• Optimal Transportation
• Winter School & Workshop: New developments in Optimal Transport, Geometry and Analysis

• Schedule
• Participants

Aim of the course it to give an overview of the fast-growing theory of metric measure spaces with Ricci curvature bounded from below. I shall discuss their definition, how to make calculus on them and how to use such calculus to establish geometric properties.

A Ricci limit space is defined by the Gromov-Hausdorff limit of a sequence of Riemannian manifolds with a lower Ricci curvature bound. It is well known that this gives a typical example of RCD spaces. In this talk we will discuss several elliptic PDEs on compact Ricci limit spaces and applications. Applications include continuities of solutions of several PDEs with respect to the Gromov-Hausdorff topology. In particular by using Poisson's equations we will study second order differential calculus on the space.
This talk is based on arXiv:1410.3296.

In this ongoing work joint with M.Fradelizi and O.Guedon, we study a Brunn-Minkowski type inequality which is a spherical version of the Euclidean covariogram defined for convex sets.

Various inequalities in analysis and geometry may often be reduced to a one-dimensional investigation. A prime example in the context of Optimal-Transport is McCann's displacement convexity principle (characterizing the convexity of a functional in L² Wasserstein space via properties of an infinitesimal particle along the 1-D transport interpolating rays), which was successfully used to prove sharp Brunn--Minkowski inequalities on a CD(K,N) weighted-manifold, and which lies at the heart of Lott-Sturm-Villani's extension of Ricci curvature to geodesic metric-measure spaces. However, with very few exceptions, this L² Optimal-Transport approach does not seem to yield sharp results for various other Sobolev-type or isoperimetric inequalities.

Our plan is to present the notion of 1-D localization - a disintegration of a measure into 1-D disjointly supported measures ("needles"), which may be used to obtain sharp results in a variety of problems, ranging from isoperimetric, Brunn-Minkowski, Sobolev-type and many more. We will see that on Riemannian manifold equipped with a density, 1-D localization preserves the CD(K,N) property. We will then proceed to describe two or three (time permitting) explicit constructions of such localizations, using different methods:

1. Geometric Measure Theory based (after P. Levy and Gromov) - a localization arising from isoperimetric minimizers, which may be used to prove sharp isoperimetric inequalities on weighted manifolds.

2. Bisection based (after Payne-Weinberger, Gromov-V. Milman and Kannan-Lovasz-Simonovits) - a localization constructed by repeatedly bisecting Euclidean space.

3. L¹ Optimal-Transport based (after Klartag) - a very recent extension of the bisection method to weighted Riemannian manifolds, by using L¹ (and not L² !) Optimal-Transport.

Free functional inequalities on the circle are developed in this talk. There are differences from the free inequalities on the line case and also from the classical counterparts. Perhaps the main novelty here is the Wasserstein distance involved in the transportation inequality which also enters the HWI inequality and is implicit also in proof of the Log-Sobolev inequality. Also the Poincare inequality has a relatively interesting twist.

I will present some old and new results on Gromov-Hausdorff tangents of metric spaces. The general question is how the local structure of the space is related to the structure of the tangents. For example, do the metric dimensions (such as Hausdorff, Assouad or Nagata dimension) of the space pass to the tangents? We will recall implications of uniqueness of tangents, and of uniform convergence to the space of tangents. As examples we will discuss Lipschitz differentiability spaces, Reifenberg vanishing-flat metric spaces, equiregular subRiemannian manifolds and metric spaces with Ricci curvature lower bounds.

Lecture I: "An Introduction to Intrinsic Flat Convergence"

Intrinsic Flat Convergence was first introduced in joint work with Stefan  Wenger building upon work of Ambrosio-Kirchheim to address a question proposed by Tom Ilmanen. In this talk, I will present an overview of the initial paper on the topic [JDG 2011]. I will briefly describe key examples and the class of limit spaces, which we call integral current spaces, posponing the rigorous details for later in the week. I will also present Wenger's Compactness Theorem [CVPDE 2011] which states that a sequence of oriented manifolds with volume and diameter and boundary volume bounded uniformly from above has a subsequence which converges in the intrinsic flat sense to an integral current space (possibly the zero space).
See https://sites.google.com/site/intrinsicflatconvergence/ for links to these papers and others that will be presented this week.

Lecture II: "Ambrosio-Kirchheim Currents on Metric Spaces"

Recall that the original notion of currents and flat convergence in Euclidean space was defined by Federer-Fleming in [Annals 1960]. In this talk, I will briefly review Federer-Fleming's work. Then I will present the key ideas behind Ambrosio-Kirchheim's paper entitled "Currents on Metric Spaces" which appeared in [Acta 2000]. This includes De Giorgi's notion of a tuple of Lipschitz functions on a metric space which replaces the notion of a smooth differential form on Euclidean space.

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.29.6534

Lecture III: "The Intrinsic Flat Distance between Integral Current Spaces"

In this talk, I will complete a more detailed presentation of the joint paper with Stefan Wenger that appeared in [JDG 2011]. We introduced the notion of an integral current space, (X,d,T), which is a metric space (X,d) endowed with an integral current structure, T, which is an integral current on the metric completion of X such that X is the set of positive density for T.
Oriented Riemannian manifolds with boundary of finite volume are integral current spaces. More generally integral current spaces include rectifiable spaces with oriented charts and rectifiable boundaries and have Borel integer valued weight functions. We define the intrinsic flat distance on this class of spaces in joint work with Wenger appearing in [JDG 2011] and we prove that if the intrinsic flat distance between two precompact integral current spaces is zero, then there is a current preserving isometry between the two spaces.

http://arxiv.org/abs/1002.1073

Lecture IV: "Scalar Curvature and Intrinsic Flat Convergence"

Gromov explored possible synthetic notions of nonnegative scalar curvature in his paper on Plateau Stein manifolds [CEJM 2014]. There he stated that intrinsic flat convergence might be the correct notion for studying limits of manifolds with nonnegative scalar curvature. There have been a number of papers concerning intrinsic flat convergence and nonnegative scalar curvature including joint work with Dan Lee [Crelle 2014], joint work with LeFloch [JFA 2015] and joint work with Dan Lee and Lan-Hsuan Huang (arxiv).
In this talk, we will discuss conjectures in this area and techniques used to prove intrinsic flat convergence. One of the techniques is an explicit estimate for the intrinsic flat distance between manifolds proven in joint work with Sajjad Lakzian [CAG 2013] which Lakzian has applied in solo work to study Ricci flow through singularities and Calabi-Yau conifolds [arxiv].

Lecture V: "Arzela-Ascoli and Bolzano-Weierstrass Theorems"

In this talk I will discuss the disappearance of points under intrinsic flat convergence either through cancellation or collapse. I will review material from the first paper with Wenger [JDG 2011] including the fact that if a sequence of integral current spaces has a Gromov-Hausdorff limit then a subsequence has an Intrinsic flat limit which is either the 0 space or a subset of the GH limit. I will also review the second joint paper with Wenger [CVPDE 2011] where we proved that for a sequence of noncollapsing manifolds with nonnegative Ricci curvature, the two notions of convergence agree. Then I will new results of mine appearing in [arxiv:1402.6066] including the Flat to Gromov-Hausdorff Convergence Theorem, two Arzela-Ascoli Theorems and a Bolzano-Weierstrass Theorem. These new results have been applied in work of Nan Li and Raquel Perales in which they prove that for noncollapsing sequences of integral current spaces of weight 1 with uniform lower Alexandrov curvature bounds the Gromov-Hausdorff and Intrinsic flat limits agree. Thay have also been applied in joint work with Zahra Sinaei and with Lan-Hsuan Huang and Dan Lee. Additional Arzela-Ascoli and Bolzano-Weierstass Theorems involving the Gromov Filling Volume will appear in upcoming joint work with Jacobus Portegies.
A complete listing of all papers concerning intrinsic flat convergence may be found at: https://sites.google.com/site/intrinsicflatconvergence/

http://arxiv.org/abs/1402.6066