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I will talk about a new distance function on the set of Hermitian positive definite (HPD) matrices. This function originally arose in an application in computer vision, but it turned out to have many remarkable properties of its own, some of which I shall present today. Consider the fundamental task of computing distances between two HPD matrices. This task can be quite nontrivial if we use distances that respect the non-Euclidean geometry of HPD matrices. For example, the widely used "natural" Riemannian distance on the manifold of HPD matrices is computationally demanding as it requires calculation of generalized eigenvalues. Such computational concerns lead us to introduce an alternative distance on HPD matrices. Our new distance respects non-euclidean geometry, while being faster to compute and "easier" to use than the Riemannian distance. But beyond just computational ease, it turns out that the new metric enjoys a number of properties that tie it to various areas: harmonic analysis on symmetric spaces, non-euclidean geometry, matrix theory, and nonconvex global optimization. I will touch upon all these connections and mention some challenging open problems.