We will discuss several results concerning quantitative rectifiability in metric spaces, which generalize Euclidean results. We will spend some time explaining both the metric space results as well as their Euclidean counterparts. An example of such a result is a structure theorem, which characterizes subsets of rectifiable curves (the Analyst's Traveling Salesman theorem). This theory is presented in terms of multi-scale analysis and multi-scale constructions, and uses a language which is analogous to that of wavelets. Some of the results we will present will be dimension free.
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