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.
