Asymptotic Problems in Representation Theory

这个项目的目的是研究在表示理论和数学物理中出现的一些相互关联的渐近问题。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对的可容许表示和研究某些量子多体系统的热力学极限。所使用的方法是G. Olshanski和PI最近发展的各种渐近方法。它们特别包括多元雅可比多项式的某些插值类比的使用和多元雅可比多项式的二项式型展开。在数学中，许多令人感兴趣的问题太复杂，无法直接解决。有两个基本原理可以帮助我们对复杂现象有所了解：对称性，它有助于理解问题的结构；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粒子量子力学系统，并利用其丰富的对称性来获得当粒子数量变得非常非常大时该系统行为的信息。这类信息非常有趣，因为大多数物理对象都是由大量非常小的组件组成的。

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.