Mathematical Sciences: Symmetry & Orthogonal Polynomials

940142 Dunkl该奖项支持对特殊函数理论中出现的问题的数学研究，强调潜在的群结构和对几个变量的扩展。该分析涉及到根系统的研究，使用与有限反射群相关的微分-差分算子的参数化代数技术。对于不同的参数值，有一个交织算子将这些代数联系起来。该项目的一个重要部分将包括通过基于相关紧李群的技术找到Weyl群的这个算子的积分公式。这一课题的进展将导致更详细的信息，如渐近性，关于Coxeter群的贝塞尔函数和傅里叶变换的某些推广。工作也将继续在Coxeter群的奇异多项式，在主题连接表示理论，微分方程的一元与群的反射超平面上的理性奇点，Hecke代数和缠结算子。近似理论和特殊函数与微分方程、算符理论、数值积分、连分式和控制理论等应用数学的许多分支有着丰富的联系。这些学科与特殊函数和相关对称群有着密切的联系，这为本研究的理论和应用方面提供了天然的协同作用。＊＊＊

940142 Dunkl This award supports mathematical research on problems arising in the theory of special functions emphasizing underlying group structures and extensions to several variables. The analysis involves the study of root systems using a technique a parametrized algebra of differential-difference operators associated to finite reflection groups. There is an intertwining operator which relates these algebras for different parameter values. An important part of the project will consist of finding integral formulas for this operator for Weyl groups by means of techniques based on the associated compact Lie groups. Progress in this topic would lead to more detailed information, such as asymptotics, about the Bessel functions of Coxeter groups and certain generalizations of the Fourier transform. Work will also continue on singular polynomials of Coxeter groups, at topic which connects representation theory, monodromy of differential equations with rational singularities on the reflecting hyperplanes of the groups, Hecke algebras and the intertwining operator. Approximation theory and special functions has fruitful connections with many branches of applied mathematics such as differential equations, operator theory, numerical integration, continued fractions and control theory to name a few. These subjects have close connections with special functions and related symmetry groups which provide a natural synergism between the theoretical and applied aspects of this research. ***