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。特别注意将支付给其他领域的数学和其他数学和物理科学，激发大多数正在考虑的问题的联系。由于基本问题中的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，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受到维数灾难的影响（问题的复杂性随着维数的增加而爆炸，因此问题很快就不再容易处理），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，只有当维数很大时才变得明显。这个项目是一个尝试，以实施这一哲学在选定的研究方向，最显着的是在那些与量子信息理论，跨学科领域，提供了理论基础的项目建设量子计算机，这是一个重大的科学和技术挑战的21世纪世纪。此外，该项目将使研究生和本科生参与深入研究，从而促进科学人力资源的开发。同样，该项目的产品之一将是一本书，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，并促进跨学科。分析是研究函数或量之间的关系，特别是它们的规律性。由于许多自然出现的关系是线性的或至少是凸的，因此对凸函数和凸集合的良好理解是理解这些关系的先决条件。 建议的研究的重点将是在高维设置。 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特性，函数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相关的几何问题，例如与正部分转置属性相关的几何问题。这些问题通常是（或可以）表达的语言几何的Banach空间或高维概率，并进行分析，主要是通过使用不同的方法，起源或发展在这些背景下。

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.