In this note we will construct an example of a quasi-geodesic in the Te-ichm uller space of a surface with the following instability property: Theorem For any hyperbolic surface S whose Teichm uller space T (S) has dimension at least 4, there is a bi-innnite quasi-geodesic L in T (S) such that 1. L projects to a compact part of the moduli space M(S), but 2. L does not lie in a bounded neighborh...

There is growing interest in using the close connection between differential geometry and statistics to model smooth manifold-valued data. In particular, much work has been done recently to generalize principal component analysis (PCA), the method of dimension reduction in linear spaces, to Riemannian manifolds. One such generalization is known as principal geodesic analysis (PGA). This paper, ...

The problem of geodesic connectedness in semi-Riemannian manifolds (i.e. the question whether each two points of the manifold can be joined by a geodesic) has been widely studied from very different viewpoints. Our purpose is to review these semi-Riemannian techniques, and possible extensions. In the Riemannian case, it is natural to state this problem on (incomplete) manifolds with (possibly n...

Measured geodesic laminations is a remarkable abstraction (due to W. P. Thurston) of many otherwise unrelated phenomena occurring in differential geometry, complex analysis and geometric topology. In this article we focus on connections of geodesic laminations with the inductive limits of finite-dimensional semi-simple C∗-algebras (AF C∗-algebras). Our main result is a bijection between combina...

Classical convexity can be naturally translated to graphs considering shortest paths, giving rise to the geodesic convexity. Moreover, if we consider induced paths between vertices, we get the so-called monophonic convexity. In this work, we present some results involving monophonic sets.

Given a family of probability measures in P (X ), the space of probability measures on a Hilbert space X , our goal in this paper is to highlight one ore more curves in P (X ) that summarize efficiently that family. We propose to study this problem under the optimal transport (Wasserstein) geometry, using curves that are restricted to be geodesic segments under that metric. We show that concept...

A symmetric random variable is called a Gaussian mixture if it has the same distribution as the product of two independent random variables, one being positive and the other a standard Gaussian random variable. Examples of Gaussian mixtures include random variables with densities proportional to e−|t| p and symmetric p-stable random variables, where p ∈ (0, 2]. We obtain various sharp moment an...

Abstract. The space which consists of measures having finite second moment is an infinite dimensional metric space endowed with Wasserstein distance, while the space of Gaussian measures on Euclidean space is parameterized by mean and covariance matrices, hence a finite dimensional manifold. By restricting to the space of Gaussian measures inside the space of probability measures, we manage to ...

We study the dynamics of the geodesic flow for a class of non-complete Riemannian metrics on a negatively curved surface M . These finite area surfaces are composed of finitely many “singular” surfaces of revolutions of the form y = x, r > 1 for 0 ≤ x ≤ 1 (thin pieces), together with connecting surfaces of bounded negative curvature (thick pieces). The curvature at every point is negative and b...

After a brief introduction on gradient flows in metric spaces and on geodesically convex functionals, we give an account of the proof (following the outline of [3, 7]) of the existence and uniqueness of the gradient flow of the Entropy in the space of Borel probability measures over a compact geodesic metric space with Ricci curvature bounded from below. Preprint SISSA 17/2014/MATE