Large Scale Geometry in Functional Analysis

泛函分析是数学的一个分支，它研究赋予向量空间某种概念的收敛和它们之间的结构保持映射。由于自然界中的许多物体都可以用这样的空间来建模，因此来自不同背景的科学家经常使用泛函分析。大尺度几何光学下的泛函分析的主题将特别令人感兴趣。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(想想远离感兴趣对象的观察者测量的几何行为)。这门课的动机是计算机科学。事实上，在处理大数据集时，这种方法有助于理解全球行为。此外，该框架提供了适当的工具来研究不同数据集之间的关系，通过大规模的几何嵌入和对象之间的等价。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(例如，算子代数、Banach空间、算子空间等)的理解，给出一些关于它们的非线性信息。它分为三个主要部分：(1)Roe代数。其目的是了解一致局部有限度量空间(或者更一般地，一致局部有限粗空间)的大规模几何中有多少被编码到它们的一致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”。我们将研究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。(2)量化。经典数学对象的量子化使人们能够将与希尔伯特空间相连的结构解释为经典对象的“非对易”或“量子”版本。在协作工作中，PI最近提出了粗空间和一致Roe代数的量子化。我们将进一步研究大规模几何性质的量子化及其与一致Roe代数的关系。(3)算子空间。虽然Banach空间的非线性几何一直受到人们的关注，特别是在最近二十年，但它的自然非交换对应关系(即算子空间的非线性理论)一直在等待着发展。该计划是为了了解算子空间的对易理论的适用程度，并将这些结果应用于严格的非对易场景。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.