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

这个项目涉及的研究领域最近被称为渐近几何分析。将特别注意与其他数学领域以及其他数学和物理科学的联系，这是所考虑的大多数问题的动机。由于潜在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有关，并且由于现实问题通常涉及非常多的参数，因此高维设置特别有趣。这对于量子理论来说尤其如此，在量子理论中，只有几个粒子组成的系统自然会导致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的诅咒（问题的复杂性随着维数的增加而爆炸，以至于问题很快就不再易于处理），但我们可以说，渐近几何分析通过识别和利用“近似对称性”来利用维数的祝福，这种“近似对称性”只有在维数很大时才会变得明显。本项目试图在选定的研究方向上实施这一理念，尤其是在与量子信息理论相关的领域，这一跨学科领域为构建量子计算机的项目提供了理论基础，量子计算机是21世纪的主要科学和技术挑战之一。此外，该项目将使研究生和本科生参与深入的研究，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，并促进跨学科的发展。分析是对函数或量之间关系的研究，特别是对它们的规律性的研究。因为很多自然出现的关系都是线性的，或者至少是凸的，所以理解凸函数和集合是理解这些关系的先决条件。建议的研究重点将放在高维设置上。要研究的样本研究课题包括：高维凸集和高维赋范空间的结构性质，泛函分析中出现的各种概率结构的非随机化，以及与运筹学相关的问题。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相关的几何问题，例如与正偏转置性质相关的几何问题。这些问题通常（或可以）用巴拿赫空间或高维概率的几何语言来表达，并主要通过使用在这些背景下产生或发展的各种方法来分析。

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.