In this talk we will survey some of the main results in the theory of bi-Lipschitz embeddings of metric spaces into certain “simple” normed spaces, such as Euclidean space. We will also sketch some of the techniques used in this field, and some of the many applications of low-distortion embeddings to geometric analysis and the design of approximation algorithms for NP-hard problems. The techniques used in this field draw on analysis, geometry, combinatorics and probability. No prerequisites beyond first year graduate mathematics will be assumed.
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