The original motivation for writing this book was rather personal. The first author, in the course of his teaching career in the Department of Pure Mathematics and Mathematical Statistics (DPMMS), University of Cambridge, and St John's College, Cambridge, had many painful experiences when good (or even brilliant) students, who were interested in the subject of mathematics and its applications and who performed well during their first academic year, stumbled or nearly failed in the exams. This led to great frustration, which was very hard to overcome in subsequent undergraduate years. A conscientious tutor is always sympathetic to such misfortunes, but even pointing out a student's obvious weaknesses (if any) does not always help. For the second author, such experiences were as a parent of a Cambridge University student rather than as a teacher.

We therefore felt that a monograph focusing on Cambridge University mathematics examination questions would be beneficial for a number of students. Given our own research and teaching backgrounds, it was natural for us to select probability and statistics as the overall topic. The obvious starting point was the first-year course in probability and the second-year course in statistics. In order to cover other courses, several further volumes will be needed; for better or worse, we have decided to embark on such a project.

Introduction

We are interested in the following question: given a compact subset K of ℝd with positive reach, and a Hausdorff approximation P of this set, is it possible to approximate Federer's curvature measures of K (see [9] or Section 12.2.2 for a definition) from P only? A positive answer to this question has been given in [8] using convex analysis. In this chapter, we show that such a result can also be deduced from a careful study of the “size” – that is, the covering numbers – of the medial axis.

The notion of medial axis, also known as ambiguous locus in Riemannian geometry, has many applications in computer science. In image analysis and shape recognition, the skeleton of a shape is often used as an idealized version of the shape, which is known to have the same homotopy type as the original shape [14].

Overview

Since the creation of Ricci flow by Hamilton in 1982, a rich theory has been developed in order to understand the behaviour of the flow, and to analyse the singularities that may occur, and these developments have had profound applications, most famously to the Poincaré conjecture. At the heart of the theory lie a large number of a priori estimates and geometric constructions, which include most notably the Harnack estimates of Hamilton, the L-length of Perelman (in the spirit of Li-Yau), the logarithmic Sobolev inequality arising from Perelman's W-entropy, and the reduced volume of Perelman, amongst others.

The objective of these lectures is to explain this theory from the point of view of optimal transportation. As I explain in Section 5.4, Ricci flow and optimal transportation combine rather well, and we will see fundamental but elementary aspects of this when we see in Theorem 5.2 how diffusions contract under reverse-time Ricci flow. However, the key to the whole theory is to realise to which object one should apply this result: not the original Ricci flow, but a new Ricci flow derived from the original one, on a base manifold of one higher dimension, that we call the canonical soliton. In this way, essentially the entire foundational theory of Ricci flow mentioned above drops out naturally.

Throughout the lectures I emphasise the intuition; the objective is to demonstrate how one can discover the theory rather than treat it as a black box that just happens to work.

Introduction and main concepts

Lyapunov conditions appeared a long time ago. They were particularly well fitted to deal with the problem of convergence to equilibrium for Markov processes; see [23, 38–40] and references therein. They also appeared earlier in the study of large and moderate deviations for empirical functionals of Markov processes (for examples, see Donsker and Varadhan [21, 22], Kontoyaniis and Meyn [33, 34], Wu [47, 48], Guillin [28, 29]), for solving the Poisson equation [24].

Their use to obtain functional inequalities is however quite recent, even if one may afterwards find hint of such an approach in Deuschel and Stroock [19] or Kusuocka and Stroock [35]. The present authors and coauthors have developed a methodology that has been successful for various inequalities: Lyapunov–Poincaré inequalities [4], Poincaré inequalities [3], transportation inequalities for Kullback information [17] or Fisher information [32], super Poincaré inequalities [16], weighted and weak Poincaré inequalities [13], or [18] for super weighted Poincaré inequalities. We finally refer to the forthcoming book [15] for a complete review. For more references on the various inequalities introduced here we refer to [1, 2, 36, 46]. The goal of this short review is to explain the methodology used in these papers and to present various general sets of conditions for this panel of functional inequalities. The proofs will of course be only schemed and we will refer to the original papers for complete statements.