Orthogonal Polynomials of Several Variables

多变量函数或构型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、数值培养和代数组合等主题有关。在这个建议中提出的问题的共同线索是一个对称群的存在。例如，一类重要的应用是由Calogero-Sutherland-Moser （CSM）系统形成的；这是一个量子力学问题，在一维空间中，许多相同的粒子具有一定的相互作用（平方反比）。对称群是坐标函数的所有排列的群（a型Weyl群）或排列和符号变化的群（b型Weyl群）；后者发生在自旋模型中。一些经典正交多项式与Weyl群和紧齐次空间有关。Dunkl发展了一种微分-差分算子理论（在数学和物理文献中称为“Dunkl算子”），这是这种分析的关键手段。这些算子是通常导数的参数化版本。利用它们构造了若干不变微分算子（证明了几种CSM模型的完全可积性）。傅里叶变换还有一个相关的推广。具体来说，这个项目涉及到与有限反射群（类型I， A， B， H）相关的多项式具有某些理想性质（例如正交性或特征函数）的生成函数的构造；研究了用非对称Jack多项式定义的广义二项式系数，寻找了能实现b型调和多项式正交分解的有用的自伴随算子（用于在球极坐标上表示CSM模型的波函数），研究了具有三体相互作用的特殊CSM模型。此外，还提出了对原微分-差分算子在双谱问题或超可积模型中的可能修正。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某些数学系统，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值和过程，这些近似值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，计算机断层扫描不能给出人体横切面的完美图像，但它确实提供了实际用途所需的所有细节。这个项目是分析的一部分，目的是在一些对称的情况下给出精确的解。这可能是不可区分粒子的量子力学问题，一种以相同方式处理每个数据点的统计分析，或者是一个晶体的分子结构，其中每个原子有六个最近的邻居，上，下，左，右，前后。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。提案中的问题可以根据对称的类型进行分类，例如那些由旋转一个圆60度的倍数（即，完整旋转的六分之一）形成的问题，或者那些与相同物体的排列有关的问题，仅举两例。该项目的目标是开发工具和发现方法，用于对称问题的多变量分析；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，以及声音和图像的数字化和后续重建技术。

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.