Abstract or Additional Information
To improve our understanding of a metric space, it is often helpful to realize the space within some Euclidean space. The embedding problem is concerned with recognizing those spaces which admit an embedding into some Euclidean space that does not distort its geometry too much. The bi-Lipschitz emebedding problem is concerned with identifying those metric spaces for which such an embedding exists. The embedding problem has recently generated great interest in theoretical computer science and, more specifically, in graphic imaging and storage and access issues for large data sets. In the first part of the talk we will examine the embeddability of two well-known sub-Riemannian manifolds, the Grushin plane and the Heisenberg group. In the second part we will discuss the embeddability of metric trees with good geometry. The talk is based on joint works with Romney (2017), Li, Chousionis, and Zimmerman (2020), David (2020), and David and Eriksson-Bique (2021).