École Normale Supérieure de Lyon

We describe the compact scaling limits of uniformly random quadrangulations with boundaries on a surface of arbitrary fixed genus. These limits, called Brownian surfaces, are homeomorphic to the surface of the given genus with or without boundaries depending on the scaling regime of the boundary perimeters of the quadrangulation. They are constructed by appropriate gluings of pieces derived from Brownian geometrical objects (the Brownian plane and half-plane). In this talk, I will review their definition and discuss possible alternative constructions. This is based on joint work with Jérémie Bettinelli.

The classical fractional crystallisation scenario for magma ocean solidification on the Moon suggests that its crust formed by flotation of light anorthite minerals on top of a liquid ocean, which has been used to explain the anorthositic composition of the lunar crust. However, this model points to rapid crustal formation over tens of million years and struggles to predict the age range of primitive ferroan anorthosites from 4.5 and 4.3 Ga. 

Here I will present a new paradigm of slushy magma ocean crystallisation in which crystals are suspended throughout the magma ocean, and the lunar crust forms by magmatic processes over several hundreds of thousand years.

We will then focus on the effects of the particular characteristics of this primary crust on the transport and eruption of magma on the Moon.

The Kruzkhov's semi-group of a scalar conservation law extends as a semi-group over $L^1$, thanks to its contraction property. M. Crandall raised in 1972 the question of whether its trajectories can be distributional, entropy solutions, or if they are only "abstract" solutions. We solve this question in the case of the multi-dimensional Burgers equation, which is a paradigm for non-degenerate conservation laws. Our answer is the consequence of dispersive estimates. We first establish $L^p$-decay rate by applying the recently discovered phenomenon of Compensated Integrability. The $L^\infty$-decay follows from a De Giorgi-style argument. This is a collaboration with Luis Sivestre (University of Chicago).

How do organisms cope with cellular variability to achieve well-defined morphologies and architectures? We are addressing this question by combining experiments with live plants and analyses of (stochastic) models that integrate cell-cell communication and tissue mechanics. During the talk, I will survey our results concerning plant architecture (phyllotaxis) and organ morphogenesis.

Several dissipative scalar conservation laws share the properties of

$L1$-contraction and maximum principle. Stability issues are naturally

posed in terms of the $L1$-distance. It turns out that constants and

travelling waves are asymptotically stable under zero-mass initial

disturbances. For this to happen, we do not need any assumption

(smallness of the TW, regularity/smallness of the disturbance, tail

asymptotics, non characteristicity, ...) The counterpart is the lack of

a decay rate.