Mathematical Sciences: Harmonic Analysis Algorithms

一个REU现场项目将侧重于使用先进的计算实验，使本科生研究人员接触到各种当前的数学研究前沿。将提供一个程序来进行非线性演化方程和椭圆系统解的定性行为的数值实验。多孔介质方程解的渐近行为的研究代表了一个特定的领域。所获得的数值数据将扩大最近关于Sobolev不等式中极值函数唯一性的理论结果。其他的工作将集中在pi变换上，这是一种将一般函数分解成简单项和的新技术。它是基于近二十年来谐波分析的重要进展。除了相对容易理解外，该变换还表现出快速傅里叶变换的某些速度特性。变换的性质使它在数值计算中很有用；了解这些特性是一项非常适合本科生研究的任务。实习生还将参与创建数值模型，用于计算几个复杂变量空间中的域的自同构群。这个高度理论化的领域可以通过某些边界微分量（称为chen - moser - tanaka不变量）来进行数值解释。将域的边界置于标准形式所需的繁琐计算现在可以进行。学生将掌握基本的代数和符号操作算法，同时学习复杂的数学结构及其分析。

An REU site project will focus on the use of advanced computational experimentation to expose undergraduate researchers to a variety of current mathematical research frontiers. A program to carry out numerical experiments on the qualitative behavior of solutions of nonlinear evolution equations and elliptic systems will be available. Studies of asymptotic behavior of solutions to the porous medium equation represents one specific area. Numerical data obtained will amplify recent theoretical results on uniqueness for extremal functions in the Sobolev inequality. Other work will concentrate on the phi-transform, a new technique developed for decomposing general functions into sums of simple terms. It is based on important advances in harmonic analysis in the last twenty years. In addition to being relatively easy to understand, this transform exhibits certain speed characteristics of the fast Fourier transform. Properties of the transform make it useful in numerical work; understanding these properties is a task well suited for undergraduate research. Student interns will also be involved in the creation of numerical models for calculation of automorphism groups for domains in the space of several complex variables. This highly theoretical field lends itself to numerical interpretation through certain boundary differential quantities known as the Chern-Moser-Tanaka invariants. Tedious calculations necessary to put the boundary of a domain in normal form can now be carried out. Students will develop the basic algebra and symbol manipulation algorithms which will be needed, learning about the complex mathematical structures and their analysis simultaneously.