The Blessing of High Dimension: Asymptotic Geometric Analysis and Its Applications

高维的祝福：渐近几何分析及其应用

• 批准号: 1600124
• 负责人: Stanislaw Szarek
• 金额: $ 21万
• 依托单位: Case Western Reserve University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600124&HistoricalAwards=false
• 关键词: Blessing; Dimension; Asymptotic; Geometric; Analysis

This project involves research in an area lately referred to as asymptotic geometric analysis. Particular attention will be paid to links with other areas of mathematics and other mathematical and physical sciences, which motivate most of the problems being considered. Since the number of free parameters in the underlying problem can often be related to the dimension of sets in the corresponding mathematical model, and since real-life problems usually involve very many parameters, the high-dimensional setting is of particular interest. This is especially true for quantum theory, where systems consisting of just several particles naturally lead to models whose dimension is from thousands to billions. While classical analysis of high-dimensional phenomena often suffers from the curse of dimensionality (the complexity of the problem explodes with the increase in dimension so that the question quickly ceases to be tractable), we may say that asymptotic geometric analysis exploits the blessing of dimensionality by identifying and exploiting "approximate symmetries", which become apparent only when the dimension is large. This project is an attempt to implement this philosophy in selected directions of research, most notably in those related to quantum information theory, the interdisciplinary area that provides theoretical underpinnings for the project of building a quantum computer, which is one of the major scientific and technological challenges of the 21st century. Additionally, the project will involve graduate and undergraduate students in intensive research, thus contributing to development of human resources in science. In the same vein, one of the products of the project will be a book surveying the interface of asymptotic geometric analysis and quantum information theory, likewise contributing to the development of scientific base and infrastructure and to the promotion of interdisciplinarity. Analysis is a study of functions, or relationships between quantities, and particularly of their regularity properties. Since very many naturally appearing relationships are linear or at least convex, a good understanding of convex functions and sets is a prerequisite for understanding those relationships. The emphasis of the proposed research will be on the high-dimensional setting. Sample research topics to be studied include: structural properties of high-dimensional convex sets and of high dimensional normed spaces, derandomization of various probabilistic constructions appearing in functional analysis, and problems motivated by links to operations research. Most notably, the project will address geometric questions related to quantum information theory and quantum computing, for example those related to the positive partial transpose property. The questions typically are (or can be) expressed in the language of the geometry of Banach spaces or of high-dimensional probability and are to be analyzed primarily by using the diverse methods that originated or were developed in those contexts.

该项目涉及最近称为渐近几何分析的领域的研究。特别关注与其他数学和其他数学和物理科学领域的联系，这激发了所考虑的大多数问题。由于基本问题中的自由参数的数量通常与相应的数学模型中集合的维度有关，并且由于现实生活中的问题通常涉及很多参数，因此高维设置特别感兴趣。对于量子理论尤其如此，在量子理论中，仅由几个粒子组成的系统自然会导致其尺寸从数千到数十亿的模型。虽然对高维现象的经典分析通常受到维数的诅咒（问题的复杂性随维度的增加而爆炸，因此问题迅速不再是可触及的），但我们可能会说，渐近几何分析利用仅通过识别和利用“近似symmetries”的尺寸，仅在较大的维度上，就可以通过识别和利用维度。该项目试图在某些研究方向上实施这种哲学，最著名的是与量子信息理论相关的哲学，这是为建造量子计算机的项目提供理论基础的跨学科领域，这是21世纪的主要科学和技术挑战之一。此外，该项目将涉及研究生和本科生参与密集研究，从而有助于科学中的人力资源发展。同样，该项目的一种产品将是一本书，调查渐近几何分析和量子信息理论的界面，同样有助于科学基础和基础设施的发展以及促进跨学科性。分析是对数量尤其是其规律性特性之间的功能或关系的研究。由于许多自然出现的关系是线性的或至少是凸的，因此对凸功能和集合的良好理解是理解这些关系的先决条件。 拟议的研究的重点将是高维环境。 要研究的样本研究主题包括：高维凸组的结构特性和高维规范空间的结构特性，在功能分析中出现的各种概率构建体的延性以及与操作研究链接所激发的问题。最值得注意的是，该项目将解决与量子信息理论和量子计算有关的几何问题，例如与正面偏置属性相关的几何问题。这些问题通常是（或可以）以Banach空间的几何形状或高维概率的语言表达，并且主要是通过使用在这些情况下起源或开发的各种方法来分析的。