Orthogonal Polynomials of Several Variables

多变量函数或配置的分析是一个重要的问题领域，与多体量子系统，多变量统计分布，特殊函数，数值立方和代数组合学等主题有关。在这个提议中提出的问题的共同线索是对称群的存在。例如，卡洛格罗-萨瑟兰-莫泽（CSM）系统就构成了一类重要的应用;这些系统是在一维空间中具有一定相互作用（例如平方反比）的若干相同粒子的量子力学问题。对称群是坐标函数的所有排列的群（A型外尔群）或排列和符号变化的群（B型外尔群）;后者出现在自旋模型中。一些经典的正交多项式与Weyl群和紧齐性空间有关。邓克尔发展了微分-差分算子理论（在数学和物理文献中称为“邓克尔算子”），这是这种分析的关键设备。这些算子是通常导数的参数化版本。它们被用来构造某些不变的微分算子（证明了几个CSM模型的完全可积性）。傅里叶变换也有一个相关的推广。特别是这个项目涉及的建设的生成函数的多项式与某些理想的性质（例如正交性或本征函数）与有限反射群有关（属于I、A、B、H类）;非对称Jack多项式定义的广义二项式系数的研究，B型调和多项式正交分解的自伴算子（这将被用来表示在球极坐标系中的线上CSM模型的波函数），一个特殊的CSM模型与三体相互作用的研究。此外，它建议调查可能的修改，原来的微分差分算子与双谱问题或超可积模型。数学分析可以被认为有两个不同的重点，一个是找到精确的公式来描述一些数学系统，如行星或钟摆的运动，或属于晶体一部分的原子的电子，另一个是找到好的和有用的近似和过程，这些近似和过程可以通过采取足够数量的步骤来尽可能接近问题的解决方案。例如，计算机断层扫描并不能给出受试者横截面的完美图像，但它确实提供了实用目的所需的所有细节。这个项目是在分析的一部分，目的是在享受一些对称性的情况下给出精确的解决方案。这可能是不可区分粒子的量子力学问题，以相同方式处理每个数据点的统计分析，或者每个原子有六个最近邻居的晶体分子结构，上，下，左，右，前和后。特别是，Dunkl开发了一种考虑对称性的微积分，从而允许 精确和强大的分析技术。该提案中的问题可以按照对称性的类型进行分类，例如通过将圆旋转60度的倍数（即完整旋转的六分之一）形成的对称性，或者与相同对象的排列相关的对称性，仅举两个例子。该项目的目标是开发工具和发现对称性问题的多变量分析方法;这些将有助于理解相互作用粒子的物理学，复杂数据的统计分析，以及数字化和随后重建声音和图像的技术。

The analysis of multi-variable functions or configurations is an important problem area with connections to topics like quantum systems of many bodies, multi-variate statistical distributions, special functions, numerical cubature, and algebraic combinatorics. The common thread of the problems posed in this proposal is the existence of a symmetry group. An important class of applications, for example, is formed by the Calogero-Sutherland-Moser (CSM) systems; these are quantum-mechanical problems of a number of identical particles in a one-dimensional space with certain interactions (inverse square, for one). The symmetry group is the group of all permutations of the coordinate functions (the type-A Weyl group) or the group of permutations and sign-changes (the type-B Weyl group); the latter occurs in spin models. Some of the classical orthogonal polynomials are associated to Weyl groups and compact homogeneous spaces. Dunkl has developed a theory of differential-difference operators (called "Dunkl operators" in both mathematics and physics literature) which are crucial devices for this analysis. These operators are a parametrized version of the usual derivatives. They are used to construct certain invariant differential operators (which prove the complete integrability of several CSM models). There is also an associated generalization of the Fourier transform. Specifically this project concerns the construction of generating functions for polynomials with certain desirable properties (orthogonality or eigenfunctions, for example) associated to finite reflection groups (of types I, A, B, H); a study of the generalized binomial coefficients defined in terms of nonsymmetric Jack polynomials, a search for useful self-adjoint operators enabling orthogonal decomposition of type-B harmonic polynomials (which would be used to express wave-functions of CSM models on the line in spherical polar coordinates), a study of special CSM models with three-body interactions. Also it is proposed to investigate possible modifications of the original differential-difference operators connected with bispectral problems or super-integrable models. Mathematical analysis can be considered as having two different emphases, one is to find exact formulae to describe some mathematical system, like the motion of the planets or of a pendulum, or an electron belonging to an atom which is part of a crystal, and the other is to find good and useful approximations and processes which can get as close as desired to the solution of a problem by taking an adequate number of steps. For example, computed tomography does not give a perfect image of a cross-section of the subject, but it does provide all the detail needed for practical purposes. This project is in the part of analysis which aims to give exact solutions in situations which enjoy some symmetry. This could be the quantum-mechanical problem of indistinguishable particles, a statistical analysis which treats each data point the same way, or the molecular structure of a crystal where each atom has six nearest neighbors, up, down, left, right, front and back. In particular, Dunkl has developed a calculus which takes the symmetry into account, thus allowing precise and powerful techniques for the analysis. The problems in the proposal can be categorized by the types of symmetry, such as those formed by rotating a circle through multiples of sixty degrees (that is, one sixth of a complete revolution), or those associated to permutations of identical objects, to name just two. The goal of the project is to develop tools and discover methods for multi-variable analysis of problems with symmetry; these will be useful in understanding the physics of interacting particles, statistical analysis of complicated data, and the techniques of digitizing and the subsequent reconstruction of sounds and images.