# Differences

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* Abstract: I will discuss the role of branching geodesics in the theory of Ricci curvature lower bounds in metric measure spaces. I will focus on the $CD(K,N)$ condition by Lott-Sturm-Villani and on the $RCD^*(K,N)$ condition by Ambrosio-Gigli-Savaré-(Erbar-Kuwada-Mondino-Rajala-Sturm). Some of the results have been obtained in collaborations with Ambrosio, Gigli, Mondino, and Sturm. \\ The $CD(K,N)$ condition is satisfied, for instance, by $\mathbb{R}^n$ equipped with any norm and the Lebesgue measure. Thus a $CD(K,N)$ space can be have a lot of branching geodesics. Regardless of branching, one can still select quite good geodesics in the space of probability measures by means of minimizing the entropy. These geodesics can be used as test plans in the Sobolev-space theory and in proving local Poincaré inequalities. Recently it was noticed that even some non-convex (in the Euclidean sense) closed subsets of $\mathbb{R}^n$ (equipped with the supremum norm and the Lebesgue measure) satisfy $CD(K,N)$. This observation shows that $CD(K,N)$ does not have the local-to-global property. \\ The $RCD^*(K,N)$ condition is more strict and it excludes non-Euclidean normed spaces. Via the EVI-formulation one can see that the $RCD^*(K,N)$ spaces are essentially non-branching. Consequently, the Brenier-McCann Theorem (stating the existence of optimal transportation maps from absolutely continuous measures) holds in $RCD^*(K,N)$ spaces and a notion of exponentiation in $RCD^*(K,N)$ spaces can be given.        * Abstract: I will discuss the role of branching geodesics in the theory of Ricci curvature lower bounds in metric measure spaces. I will focus on the $CD(K,N)$ condition by Lott-Sturm-Villani and on the $RCD^*(K,N)$ condition by Ambrosio-Gigli-Savaré-(Erbar-Kuwada-Mondino-Rajala-Sturm). Some of the results have been obtained in collaborations with Ambrosio, Gigli, Mondino, and Sturm. \\ The $CD(K,N)$ condition is satisfied, for instance, by $\mathbb{R}^n$ equipped with any norm and the Lebesgue measure. Thus a $CD(K,N)$ space can be have a lot of branching geodesics. Regardless of branching, one can still select quite good geodesics in the space of probability measures by means of minimizing the entropy. These geodesics can be used as test plans in the Sobolev-space theory and in proving local Poincaré inequalities. Recently it was noticed that even some non-convex (in the Euclidean sense) closed subsets of $\mathbb{R}^n$ (equipped with the supremum norm and the Lebesgue measure) satisfy $CD(K,N)$. This observation shows that $CD(K,N)$ does not have the local-to-global property. \\ The $RCD^*(K,N)$ condition is more strict and it excludes non-Euclidean normed spaces. Via the EVI-formulation one can see that the $RCD^*(K,N)$ spaces are essentially non-branching. Consequently, the Brenier-McCann Theorem (stating the existence of optimal transportation maps from absolutely continuous measures) holds in $RCD^*(K,N)$ spaces and a notion of exponentiation in $RCD^*(K,N)$ spaces can be given.
-    * Antoine Gournay: //{{:grenoble_gournay.pdf|"Transport problems and vanishing of $l^p$ cohomology"}}//+    * Antoine Gournay: //{{:grenoble_gournay2.pdf|"Transport problems and vanishing of $l^p$ cohomology"}}//
*Abstract: In this talk, I will explain how solving transport problems between measures on graphs can be used to get new results on their $l^p$ cohomology. The (reduced) $l^p$ cohomoglogy of an infinite graph of bounded valency is the quotient of the space of functions with $l^p$ gradients by the (closure of) the gradients of functions in $l^p$. It turns out this quotient carries interesting geometric information about the graph, e.g. when $p=1$ this identifies to a function space on ends, e.g. when the graph is hyperbolic it identifies to a certain space of functions on the hyperbolic boundary. Cheeger & Gromov showed the reduced $l^2$-cohomology of (Cayley graphs of) amenable groups is trivial. That the same can be said for other $p$ remains the topic of a conjecture. Finding bounds for $W_1$ and some less traditionnal metrics allows to extend the range where this conjecture holds ( e.g. in certain wreath products and for all $p$, e.g. in all amenable groups for all $p<2$, e.g. in groups of intermediate growth for all $p$ , ...).        *Abstract: In this talk, I will explain how solving transport problems between measures on graphs can be used to get new results on their $l^p$ cohomology. The (reduced) $l^p$ cohomoglogy of an infinite graph of bounded valency is the quotient of the space of functions with $l^p$ gradients by the (closure of) the gradients of functions in $l^p$. It turns out this quotient carries interesting geometric information about the graph, e.g. when $p=1$ this identifies to a function space on ends, e.g. when the graph is hyperbolic it identifies to a certain space of functions on the hyperbolic boundary. Cheeger & Gromov showed the reduced $l^2$-cohomology of (Cayley graphs of) amenable groups is trivial. That the same can be said for other $p$ remains the topic of a conjecture. Finding bounds for $W_1$ and some less traditionnal metrics allows to extend the range where this conjecture holds ( e.g. in certain wreath products and for all $p$, e.g. in all amenable groups for all $p<2$, e.g. in groups of intermediate growth for all $p$ , ...).