Recently, hyperbolic versions of uniform planar maps have attracted a great deal of attention. These maps are conjectured to be local limits of uniform maps embedded on high genus surfaces. First, I will describe a resolution of this conjecture for unicellular (or one-face) maps. Although for other cases this still remains a conjecture, several possible candidates have been constructed. I will give a brief overview of these models, their construction and geometric properties. I will also discuss behaviour of random walks (e.g. their speed) on them and how the ''final behaviour" of random walks on them can be nicely described via their circle packings. Parts of these works are joint with Omer Angel, Guillaume Chapuy, Nicolas Curien, Tom Hutchcroft and Asaf Nachmias.

We investigate two aspects of large random trees built by linear preferential attachment, also known in the literature as Barabasi-Albert trees. Starting with a given tree (called the seed), a random sequence of trees is built by adding vertices one by one, connecting them to one of the existing vertices chosen randomly with probability proportional to its degree. Bubeck, Mossel and Racz conjectured that the law of the trees obtained after adding a large number of vertices still carries information about the seed from which the process started. We confirm this conjecture using an observable based on the number of ways of embedding a given (small) tree in a large tree obtained by preferential attachment. Next we study scaling limits of such trees. Since the degrees of vertices of a large preferential attachment tree are much higher than its diameter, a simple scaling limit would lead to a non locally compact space that fails to capture the structure of the object. Yet, for a planar version of the model, a much more convenient limit may be defined via its loop tree. The limit is a new object called the Brownian tree, obtained from the CRT by a series of quotients.

In this talk, I will present a rigorous probabilistic construction of Liouville Field Theory on the Riemann sphere with positive cosmological constant, as considered by Polyakov in his 1981 seminal work "Quantum geometry of bosonic strings". Then, I will explain some of the fundamental properties of the theory like conformal covariance under PSL$_2(C)$-action, Seiberg bounds, KPZ scaling laws, the KPZ formula and the Weyl anomaly (Polyakov-Ray-Singer) formula. If time permits, I will also explain the construction in the disk. This is based on joint works (some on arxiv and others in progress) with F. David, Y. Huang, A. Kupiainen, R. Rhodes.

Séminaire Laurent Schwartz — EDP et applications

Séminaire Laurent Schwartz — EDP et applications

Séminaire Laurent Schwartz — EDP et applications

We will review modified gravity theories and in particular scalar tensor theories, where we have an additional scalar field coupling to the metric tensor. By means of a theorem given by Horndeski back in 1974 we will briefly discuss the most general of these theories. We will examine a particular sub-class of Horndeski theory which has interesting properties with respect to the cosmological constant problem. We will then find black hole solutions of this subclass which in some cases will be identical to GR solutions. The novel ingredient will be the presence of a time and space dependent scalar field. As we will see time dependence and higher order Galileon terms will bifurcate no hair theorems and provide scalar tensor black holes with a non trivial scalar field.

Many cellular and developmental processes are tightly controlled by regulatory logic that is similar across different species. Is this similarity the result of common ancestry or is it due to convergent evolution?
We address this question using in silico modeling of genetic networks controlling organ development in flowers. First, we determine the number of networks implementing a given logic and point to an open mathematical problem. Second, we take a computational approach based on Markov Chain Monte Carlo and sample uniformity that highly constrained space of networks. The production of a large number of samples in that ensemble reveals how structural features of genetic networks are shaped by the imposed regulatory logic. Thus, network structural features are expected to be shared across different species through convergent evolution.

The substrate for heredity, DNA, is chemically rather inert. However, it bears one of the elements of information that specify the form of the organism. How can a form be specified, starting from DNA ? Recent observations indicate that the dynamics of transcription — the process that decodes the hereditary information — can imprint forms of a certain topological class onto DNA. This topology allows both to optimize transcription and to facilitate the concerted change of the transcriptional status in response to environmental modifications. To the best of our knowledge, this morphogenetic event is first on the path from DNA to organism. It inspires new rules to optimize networks of transcriptional interactions 'à la manière de' synthetic biology. The latter new interdisciplinary domain will be briefly presented.