mms&convergence seminar

October 9, 2020: 

Daniele Semola (Scuola Normale Superiore)

Video 

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.

October  2, 2020:  

Flavia Santarcangelo (SISSA)

Video 

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.

September 25, 2020: Andrea Mondino (University of Oxford)

 An optimal transport formulation of the Einstein equations of general relativity

Video 

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.

September  11, 2020: 

Danka Lučić (Jyväskylä)

Techniques for proving infinitesimal Hilbertianity

Video 

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).