The aim of this talk is to introduce the fundamental ideas of the Optimal Transport theory. In particular, after a brief heuristic part to present the fundamental concepts, we formalise in a rigorous way the Optimal Transport problem. We present then the Monge and Kantorovich's formulations, describing their strengths and weaknesses. After stating an important result about compactness of families of probability measures (Prokhorov's theorem), we show the well-posedness of Kantorovich's problem. In the last part, we state the Brenier's theorem and we show an interesting application of it to the isoperimetric inequality.
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