We investigate how deeply nested are the loops in the O(n) model on random maps. In particular, we find that the number P of loops separating two points in a planar map in the dense phase with V >> 1 vertices is typically of order c(n) ln V for a universal constant c(n), and we compute the large deviations of P. The formula we obtain shows similarity to the CLE_{kappa} nesting properties for n = 2cospi(1 – 4/kappa). The results can be extended to all topologies using the topological recursion.

This is based on a joint ongoing work with J. Bouttier

 

Locally analytic representations of p-adic Lie groups are of interest in several branches of arithmetic algebraic geometry, notably the p-adic local Langlands program. I will discuss some work in progress towards a Beilinson-Bernstein style localisation theorem for admissible locally analytic representations of semisimple compact p-adic Lie groups using equivariant formal models of rigid analytic flag varieties.

Page web du séminaire

The four billion years old genetic code is quasi-universal among living organisms. Mathematical insights into the genetic code can be gained by studying its symmetries. Symmetries by base substitutions were identified for the 2' and 3' sets of aminoacyl-tRNAs with Soulé in 2007. Half a century ago, Rumer described symmetries by base substitutions for degeneracy, the number of base triplets coding for each of the twenty canonical amino acids found in natural proteins. Apart from base substitutions, another class of mutations consists of frameshift mutations. The most frequent frameshift mutations are single-base deletions which are highly deleterious within genes as they alter the reading frame. The central role of mutations which are errors induced by biological catalysts, will be analyzed in the context of the minimization of their deleterious effects, the codon assignment of amino acids and chain termination signals in the genetic code and molecular evolution.
 

The central axis of the famous DNA double helix is often constrained or even circular.   The topology of this axis can influence which proteins interact with the underlying DNA. Subsequently, in all cells there are proteins whose primary function (type II topoisomerases)  is to change the DNA axis topology — for example converting a torus link into an unknot. Additionally, there are several protein families (most importantly, site-specific recombinases) that change the axis topology as a by-product of their interaction with DNA.

This talk will describe some typical DNA conformations, and the families of proteins that change these conformations. I'll present a few examples illustrating how 3-manifold topology (including Dehn surgery and Heegaard Floer homology) have been useful in understanding certain DNA-protein interactions, and discuss the most common techniques used to attack these problems.

Une des définitions possibles des nombres de Hurwitz est qu'ils comptent certaines cartes étiquetées. De même que les cartes planaires sont en bijection avec des arbres plans (ou ordonnés), ces cartes étiquetées sont en bijection avec des arbres étiquetés de type Cayley. Le but de l'exposé sera de montrer comment l'étude de ces arbres permet de d'obtenir de nouvelles formules pour les nombres de Hurwitz doubles ainsi que des propriétés de polynomialité.

Considérons une marche aléatoire simple sur un arbre de Galton-Watson critique conditionné à avoir une hauteur supérieure à $n$. La loi du point d'atteinte de la hauteur $n$ par la marche aléatoire s'appelle la mesure harmonique au niveau $n$. Il est bien connu que le cardinal de l'ensemble des sommets de l'arbre au niveau $n$ est de l'ordre de $n$. En 2013 Curien et Le Gall ont prouvé qu'il existe une constante $beta=0.78…$ telle que la mesure harmonique est portée, à un ensemble de masse arbitrairement petite près, par un ensemble de cardinal de l'ordre de $n^beta$. Dans cet exposé, nous présentons l'existence d'une nouvelle constante universelle $lambda=1.21…$ telle que, avec grande probabilité, la mesure harmonique portée par un sommet typique à la hauteur $n$ est de l'ordre $n^{-lambda}$.

In 1960, Higman asked whether the number f(n,q) of conjugacy classes of n x n unitriangular matrices U(n,q) over the finite field with q elements, is polynomial in q for every n.  I will survey what is known about this problem, explain the connections to realization spaces of matroids, enumerative combinatorics and group representation theory.  I will then describe our recent efforts to prove the conjecture for small n and disprove it for large n.

Joint work with Andrew Soffer.

I will show a simple example of fast-slow partially hyperbolic systems in which the fast variable acts similarly to a Gaussian  noise and will obtain, for such a deterministic setting, results similar to the one obtained in Freidlin–Wentzell theory. In particular, I will show conditions under which there exists a unique SRB measure with exponential decay of correlations.
 
(Work in collaboration with Jacopo de Simoi).

 

Information plays a critically important role in ecology and evolution but is very often subjective or analog or both. This is a problem because most information theory has been developed for objective and discrete information. Can information theory be extended this theory to incorporate multiple forms of information, each with its own (physical) carriers and dynamics? Here I will not review all the possible roles information can play, but rather what conditions an appropriate theory should satisfy.
 

Séminaire Laurent Schwartz — EDP et applications