From bispectrality to integrable systems, orthogonal polynomials, heat equations and W-algebras

从双谱到可积系统、正交多项式、热方程和 W 代数

• 批准号: 0901092
• 负责人: Plamen Iliev
• 金额: $ 14.09万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2012-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901092&HistoricalAwards=false
• 关键词: bispectrality; integrable; systems; orthogonal; polynomials

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The bispectral problem appeared first in the pioneering work of Duistermaat and Grunbaum. It asks to classify spectral problems possessing a hidden symmetry between the space and the spectral variables. The goals of this proposal is to establish and explore new connections between integrable systems, orthogonal polynomials, heat equations and Lie algebras, closely related with each other and mysteriously linked to the bispectral property. The first part of the proposal aims at the construction and the classification of multivariable classical orthogonal polynomials; their duality properties (between the continuous and the discrete variables) and relations to integrable systems and algebraic geometry. Another part of the proposal explores the connection between rank one bispectral operators and dynamical systems of particles on the line (such as Calogero-Moser and Ruijsenaars-Schneider). The bispectral property in this case leads to the construction of new interesting W-algebras and their representations. The PI has also proposed a new approach to study heat equations based on soliton equations (such as KdV and Toda lattice) and Sato?s Grassmannian, and to use this approach to characterize the bispectral operators by a finiteness property of the heat kernel.The classical (one dimensional) orthogonal polynomials have played a crucial role in mathematics and physics in the last centuries. The construction and the classification of such polynomials in more than one variable is an old and important problem and its solution will have numerous applications in both pure and applied mathematics. The connection with algebraic geometry and integrable systems serves as a useful bridge which translates powerful techniques, explains the bispectrality from duality and leads to various extensions. Heat equations, arising in many physical applications, can be studied in a uniform way, by connecting them to soliton equations. The characterization of bispectrality in terms of representation theory of Lie algebras brings the purely algebraic side of the proposal. The interdisciplinary nature of this project will establish new links between the different branches of mathematics and physics and will stimulate communications and collaborations between specialists in the various areas involved. Research projects are also designated for students, whose involvement in the project will advance discovery and understanding while promoting teaching and training.

该奖项是根据2009年美国复苏和再投资法案(公法111-5)资助的。双谱问题最早出现在Duistermaat和Grunbaum的开创性工作中。它要求对在空间和光谱变量之间具有隐藏对称性的光谱问题进行分类。这个建议的目的是建立和探索可积系统、正交多项式、热方程和李代数之间的新联系，这些联系彼此密切相关，并神秘地与双谱性质联系在一起。该方案的第一部分致力于多元经典正交多项式的构造和分类，它们的对偶性质(连续变量和离散变量之间)以及与可积系统和代数几何的关系。该提案的另一部分探索了一阶双谱算子和直线上的粒子动力系统(如Calogero-Moser和Ruijsenaars-Schneider)之间的联系。这种情况下的双谱性质导致了新的有趣的W-代数及其表示的构造。PI还提出了一种新的方法来研究基于孤子方程(如KdV和Toda格子)和Sato？S格拉斯曼方程的热方程，并利用这种方法利用热核的有限性质来刻画双谱算子。在过去的几个世纪里，经典的(一维)正交多项式在数学和物理中发挥了至关重要的作用。一元多项式的构造和分类是一个古老而重要的问题，它的求解在理论数学和应用数学中都有广泛的应用。与代数几何和可积系统的联系是一座有用的桥梁，它转换了强大的技巧，解释了对偶性中的双谱，并导致了各种扩展。在许多物理应用中出现的热方程，通过将它们与孤子方程联系起来，可以以统一的方式进行研究。根据李代数的表示理论对双谱的刻画带来了该提议的纯代数方面。该项目的跨学科性质将在数学和物理的不同分支之间建立新的联系，并将促进有关领域的专家之间的交流和合作。研究项目也是为学生指定的，他们参与项目将促进发现和理解，同时促进教学和培训。