Geometry of the space of probability measures and its applications

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We studied the geometry of Wasserstein spaces, which are the metric spaces consisting of probability measures on metric spaces. One of our achievements is the investigation of the geometric structure of the Wasserstein space over a possibly highly singular space (an Alexandrov space), and the establishment of the theory of gradient flows in such a space. We also generalized the known equivalence between the convexity of a certain entropy on the Wasserstein space and a curvature bound of the underlying space to a wider class of spaces (Finsler manifolds).