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: 0801275
• 负责人: Stanislaw Szarek
• 金额: $ 57.42万
• 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: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801275&HistoricalAwards=false
• 关键词: Blessing; Dimension; Asymptotic; Geometric; Analysis

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.

这个项目涉及的几何，概率和组合方面的研究功能分析和凸性理论，松散定义的领域，最近被称为？渐近几何分析 特别注意将支付给非交换对象和现象，并与其他领域的数学和其他数学科学，这激发了大多数正在研究的问题的联系。样本研究主题包括：高维凸体和高维赋范空间的结构性质，度量熵（凸集或线性算子）和这种熵的对偶性，泛函分析中出现的各种概率结构的去随机化，与量子信息理论和量子计算有关的一些几何问题，以及与数学规划有关的问题。通常情况下，这些问题是（或可以）表达的语言几何的巴拿赫空间（高，但有限维），并进行分析，使用不同的方法，起源或发展在这种情况下。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“过于严格”的方法。最后，为了解释我们对非对易性的强调，我们指出，它仅仅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;这一原理最著名的但不是唯一的表现是量子力学。 在初级水平上，分析是对函数的研究，或者是量与它们所依赖的参数之间的关系。由于许多自然出现的关系是线性的或至少是凸的，因此对凸函数以及凸集的良好理解是理解这些关系的先决条件。底层问题中的自由参数的数量通常与相应数学模型中对象的维数有关。由于现实生活中的系统或过程（比如物理的、生物的或经济的）通常表现出非常多的自由度，因此高维环境特别令人感兴趣。这正是渐近几何分析的领域，渐近几何分析研究当维数达到无穷大时各种几何结构的定量性质。虽然对高维现象的研究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，导致棘手），但我们可以说渐近方法利用了维数的好处，只有当维数很大时，问题的对称性才变得明显。虽然这是一个纯数学项目，但许多研究课题都是在其他领域出现的，或者是由其他领域激发的，例如数学物理，运筹学，控制理论，计算机科学或概率统计。因此，任何进展都有可能产生跨学科的影响。一方面，本研究为应用领域的发展提供了新的视角，可以想象吗从而取得突破 另一方面，在这些领域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，并可能开辟全新的研究方向。此外，该项目将使研究生和本科生参与深入研究，从而促进科学基础和基础设施的发展。