## Presentation

 Sourav Chatterjee (Courant Institute, New York University, USA) Combinatorial properties of convex sets. The course will cover several topics in combinatorial convexity, where theorems of Caratheodory, Helly, Radon, and Tverberg are the typical and classical results. We plan to investigate weak epsilons-nets, halving lines and planes, the $(p,q)$ problem and its solution, extensions to lattice convex sets, and colourful versions of the above mentioned classical theorems. Further possible topics are transversals, lattice polytopes and random polytopes. The methods here use tools from linear algebra, combinatorics, topology, geometry, probability theory, and geometry of numbers. Thresholds. Thresholds for increasing properties are a central concern in probabilistic combinatorics and elsewhere. (An increasing property, say F, is a superset- closed family of subsets of some (here finite) set X, and the threshold question for such an F asks, roughly, about how many random elements of X should one choose to make it likely that the resulting set lies in F? For example: about how many random edges from the complete graph Kn are typically required to produce a Hamiltonian cycle?) These lectures will primarily focus on recent progress (and lack thereof) on threshold and related questions. Lecture Notes in PDF: Day1 Day2 Day3 ZIP of HTML files

These autumn days are part of the the A.N.R. project GeMeCoD. They will consist in two courses:

• “Analysis and geometry on spaces with Ricci curvature bounded from below” by Nicola Gigli
• “Ricci curvature and functional inequalities in discrete spaces” by Cyril Roberto

and talks by:

• Michel Bonnefont: “Measure doubling property and Poincaré inequality in sub-elliptic geometry under a curvature-dimension criterion”
• Abstract: In this talk, I will present a curvature-dimension criterion in sub-elliptic geometry that generalizes the one of Bakry-Emery. With this criterion one can obtain Li-Yau type gradient estimates and a reverse log-sobolev inequality for the semigroup. Finally, in the non-negative curvature case, one can then deduce the measure doubling property, gaussian estimates for the heat kernel and the Poincaré inequality on balls. This is joint work with F. Baudoin and N. Garofalo.
• Erwan Hillion: “Benamou-Brenier curves on Z”
• Abstract: Let $(f_t)_{t \in [0,1]}$ be a smooth family of probability measures on Z. We set $g_t(k)=-\sum_{l \leq k} f_t'(l)$ and $h_t(k)= -\sum_{l \leq k} g_t'(l)$. The family $(f_t)$ is said to be a Benamou-Brenier curve (or BB-curve), if $f_t(k)h_t(k-1)=g_t(k)g_t(k-1)$. During this talk, we will explain why BB-curves can be seen as a discrete version of W_2 geodesics, then we will construct, under a stochastic domination assumption, a BB-curve joining two fixed measures $f_0$ and $f_1$. We will finish by proving the convexity of some functionals along BB-curves.
• Tapio Rajala: “Ricci curvature lower bounds and branching geodesics”
• Abstract: I will discuss the rôle 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: “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 $p$ , …).
• Laurent Veysseire: “Coarse Ricci curvature for Markov processes and Poincaré inequality”
• Abstract: In this talk, I will introduce Ollivier's notion of coarse Ricci curvature for Markov chains on metric spaces, and its generalization to continuous time Markov processes. If the process is reversible and if this curvature is bounded from below by a positive constant $k$, $k$ is a lower bound for the spectral gap of the generator, or equivalently, we have a Poincaré inequality with constant $\frac{1}{k}$. In the case of diffusions on Riemannian manifolds, we can get a better bound on the spectral gap: the harmonic mean of the curvature. To define the coarse Ricci curvature, one can choose the metric independently of the generator. I will give some examples of diffusions where the choice of a metric different from the canonical one associated with the generator provides a better bound for the spectral gap.