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Large Scale Geometry in Functional Analysis

泛函分析中的大尺度几何

基本信息

批准号：
2054860
负责人：
Bruno de Mendonca Braga
金额：
$ 9.88万
依托单位：
University of Virginia Main Campus
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-15 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054860&HistoricalAwards=false
关键词：
Large Scale Geometry Functional Analysis

项目摘要

Functional analysis is a branch of mathematics that investigates vector spaces endowed with some notion of convergence and structure-preserving maps between them. Since many objects in nature can be modeled by such spaces, functional analysis is frequently used by scientists from different backgrounds. Topics in functional analysis under the optics of large-scale geometry will be of particular interest. In a nutshell, large-scale geometry is the study of the global structure of certain mathematical objects (think of geometric behavior measured by an observer far away from the object of interest). This subject is motivated by computer science. Indeed, when working with large data sets, such methods aid the understanding of global behavior. Moreover, this framework provides the appropriate tools to study the relation between different data sets through large-scale geometric embeddings and equivalences between those objects. The PI will continue his engagement with undergraduate and graduate students, and seminar/conference organization. This project will improve our understanding of certain linear objects (for instance, operator algebras, Banach spaces, operator spaces, etc.), given some nonlinear information about them. It is divided into three main parts: (1) Roe algebras. The goal is to understand how much of the large-scale geometry of a uniformly locally finite metric space (or more generally, of a uniformly locally finite coarse space) is encoded in their uniform Roe algebras and Roe algebras. The questions in this area are often referred to as ‘rigidity problems’ for Roe algebras. Embeddings and isomorphisms between those algebras and how their existence affects the geometry of the metric spaces will be studied. (2) Quantization. The quantization of classic mathematical objects allows one to interpret structures connected to Hilbert spaces as "noncommutative" or "quantum" versions of their classical counterparts. In collaborative work, the PI has recently proposed a quantization of coarse spaces and uniform Roe algebras. Further developments in the quantization of large-scale geometric properties and their relation with uniform Roe algebras will be investigated. (3) Operator spaces. Although the nonlinear geometry of Banach spaces has been receiving attention, especially in the last two decades, its natural noncommutative counterpart (that is, the nonlinear theory of operator spaces) has been waiting to be developed. The plan is to understand how much of the commutative theory holds for operator spaces and apply those results to the strictly noncommutative scenario.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.

泛函分析是数学的一个分支，研究具有收敛性和结构保持映射的向量空间。由于自然界中的许多物体都可以用这样的空间来建模，因此泛函分析经常被来自不同背景的科学家使用。在大尺度几何光学下的泛函分析的主题将特别感兴趣。简而言之，大尺度几何是对某些数学对象的全局结构的研究（考虑由远离感兴趣对象的观察者测量的几何行为）。这门课是由计算机科学激发的。事实上，在处理大型数据集时，这种方法有助于理解全局行为。此外，该框架提供了适当的工具，通过大规模的几何嵌入和这些对象之间的等价关系，研究不同的数据集之间的关系。PI将继续与本科生和研究生接触，并组织研讨会/会议。这个项目将提高我们对某些线性对象的理解（例如，算子代数，Banach空间，算子空间等），给出了一些非线性信息。主要分为三个部分：（1）Roe代数。我们的目标是了解一致局部有限度量空间（或更一般地，一致局部有限粗糙空间）的大尺度几何有多少被编码在它们的一致Roe代数和Roe代数中。这方面的问题通常被称为Roe代数的“刚性问题”。这些代数之间的嵌入和同构以及它们的存在如何影响度量空间的几何将被研究。(2)量化。经典数学对象的量子化允许人们将与希尔伯特空间相连的结构解释为它们的经典对应物的“非交换”或“量子”版本。在合作工作中，PI最近提出了粗糙空间和均匀Roe代数的量化。进一步的发展，在量子化的大规模几何性质和它们的关系与一致的罗伊代数将进行调查。(3)运算符空间。虽然Banach空间的非线性几何一直受到人们的关注，特别是在过去的二十年里，它的自然非交换对应物（即算子空间的非线性理论）一直有待发展。该计划是为了了解有多少交换理论适用于算子空间，并将这些结果应用于严格的非交换场景。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。