# mms&

# convergence

The mms&convergence talks took place last year 2020. They were mostly related to Sectional/Ricci lower bounds, Gromov-Hausdorff and Intrinsic Flat convergence and, metric measure spaces. Watch out for future mms&convergence talks!

---> **Coming back in September 2021** <---

June 5:

Title: A volume comparison theorem for characteristic numbers

Abstract: We show that assuming lower bounds on the Ricci curvature and the injectivity radius the absolute value of any characteristic number of a Riemannian manifold M is bounded proportional to the volume, i.e. bounded by Cvol(M) where C depends only on the characteristic number, the dimension of M, and both bounds. The proof relies on the definition of a connection for an harmonic Hölder regular metric tensor as they appear for instance as Gromov-Hausdorff limits of Riemannian manifolds

June 12:

Title: Applications of needle decomposition for metric measure spaces

Abstract: In this talk I show how one can formulate and prove the Heintze-Karcher inequality in the context of nonsmooth spaces that satisfy a Ricci curvature bound in the sense of Lott, Sturm and Villani. As a by-product one obtains a notion of mean curvature for the boundary of Borel sets in such spaces. My approach is based on the needle decomposition method introduced for this framework by Cavalletti and Mondino.

June 19:

Sergio Zamora

Fundamental Groups and Limits of Almost Homogeneous Spaces

Abstract: We show that for a sequence of proper length spaces X_n with groups Gamma_n acting discretely and almost transitively by isometries, if they converge to a proper finite dimensional length space X, then X is a nilpotent Lie group with an invariant sub-Finsler Carnot metric. Also, for large enough n, there are subgroups Lambda_n \leq pi_1(X_n) and surjective morphisms Lambda_n to pi_1(X).

June 26:

Ricci limit spaces : An introduction to the tools of Cheeger-Jiang-Naber's work

The goal of this expository talk is to explain parts of the work of J. Cheeger, W. Jiang and A. Naber: https://arxiv.org/abs/1805.07988 . For a converging, non-collapsing sequence of Riemannian manifolds with a uniform Ricci lower bound, they proved that singular strata of the limit space are rectifiable. Some of the key tools in the proof include quantitative stratification, which was first introduced in previous work of Cheeger-Naber, and new related volume estimates, together with a precise study of neck regions. After a brief review of Cheeger-Colding theory, the talk will focus on explaining the notions of quantitative stratifications, neck regions and their role in the proof.

July 3:

Null Distance and Convergence of Warped Product Spacetimes

Abstract: The null distance was introduced by Christina Sormani and Carlos Vega as a way of turning a spacetime into a metric space. This is particularly important for geometric stability questions relating to spacetimes such as the stability of the positive mass theorem. In this talk, we will describe the null distance, present properties of the metric space structure, and examine the convergence of sequences of warped product spacetimes equipped with the null distance. This is joint work with Annegret Burtscher.

September 4

Gilles Carron

Euclidean heat kernel rigidity

Abstract : This is joint work with David Tewodrose (Cergy). I will explain that a metric measure space with Euclidean heat kernel are Euclidean. An almost rigidity result comes then for free, and this can be used to give another proof of Colding's almost rigidity for complete manifold with non negative Ricci curvature and almost Euclidean growth.

September 11:

Danka Lučić (Jyväskylä)

Techniques for proving infinitesimal Hilbertianity

Abstract: A metric space is said to be "universally infinitesimally Hilbertian" if, when endowed with any arbitrary Radon measure, its associated 2-Sobolev space is Hilbert. For instance, all (sub)Riemannian manifolds and CAT(K) spaces have this property. In this talk, we will illustrate three different strategies to prove the universal infinitesimal Hilbertianity of the Euclidean space, which is the base case and where all the known approaches work.

The motivations come, among others, from the study of rectifiable metric measure spaces, of metric-valued harmonic maps, and of variational problems (such as models representing low-dimensional elastic structures).

September 25: Andrea Mondino (University of Oxford)

An optimal transport formulation of the Einstein equations of general relativity

Abstract: In the seminar I will present a recent work joint with S. Suhr (Bochum) giving an optimal transport formulation of the full Einstein equations of general relativity, linking the (Ricci) curvature of a space-time with the cosmological constant and the energy-momentum tensor. Such an optimal transport formulation is in terms of convexity/concavity properties of the Shannon-Bolzmann entropy along curves of probability measures extremizing suitable optimal transport costs. The result, together with independent work by McCann on lower bounds for Lorentzian Ricci Curvature, gives a new connection between general relativity and optimal transport; moreover it gives a mathematical reinforcement of the strong link between general relativity and thermodynamics/information theory that emerged in the physics literature of the last years.

October 2:

Flavia Santarcangelo (SISSA)

Independence of synthetic Curvature Dimension conditions on transport distance exponent

Abstract: The celebrated Lott-Sturm-Villani theory of metric measure spaces furnishes synthetic notions of a Ricci curvature lower bound $K$ joint with an upper bound $N$ on the dimension.

Their condition, called the Curvature-Dimension condition and denoted by $\mathsf{CD}(K,N)$, is formulated in terms of a modified displacement convexity of an entropy functional along $W_{2}$-Wasserstein geodesics. In a joint work with A. Akdemir, F. Cavalletti, A. Colinet and R. McCann, we show that the choice of the squared-distance function as transport cost does not influence the theory. In particular, by denoting with $\mathsf{CD}_{p}(K,N)$ the analogous condition but with the cost given by the $p^{th}$ power of the distance, we prove that $\CD_{p}(K,N)$ are all equivalent conditions for any $p>1$ --- at least in spaces whose geodesics do not branch.

Following the strategy introduced in the work by Cavalletti-Milman, we also establish the local-to-global property of $\mathsf{CD}_{p}(K,N)$ spaces.

Finally, we will present a result obtained in collaboration with F. Cavalletti and N. Gigli that, combined with the one previously described, allows to conclude that for any $p\geq1$, all the $\mathsf{CD}_{p}(K,N)$ conditions, when expressed in terms of displacement convexity, are equivalent, provided the space $X$ satisfies the appropriate essentially non-branching condition.

October 9:

Daniele Semola (Scuola Normale Superiore)

Rectifiability of RCD(K,N) spaces via delta-splitting maps

Abstract: The theory of metric measure spaces verifying the Riemannian-Curvature-Dimension condition RCD(K,N) has attracted a lot of interest in the last years. They can be thought as a non smooth counterpart of the class of Riemannian manifolds with Ricci curvature bounded from below by K and dimension bounded from above by N.

In this talk, after providing some background and motivations, I will describe a simplified approach to the structure theory of these spaces relying on the so-called delta-splitting maps. This tool, developed by Cheeger-Colding in the study of Ricci limits, has revealed to be extremely powerful also more recently in the study of RCD spaces.

The seminar is based on a joint work with Elia Brue' and Enrico Pasqualetto.