制及其有限维近似以及其他相关结构，例如Hilbert C* - 模块。对这些投资产生的一些单独的量子度量空间的重点观察产生了与度量几何和量子信息理论的联系。该奖项反映了NSF的法定任务，并通过使用该基金会的知识分子的优点和更广泛的影响来审查标准，被认为是通过评估来获得的支持。