The geometry of Metric spaces equipped with a probability measure is a very dynamic field. One motivation for the study of such spaces is that they are the natural limits of Riemannian manifolds in many contexts. In this talk, I will introduce basic properties of metric measure spaces and the Gromov-Prohorov distance on them. I will also discuss joint work with Manjunath Krishnapur in which we show that independently sampling points according to the given measure gives an asymptotically bi-Lipschitz correspondence between Metric measure spaces and Random matrices. Finally, I will briefly discuss work with Divakaran in which we study the compactification of the Moduli space of Riemann surfaces in terms of metric measure spaces.

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