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

这个项目涉及泛函分析和凸性理论的几何、概率和组合方面的研究，这是最近被称为渐近几何分析的松散定义的领域。将特别注意非对易对象和现象，以及与数学和其他数学科学的其他领域的联系，这是正在研究的大多数问题的动机。样本研究主题包括：高维凸体和高维赋范空间的结构性质，(凸集或线性算子的)度量熵及其对偶性，泛函分析中出现的各种概率结构的去随机化，一些与量子信息论和量子计算有关的几何问题，以及与数学规划有关的问题。通常，问题是(或可以用)Banach空间的几何语言表达的(高维，但有限维)，并将使用在该背景下产生或发展的各种方法进行分析。这种方法依赖于识别和利用各种问题的近似对称性，这些问题逃脱了早期经典泛函分析和经典几何的“过于定性”或“过于僵化”的方法。最后，为了解释我们对非对易性的强调，我们指出，它只是反映了这样一个事实，即一个过程的最终结果可能取决于所涉及的操作的顺序；这一原理最著名但迄今不是唯一的表现是量子力学。在初级水平上，分析是对函数或量与它们所依赖的参数之间的关系的研究。由于许多自然出现的关系都是线性的，或者至少是凸的，因此很好地理解凸函数和凸集是理解这些关系的先决条件。基本问题中的自由参数的数量通常可以与相应数学模型中的对象的维度相关。由于现实生活中的系统或过程(例如，物理的、生物的或经济的)通常表现出非常多的自由度，高维的设置尤其令人感兴趣。这正是渐近几何分析的领域，它研究各种几何结构随着维度变得无限大而具有的定量性质。虽然高维现象的研究经常受到维度灾难的影响(问题的复杂性随着维度的增加而爆炸，并导致难以解决)，但我们可以说，渐近方法利用了维度的好处，只有当维度很大时，问题的对称性才会变得明显。虽然这是一个纯数学的项目，但许多研究课题都出现在其他领域，或受到其他领域的推动，如数学物理、运筹学、控制论、计算机科学或概率与统计。因此，任何进展都会产生跨学科的潜在影响。一方面，这一研究提供了一个新的视角和思路，有助于推进应用领域。可以想象吗？取得突破性进展。另一方面，这些领域中出现的问题和想法将反馈到基础研究中，有助于保持数学的活力，并可能开辟全新的探究方向。此外，该项目将让研究生和本科生参与密集的研究，从而促进科学基础和基础设施的发展。

This project involves research on the geometric, probabilistic and combinatorial aspects of functional analysis and convexity theory, the loosely defined area that has been lately referred to as ?asymptotic geometric analysis.? Particular attention will be paid to non-commutative objects and phenomena, and to links with other areas of mathematics and other mathematical sciences, which motivate most of the problems that are being studied. Sample research topics include: structural properties of high-dimensional convex bodies and of high-dimensional normed spaces, metric entropy (of convex sets or of linear operators) and duality of such entropy, derandomization of various probabilistic constructions appearing in functional analysis, some geometric questions related to quantum information theory and quantum computing, and problems motivated by links to mathematical programming. Typically, the questions are (or can be) expressed in the language of geometry of Banach spaces (of high, but finite dimension) and are to be analyzed using the diverse methods that originated or were developed in that context. The approach depends on identifying and exploiting approximate symmetries of various problems that escaped the earlier "too qualitative" or "too rigid" methods of classical functional analysis and classical geometry. Finally, to explain our emphasis on non-commutativity, we point out that it simply reflects the fact that the final outcome of a process may depend on the order of operations involved; the best known, but by far not the only manifestation of that principle is quantum mechanics. On the elementary level, Analysis is a study of functions, or relationships between quantities and the parameters on which they depend. Since very many naturally appearing relationships are linear or at least convex, a good understanding of convex functions and, consequently, of convex sets is a prerequisite for understanding those relationships. The number of free parameters in the underlying problem can often be related to the dimension of objects in the corresponding mathematical model. Since real-life systems or processes (say, physical, biological or economic) usually exhibit very many degrees of freedom, the high-dimensional setting is of particular interest. This is exactly the domain of asymptotic geometric analysis, which studies quantitative properties of various geometric structures as the dimension goes to infinity. While investigation of high-dimensional phenomena often suffers from the curse of dimensionality (the complexity of the problem exploding with the increase in dimension and resulting in intractability), we may say that the asymptotic approach exploits the blessing of dimensionality, with symmetries of the problem becoming apparent only when the dimension is large. While this is a project in pure mathematics, many of the research topics arose in, or were motivated by, other fields such as mathematical physics, operations research, control theory, computer science or probability and statistics. Accordingly, any progress has a potential impact cutting across disciplines. On the one hand, this research may help to advance the applied areas by providing a new perspective and ? conceivably ? leading to breakthroughs. On the other hand, the problems and ideas arising in those areas will feed back into the fundamental research, contribute to maintaining the vitality of mathematics and potentially open completely new directions of inquiry. Additionally, the project will involve graduate and undergraduate students in intensive research, thus contributing to the development of scientific base and infrastructure.