Totally positive matrices are characterized by having all their minors positive. They appear in various areas of physics and mathematics, including oscillations in mechanical systems, quantum groups, and algebraic geometry. It has been known since the work of Fomin that the two-point correlations functions of the two-dimensional Gaussian free field satisfy total positivity. I will present an analogous result for the correlations of the planar Ising model. The idea is to prove that determinants of such correlations have interpretations in terms of probabilities of events in the random current model. A natural open question is to identify all planar totally positive spin systems.

Soit G un groupe hyperbolique sans torsion, soit S une partie génératrice finie de G, et soit f un automorphisme de G. Nous cherchons à comprendre les taux de croissance possibles pour la longueur d’un élément g du groupe G (écrit comme un mot en les générateurs dans S) sous l’itération de f. Le cas où G est un groupe de surface ou un groupe libre est compris grâce à des résultats de Thurston et Bestvina-Handel. Nous montrons qu’en général, il n’y a qu’un nombre fini de taux de croissance exponentiels possibles lorsque l’élément g parcourt G. Par ailleurs, nous montrons la dichotomie suivante : tout élément a une croissance qui est soit exponentielle, soit polynomiale. Ceci est un travail en commun avec Rémi Coulon, Arnaud Hilion et Gilbert Levitt.

We study the gradient field models with uniformly convex potential (also known as the Ginzburg-Landau field) in two dimension. These log-correlated non-Gaussian random fields arise in different branches of statistical mechanics. Existing results were mainly focused on the CLT for the linear functionals. In this talk I will describe some recent progress on the global maximum and local CLT for the field, thus confirming they are in the Gaussian universality class in a very strong sense. The proof uses a random walk representation (a la Helffer-Sjostrand) and an approximate harmonic coupling (by J. Miller).

In vitro branched actin networks have been shown to generate a pushing force. In vivo these networks have been identified at different subcellular locations to drive membrane protrusion and remodeling during intracellular traffic. Here we will discuss the integration of positive and negative regulation of membrane protrusions and how it impacts the essential parameters of cell migration. Surprisingly we found that the signaling pathway that generates branched actin at membrane protrusions, Rac-WAVE-Arpin, also controls cell cycle progression.

In fact, this signaling pathway has all the expected properties of a cell cycle checkpoint. It is required in normal cells but this requirement is lost in cancer cells. The branched actin network of membrane protrusions integrates growth factor stimulation with mechanotransduction of cell adhesion to instruct the cell that its environment is permissive for migration and cell cycle progression.

Séminaire Laurent Schwartz — EDP et applications

Séminaire Laurent Schwartz — EDP et applications

Nous montrerons que, pour de telles représentations ρ : π1Σ → G, l'adhérence de Zariski de ρ(Γ) contient le SL2 principal sauf lorsque Γ est un sous-groupe monogène de π1Σ, auquel cas cette adhérence de Zariski est un sous-groupe abélien régulier connexe. Ceci entraîne la classification des adhérences de Zariski possibles puisque la classification des groupes algébriques contenant le SL2 principal est connue. Enfin dans le cas où Γ = π1Σ (ou un sous-groupe d'indice fini) les paramètres de Hitchin déterminent l'adhérence de ρ(Γ).

I will talk about the construction of the continuous series for affine sl(2,R), possible relation to the "continuous tensor categories" of modular double of Uq(sl(2,R)) and Virasoro algebra.

We investigate the degree to which the geometry of a compact real projective manifold with boundary is reflected in the associated holonomy representation, a representation of the fundamental group in the projective general linear group PGL(n,R) which in general need not have any nice properties.

We show that if the projective manifold is strictly convex, then its holonomy representation is projective Anosov, a condition which generalizes the dynamical properties of convex cocompact representations in rank one (e.g. hyperbolic) geometry. Conversely, a strictly convex projective manifold may be constructed from a projective Anosov representation that preserves a properly convex set in projective space. Applications include new examples of both convex projective manifolds and Anosov representations. Joint work with François Guéritaud and Fanny Kassel.