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 $n^{-lambda}$.

I will describe general ideas that lead to the explicit constructions of models of emergent space. In particular, I will solve a pre-geometric quantum mechanical model for D-particles in a presence of a large number of background D4 branes, and show that it is equivalent to the classical motion of the particles in the ten dimensional geometry sourced by the D4 branes. If time allows, other cases will be briefly discussed as well.

Résumé : En 1948 H.S. Wall a publié ses résultats sur la théorie analytique des fractions continues. Dans la première partie de cet exposé nous présentons une classe remarquable de fractions continues introduite par Wall et appelées g-fractions. Nous montrerons comment elles peuvent être utilisées pour approcher certaines applications analytiques bornées réelles.
La deuxième partie de l’exposé sera consacrée aux applications de cette méthode. Nous discuterons de la conjecture de Ramanujan portant sur la convergence d’une des fractions continues, la théorie de renormalisation des applications unimodales (l’approche herglotzien de H. Epstein), et la sommabilité de séries de Poincaré-Sundman dans le problème des 3 corps.

Abstract: In 1948 Hubert Wall introduced the particular class of analytic continued fractions called g-fractions. In the first part of this talk we describe briefly the definition and its principal properties. We show how g-fractions arise naturally in some problems related to dynamical systems theory and random walks. We introduce then the Ramanujan conjecture from the theory of limit periodic continued fractions and its solution given with help of g-fractions. The g-fractions can be effectively used in approximation theory. Here we explain how the anti-Herglotz functions approach of Henri Epstein can inspire us to use g-fractions in approximation of fixed points of renormalization operators in functional spaces. Finally, some applications of g-fractions to Sundman-Poincaré series from celestial mechanics are discussed.

A formal approach (mathematical models) describing the first two hours of development of the embryo of Drosophila (fruit fly) is presented. The first part of the talk is centered on the biological and biochemical events and processes that determine the form and shape (morphogenesis) of the embryo of Drosophila. Then we show the basic equations associated to some of the developmental processes and discuss the calibration and validation of the mathematical models with real experimental data. The models describe all the biochemical morphogenetic parameters measured in the Drosophila embryo with a relative error below 5% and account for the observed phenotypic plasticity and resilience. This approach leads to a precise mathematical definition of resilience, which we associate to the existence of a continuous family of Pareto optimal solutions of the models. Problems associated with conservation laws (symmetries) and optimization for ill posed problems and noisy objective functions will also be discussed.

I will explain how General Relativity in four space-time dimensions (with cosmological constant) can be reformulated as an SU(2) gauge theory of a certain type. The action is a (diffeomorphism invariant) functional of an SU(2) connection; no metric is present in the formulation of the theory. This formulation of GR is in many ways analogous to the one proposed many years ago by Eddington (Eddington’s Lagrangian is a function of just the affine connection and is given by the square root of the determinant of the Ricci tensor). The new formulation has some remarkable properties. First, on a 4-manifold, the space of SU(2) connections modulo gauge transformations has just 9 components per space-time point, as compared to 10 components of a metric tensor. Correspondingly, the action of the new formulation can be interpreted as a functional on the space of conformal classes of metrics. Thus, the conformal factor is not free to propagate in this formulation even off-shell. This has the effect that the action functional is (at least locally) convex – there is no conformal mode problem of the usual metric formulation of GR. Another remarkable property of the new formulation is that diffeomorphisms are very easy to deal with. The most natural gauge-fixing of these does not make the corresponding components of the connection propagate. As a result, in (linearized) quantum theory only 8 components of the connection propagate as compared to 10 metric components in the usual formulation. All these 8 components fit into a single irreducible representation of the Lorentz group, which makes the propagator very simple. There are also some remarkable simplifications in the structure of the interaction vertices. The full off-shell 3-vertex (in de Sitter space) contains just 3 terms as compared to a couple of dozen terms in the metric formulation. The 4-vertex is a couple of lines as compared to a couple of pages in the standard description. As an illustration of the formalism I will describe how the graviton scattering amplitudes are computed in this approach.

The new gauge-theoretic reformulation of GR also leads to (an infinite-parameter) family of modified gravitational theories, all propagating just two polarizations of the graviton, as GR. This leads to a rather strong claim that, in spite of the standard GR uniqueness theorems, General Relativity is not the only interacting theory of massless spin two particles. However, GR appears to be the only parity-invariant gravitational theory, as all the « deformations of GR » can be shown to be parity violating. I will also describe how matter is coupled in this approach, and give some speculations as to a possible UV completion of gravity.

Let K be a CDVF of mixed characteristic (0,p) and G the absolute Galois group of K. When the residue field of K is perfect, Laurent Berger constructed a p-adic differential equation NdR(V) for any de Rham representation V of G. In this talk, we will generalize his construction when the residue field of K is not perfect. We also explain some ramification properties of our NdR , which are due to Adriano Marmora in the perfect residue field case.

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Computational anatomy has emerged as a new subject focusing on the quantitative analysis of the variability of biological shapes with interesting challenges both from the mathematical and computational point of view. In this talk I will explain how a geometrical point of view based on the idea of shapes as structured spaces, actions of groups of diffeomorphisms and (sub)-Riemannian geometry provide nice vehicles to build a full processing framework called here diffeomorphometry.

In 1999, C Sonnenschein and AM Soto proposed the tissue organization field theory (TOFT). The TOFT posits 1) that cancer is a tissue-based disease whereby carcinogens (directly) and germ-line mutations (indirectly) alter normal interactions between the stroma and adjacent epithelium; and 2) the default state of all cells is proliferation with variation and motility. This later premise is relevant to and compatible with evolutionary theory, and is diametrically opposed to that of the somatic mutation theory.

This theoretical change is incompatible with the reductionist genocentric perspective generated by the molecular biology revolution. Rather than forcing a “bricolage” we decided to frontally attack the problem by joining efforts with philosophers, mathematicians and theoretical biologists to search for principles upon which to build a theory of organisms .While the theory of evolution has provided an increasingly adequate explanation of phylogeny, biology still lacks a theory of organisms that would encompass ontogeny and life cycles, and thus phenomena on a conception to death time-scale.

To achieve this goal we propose that theoretical extensions of physics are required in order to grasp the living state of matter. Such extensions will help to describe the proper biological observables, i.e. the phenotypes. Biological entities must also follow the underlying principles used to understand inert matter. However, these physical laws and principles may not suffice to make the biological dynamics intelligible at the phenotypic level. By analogy with classical mechanics, where the principle of inertia is the default state of inert matter, we are proposing two aspects of the default state in biology, and a framing principle, namely: i) Default state: cell proliferation with variation as a constitutive property of the living. Variation is generated by the mere fact that cell division results in two overall similar, but not identical cells. ii) Default state: motility, which encompasses cell and organismic movements as well as movement within cells. iii) Framing principle: life phenomena exhibit never identical iterations of a morphogenetic process. Organisms are the consequence of the inherent variability generated by proliferation, motility and auto-organization which operate within the framing principle. From these basic premises, we will elaborate on the generation of robustness, the structure of theoretical determination, and the identification of biological proper observables.