Banach Spaces and Graphs: Geometric Interactions and Applications

可以说，一个人生活的世界本质上是几何的。许多日常生活中的实际问题，以及基本的科学奥秘，可以用几何术语来表达。例如，网络在我们的现代社会中无处不在。从万维网及其强大的搜索引擎到社交网络，从电信网络到经济系统，网络代表了广泛的真实的世界系统。通过考虑连接两个节点的最短路径的边的数量作为测量它们的接近度的量，它们可以自然地被视为几何对象。对物理定律的研究也导致了一个精确的数学框架的发展，在这个框架中，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学基础的对称性。度量空间的概念是一个中心概念，在网络优化问题的数学模型中以及在广泛的应用领域中至关重要，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推广了欧几里得空间的经典概念，其中从点A到点B的距离被计算为连接它们的假想直线的长度。问题的核心通常归结为理解一个给定的度量空间，特别是一个配备了最短路径距离的图，是否可以在一个更结构化的空间中忠实地表示，通常是一个具有某些理想性质的Banach空间。我们这样做的能力通常有巨大的应用。在这个项目中研究的问题是由他们在理论物理和理论计算机科学的潜在应用的动机。大多数的问题被认为找到他们的起源要么在实际问题（例如有效的近似算法的设计），或在拓扑或非交换几何的基本数学问题（例如诺维科夫猜想（S），鲍姆-康纳斯猜想（S））。嵌入的问题，出现在连接到这些问题已被认为是独立的几组数学家（巴拿赫空间geometers，几何群论，计算机科学家.）。这一建议的一个突出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存在的开放问题（例如Enflo问题，一致光滑性的度量重新表述，群和扩展图到超自反Banach空间的粗嵌入性.）将从不同的角度用新的和创新的技术来解决。特别是，该项目将开发一定的Banach空间的渐近理论，并探索其连接到无限图形的几何。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，以便获得新的见解并设计新的方法。这种方法的动机是这样一个事实，即渐近设置通常提供了一个更好的图片，在某些情况下更好地理解，并需要完全不同的技术。该项目研究方法的总体概述是利用周边领域的强大工具（图论、概率论、拉姆齐理论.），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。

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.