Combinatorics, Banach Spaces, Probability, Spin Glasses

组合学、Banach 空间、概率、自旋玻璃

• 批准号: 9703879
• 负责人: Michel Talagrand
• 金额: --
• 依托单位: Ohio State University Research Foundation -DO NOT USE
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9703879&HistoricalAwards=false
• 关键词: Combinatorics; Banach; Spaces; Probability; Spin

9703879 Talagrand This research centers on two main tools of abstract probability, concentration of measure (a versatile tool to bound the fluctuations of complicated functions) and the theory of majorizing measures (a versatile tool to find upper and lower bounds for the expected values of complicated functions). The investigator will try to pursue the development of the underlying theories, and will also try to find new applications, in particular to the geometry of Banach spaces and Hilbert spaces. He will also try to use abstract methods to achieve progress on the understanding of mean field models for spin glasses, a subject where rigorous results are very hard to obtain. A separate project consists in the study of the subsets of the discrete cube of large dimension, and of their extremal properties. Real life phenomenon are usually affected by random events outside our control. Random variables that depend on many independent influences are therefore a good model for the outcome of complicated real procedures. Surprisingly general tools have recently been developed to rigorously control the fluctuations of these random variables in a wide variety of situations, and they demonstrate the power and the relevance of abstract methods to concrete situations. The development of these methods and the research of new applications will be continued. A particular effort will be made to use these methods to obtain rigorous results for spin glass models. These are mathematical models for some very intriguing states of condensed matter, that are of considerable interest in the physics community.

9703879 Talagrand 这项研究集中在抽象概率的两个主要工具上，测度集中（限制复杂函数波动的通用工具）和主要测度理论（寻找复杂函数期望值的上限和下限的通用工具）。研究者将努力追求基础理论的发展，并将尝试寻找新的应用，特别是巴纳赫空间和希尔伯特空间的几何。他还将尝试使用抽象方法来加深对自旋玻璃平均场模型的理解，这是一个很难获得严格结果的学科。一个单独的项目包括研究大维离散立方体的子集及其极值属性。 现实生活中的现象通常会受到我们无法控制的随机事件的影响。 因此，依赖于许多独立影响的随机变量是复杂实际过程结果的良好模型。令人惊讶的是，最近开发了一些通用工具来严格控制这些随机变量在各种情况下的波动，它们证明了抽象方法与具体情况的威力和相关性。这些方法的开发和新应用的研究将继续进行。我们将特别努力使用这些方法来获得旋转玻璃模型的严格结果。这些是一些非常有趣的凝聚态物质状态的数学模型，在物理学界引起了相当大的兴趣。