Mathematical Sciences: Symmetry & Orthogonal Polynomials

小行星940142 该奖项支持数学研究中出现的问题，在理论的特殊功能，强调潜在的群体结构和扩展到几个变量。 分析涉及根系的研究，使用的技术参数化代数的微分差分算子与有限反射群。 有一个交织运营商，涉及这些代数为不同的参数值。 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为Weyl群找到该算子的积分公式。 在这一主题的进展将导致更详细的信息，如渐近性，贝塞尔函数的考克斯特群和某些推广的傅里叶变换。 工作也将继续奇异多项式的考克斯特组，在主题连接代表性理论，monodromy微分方程与理性奇异性的反映超平面的群体，赫克代数和交织运营商。 逼近理论和特殊函数与应用数学的许多分支有着富有成效的联系，例如微分方程，算子理论，数值积分，连分数和控制理论等等。 这些主题与特殊函数和相关的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自然的协同作用。 ***

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. ***