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.

该项目涉及最近被称为渐近几何分析的领域的研究。将特别关注与数学其他领域以及其他数学和物理科学的联系，这激发了正在考虑的大多数问题。由于底层问题中自由参数的数量通常与相应数学模型中集合的维数相关，并且由于现实生活中的问题通常涉及非常多的参数，因此高维设置特别令人感兴趣。对于量子理论来说尤其如此，其中仅由几个粒子组成的系统自然会产生维度从数千到数十亿的模型。虽然高维现象的经典分析经常遭受维数灾难（问题的复杂性随着维数的增加而爆炸，使得问题很快变得难以处理），但我们可以说渐近几何分析利用了维数的祝福通过识别和利用“近似对称性”，这种对称性只有在维度很大时才变得明显。该项目试图在选定的研究方向中实施这一理念，特别是在与量子信息论相关的领域，量子信息论是一个跨学科领域，为构建量子计算机的项目提供理论基础，量子计算机是主要科学技术之一。 21世纪的挑战。此外，该项目还将让研究生和本科生参与深入研究，从而促进科学人力资源的发展。同样，该项目的产品之一将是一本调查渐近几何分析和量子信息理论接口的书，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。分析是对函数或数量之间的关系，特别是它们的规律性特性的研究。由于许多自然出现的关系是线性的或至少是凸的，因此对凸函数和集合的良好理解是理解这些关系的先决条件。 拟议研究的重点将放在高维环境上。 要研究的样本研究主题包括：高维凸集和高维赋范空间的结构特性、泛函分析中出现的各种概率结构的去随机化，以及由运筹学链接引发的问题。最值得注意的是，该项目将解决与量子信息论和量子计算相关的几何问题，例如与正部分转置性质相关的问题。这些问题通常是（或可以）用巴拿赫空间的几何语言或高维概率的语言来表达，并且主要通过使用在这些背景下起源或开发的各种方法来进行分析。