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Banach Spaces and Graphs: Geometric Interactions and Applications

Banach 空间和图：几何相互作用和应用

基本信息

批准号：
1800322
负责人：
Florent Baudier
金额：
$ 11.28万
依托单位：
Texas A&M University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2022-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800322&HistoricalAwards=false
关键词：
Banach Spaces Graphs Geometric Interactions

项目摘要

It can arguably be said that the world one lives in is geometric in nature. Numerous practical everyday-life issues, as well as fundamental scientific mysteries, can be expressed in geometric terms. For instance, networks are ubiquitous in our modern society. From the World Wide Web and its powerful search engines to social networks, from telecommunication networks to economic systems, networks represent a wide range of real world systems. They can naturally be seen as geometric objects by considering the number of edges of the shortest path connecting two nodes as a quantity measuring their proximity. The study of physical laws has led as well to the development of a refined mathematical framework where elaborate geometric structures are able to depict and model the interactions of elementary particles and the symmetries underlying quantum physics. The notion of a metric space is a central concept that is pivotal in mathematical models of optimization problems in networks, and in a vast range of application areas, including computer vision, computational biology, machine learning, statistics, and mathematical psychology, to name a few. This extremely useful abstract concept generalizes the classical notion of an Euclidean space, where the distance from point A to point B is computed as the length of an imaginary straight line connecting them. The heart of the matter usually boils down to understanding whether a given metric space, in particular a graph equipped with its shortest path distance, can be faithfully represented in a more structured space, typically a Banach space with some desirable properties. Our ability to do so usually has tremendous applications.The problems investigated in this project are motivated by their potential applications in theoretical physics and theoretical computer science. Most of the problems considered find their origins either in practical issues (e.g. the design of efficient approximation algorithms), or in fundamental mathematical problems in topology or noncommutative geometry (e.g. the Novikov conjecture(s), the Baum-Connes conjecture(s)). Embedding problems that arise in connection to these problems have been considered independently by several groups of mathematicians (Banach space geometers, geometric group theorists, computer scientists...). An underlined aspect of this proposal is to consider these embeddings problems from a unified and global standpoint. Fundamental and long-standing open problems in quantitative metric geometry (e.g. Enflo's problem, a metric reformulation of uniform smoothability, the coarse embeddability of groups and expander graphs into super-reflexive Banach spaces...) will be tackled from a different angle with new and innovative techniques. In particular, the project will develop a certain asymptotic theory of Banach spaces and explore its connections to the geometry of infinite graphs. The approach here to solve the local problems above, is to study asymptotic counterparts of the local properties involved, in order to gain new insights and to devise new approaches. This approach is motivated by the fact that the asymptotic setting usually provides a finer picture, is on some occasion better understood, and requires completely different techniques. A general outline of the research methodology of this project is to utilize powerful tools from surrounding fields (graph theory, probability theory, Ramsey theory...), and cross-over techniques (e.g. techniques from theoretical computer science to solve geometric group theoretic problems).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.

可以说，一个人生活的世界本质上是几何的。许多实际的日常生活问题，以及基本的科学奥秘，都可以用几何术语来表达。例如，网络在现代社会中无处不在。从万维网及其强大的搜索引擎到社会网络，从电信网络到经济系统，网络代表了广泛的现实世界系统。通过考虑连接两个节点的最短路径的边数作为测量其接近程度的数量，它们自然可以被视为几何对象。对物理定律的研究也导致了精细数学框架的发展，其中精细的几何结构能够描述和模拟基本粒子的相互作用和量子物理的对称性。度量空间的概念是一个核心概念，在网络优化问题的数学模型中至关重要，在广泛的应用领域，包括计算机视觉，计算生物学，机器学习，统计学和数学心理学，仅举几例。这个非常有用的抽象概念概括了欧几里得空间的经典概念，在欧几里得空间中，从A点到B点的距离被计算为连接它们的虚直线的长度。问题的核心通常归结为理解给定度量空间，特别是具有最短路径距离的图，是否可以在更结构化的空间中忠实地表示，通常是具有某些理想属性的Banach空间。我们这样做的能力通常有巨大的应用。本项目研究的问题是由它们在理论物理和理论计算机科学中的潜在应用所激发的。所考虑的大多数问题都可以在实际问题（例如有效近似算法的设计）或拓扑或非交换几何中的基本数学问题（例如诺维科夫猜想，Baum-Connes猜想）中找到它们的起源。与这些问题相关的嵌入问题已经被几组数学家（巴拿赫空间几何学家、几何群理论家、计算机科学家……）独立地考虑过。该建议强调的一个方面是从统一和全球的角度考虑这些嵌入问题。定量度量几何中的基本和长期开放问题（例如Enflo问题，均匀光滑性的度量重新表述，群的粗嵌入性和扩展图到超自反巴拿赫空间……）将从不同的角度用新的创新技术来解决。特别是，该项目将发展Banach空间的某种渐近理论，并探索其与无限图几何的联系。这里解决上述局部问题的方法是研究所涉及的局部性质的渐近对应物，以获得新的见解并设计新的方法。这种方法的动机是渐近设置通常提供更清晰的画面，在某些情况下更容易理解，并且需要完全不同的技术。该项目的研究方法概述是利用周围领域的强大工具（图论，概率论，拉姆齐理论……）和交叉技术（例如，从理论计算机科学中解决几何群论问题的技术）。该奖项反映了美国国家科学基金会的法定使命，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。