This talk will present recent results showing the equivalence of two very different ways of generalising Rademacher's theorem to metric measure spaces. The first was introduced by Cheeger and is based upon differentiation with respect to another, fixed, chart function. The second approach is new for this generality and originates in some ideas of Alberti. It is based upon forming partial derivatives along a very rich structure of Lipschitz curves, analogous to the differentiability theory of Euclidean spaces. By examining this structure further, we naturally arrive to several descriptions of Lipschitz differentiability spaces.
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