We define the notion of a Ricci curvature lower bound for parametrized statistical models. Following the seminal ideas of Lott–Sturm–Villani, we define this notion based on the geodesic convexity of the Kullback–Leibler divergence in a Wasserstein statistical manifold, that is, a manifold of probability distributions endowed with a Wasserstein metric tensor structure. Within these definitions, which are based on Fisher information matrix and Wasserstein Christoffel symbols, the Ricci curvature is related to both, information geometry and Wasserstein geometry. These definitions allow us to formulate bounds on the convergence rate of Wasserstein gradient flows and information functional inequalities in parameter space. We discuss examples of Ricci curvature lower bounds and convergence rates in exponential family models.

Motivated by results on the simplicial volume of locally symmetric spaces of finite volume, in this note, we observe that the simplicial volume of the moduli space Mg,n is equal to 0 if g  2; g = 1, n  3; or g = 0, n  6; and the orbifold simplicial volume of Mg,n is positive if g = 1, n = 0, 1; g = 0, n = 4. We also observe that the simplicial volume of the Deligne-Mumford compactification of Mg,n is equal to 0, and the simplicial volumes of the reductive Borel-Serre compactification of arithmetic locally symmetric spaces ..\X and the Baily-Borel compactification of Hermitian arithmetic locally symmetric spaces ..\X are also equal to 0 if the Q-rank of ..\X is at least 3 or if ..\X is irreducible and of Q-rank 2.

In this paper, we establish various L 2 -estimates for the exterior differential operator on p-convex Riemannian manifolds in the sense of Harvey and Lawson. As applications, we establish a Carleman type estimate which is uniform with respect to both weight functions and domains, and we also obtain topological restrictions for a Riemannian manifold to be p-convex.

In this paper, we will study the (linear) geometric analysis on metric measure spaces. We will establish a local Li-Yau’s estimate for weak solutions of the heat equation and prove a sharp Yau’s gradient gradient for harmonic functions on metric measure spaces, under the Riemannian curvature-dimension condition RCD(K; N).

No abstract uploaded!