Related papers: Deriving Perelman's entropy from Colding's monoton…

Under the definition of Ricci curvature bounded below for Alexandrov spaces introduced by Zhang-Zhu, we generalize a result by Colding that an n dimentional manifold with Ricci curvature greater or equal to n minus 1 and volume close to…

Let $M$ be a Riemannian manifold with dimension greater or equal to $3$ which admits a complete, finite-volume Riemannian metric $g_0$ locally isometric to a rank-1 symmetric space of non-compact type. The volume entropy rigidity theorem…

Spatial random permutations were originally studied due to their connections to Bose-Einstein condensation, but they possess many interesting properties of their own. For random permutations of a regular lattice with periodic boundary…

The purpose of this report is to acknowledge the influence of M. Gromov's vision of geometry on our own works. It is two-fold: in the first part we aim at describing some results, in dimension 3, around the question: which open 3-manifolds…

This paper provides tight bounds on the R\'enyi entropy of a function of a discrete random variable with a finite number of possible values, where the considered function is not one-to-one. To that end, a tight lower bound on the R\'enyi…

We prove a sharp dimension-free isoperimetric inequality, involving the volume entropy, in non-compact metric measure spaces with non-negative synthetic Ricci curvature.

By Gromov's compactness theorem for metric spaces, every uniformly compact sequence of metric spaces admits an isometric embedding into a common compact metric space in which a subsequence converges with respect to the Hausdorff distance.…

This paper studies the behavior of the entropy numbers of classes of functions with bounded integral norms from a given finite dimensional linear subspace. Upper bounds of these entropy numbers in the uniform norm are obtained and applied…

We introduce a flow of Riemannian metrics and positive volume forms over compact oriented manifolds whose formal limit is a shrinking Ricci soliton. The case of a fixed volume form has been considered in our previous work. We still call…

We show that the systolic constant, the minimal entropy, and the spherical volume of a manifold depend only on the image of the fundamental class under the classifying map of the universal covering. Moreover, we compute the systolic…

The various entropy bounds that exist in the literature suggest that spacetime is fundamentally discrete, and hint at an underlying relationship between geometry and "information". The foundation of this relationship is yet to be uncovered,…

In any static spacetime the quasi-local Tolman mass contained within a volume can be reduced to a Gauss-like surface integral involving the flux of a suitably defined generalized surface gravity. By introducing some basic thermodynamics and…

We study some new isoperimetric inequalities on graphs. We etablish a relation between the volume entropy (or asymptotic volume), the systole and the first Betti number of weighted graphs. We also find bounds for the volume, associated to…

A dynamical estimate is given for the Boltzmann entropy of the Universe, under the simplifying assumptions provided by Newtonian cosmology. We first model the cosmological fluid as the probability fluid of a quantum-mechanical system. Next,…

We derive a family of weighted scalar curvature monotonicity formulas for generalized Ricci flow, involving an auxiliary dilaton field evolving by a certain reaction-diffusion equation motivated by renormalization group flow. These scalar…

We show, by using Regge calculus, that the entropy of any finite part of a Rindler horizon is, in the semi-classical limit, one quarter of the area of that part. We argue that this result implies that the entropy associated with any horizon…

The entropy of a thermally isolated system should not decrease after a quench or external driving. For a classical system following Hamiltonian dynamics, we show how this statement emerges for a large system in the sense that the extensive…

This is the second paper in a series of works devoted to nonholonomic Ricci flows. By imposing non-integrable (nonholonomic) constraints on the Ricci flows of Riemannian metrics we can model mutual transforms of generalized Finsler-Lagrange…

Experimental and theoretical results about entropy limits for macroscopic and single-particle systems are reviewed. It is clarified when it is possible to speak about a quantum of entropy, given by the Boltzmann constant k, and about a…

We explain a characterization of Einstein-Fano manifolds in terms of the lower bound of the density of the volume of the K\"ahler-Ricci Flow. This is a direct consequence of Perelman's uniform estimate for the K\"ahler-Ricci Flow and a…