Asymptotic Problems in Representation Theory

表示论中的渐近问题

• 批准号: 9801466
• 负责人: Andrei Okounkov
• 金额: $ 6.18万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-08-01 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9801466&HistoricalAwards=false
• 关键词: Asymptotic; Problems; Representation; Theory

Abstract Okounkov The aim of this project is to investigate a number of interrelated asymptotic problems arising in representation theory and mathematical physics. The two main parts of the project are the study of admissible representations of Gelfand pairs associated to infinite-dimensional symmetric spaces and the investigation of the thermodynamic limit of certain quantum many-body systems. The methods to be used are the various asymptotic methods developed recently by G. Olshanski and the PI. They include, in particular, the use of certain interpolation analogs of the multivariate Jacobi polynomials and binomial-type expansions of the multivariate Jacobi polynomials. In mathematics, many problems of interest are just too complicated to attack them directly. There are two fundamental principles which help gain some insight into complicated phenomena: the symmetry, which helps understand the structure of the problem, and the law of large numbers, which forces systems of a very large size behave, in a sense, in a simplified and predictable fashion. The interaction of these principles plays the key role in the project. For example, we study a certain quantum mechanical system of particles and use its rich symmetries to obtain information about the behavior of this system as the number of particles becomes very, very large. This sort of information is of great interest because most physical objects consist of a huge number of very small components.

抽象的Okounkov 这个项目的目的是研究在表示论和数学物理中出现的一些相互关联的渐近问题。的 该项目的两个主要部分是研究与无限维对称空间相关的Gelfand对的可接受表示以及 研究某些量子多体系统的热力学极限。 所使用的方法是最近由 G.奥尔尚斯基和私家侦探。它们包括，特别是，使用某些插值类似物的多元雅可比多项式和二项式型 多变量Jacobi多项式的展开式 在数学中，许多感兴趣的问题太复杂了，无法直接解决。有两个基本原则有助于深入了解复杂的现象：对称性，它有助于理解结构 这个问题的答案，以及大数定律，它迫使一个非常大的系统在某种意义上以一种简化的和可预测的方式运行。这些原则的相互作用在项目中起着关键作用。 例如，我们研究一个特定的量子力学粒子系统 并利用它丰富的对称性来获得关于这个系统的行为的信息，因为粒子的数量变得非常非常大。这类信息非常有趣，因为大多数物理对象都由大量非常小的组件组成。