Mathematical Sciences: RUI: Quasisymmetric Embeddings

在这个项目中，首席研究员将研究准对称嵌入和相关类型的映射。特别是，她会研究p维欧几里德空间到n维空间的映射，其中p在1到n-1之间。研究的一个主要目的是对所有准对称常数等于1的嵌入进行分类。目前已知的这个方向的结果是在p=n和p=1, n=2的情况下。可以预期，对于p=n-1的情况，唯一这样的嵌入是仿射（线性+常数）嵌入。复变量的大部分主题都与所谓的“几何函数理论”有关，在这个子领域中，人们根据函数的定义域（函数定义的点的集合）和函数的值域（函数的值的集合）的几何来研究函数或映射的性质。在这个项目中，首席研究员将研究以规定方式扭曲空间的不同维度欧几里得空间之间的某些类型的映射。研究的目的是对具有给定失真的所有类型的函数进行分类。

In this project the principal investigator will study quasisymmetric embeddings and related classes of mappings. In particular, she will look at mappings of p-dimensional euclidean space into n-dimensional space, where p is between 1 and n-1. A major objective of the research is to classify all such embeddings that have a quasisymmetry constant equal to one. Currently results in this direction are known for the cases p=n and p=1, n=2. It is expected that for the case p=n-1 the only such embeddings are affine (linear + constant) embeddings. Much of the subject of complex variables is concerned with what is called "geometric function theory", the subarea in which one studies the properties of functions or mappings in terms of the geometry of the domain of the function (the set of points on which the function is defined) and the range of the function (the set of values of the function). In this project the principal investigator will study certain types of mappings between euclidean spaces of different dimensions that distort the spaces in a prescribed way. The goal of the research is to classify all types of functions that have a given distortion.