Erwin Bolthausen (Zürich): On the Derrida-Ruelle cascades, and the ultrametricity issue in spin glass theory

Derrida and Ruelle introduced a class of spin glass models, called the "generalized random energy models" (GREM for short), which exhibit some remarkable properties, for instance invariance properties under a transformation by a cavity field. For these reasons, the models are considered to be universal limiting objects in spin glass theory, at least in the mean-field case. However, for no real spin glass model, except one, the limiting GREM-type behavior has been mathematically proved. The one exception is the p-spin SK model investigated by Talagrand. We discuss these issues, and present some new examples which (hopefully) shed some new light on the ultrametricty and chaos problem in spin glass theory. (Joint work with Nicola Kistler).

Eric Brunet (Paris): Effect of noise on front propagation

The Fisher or Kolmogorov-Petrovsky-Piscounov equation (FKPP) describes the propagation of a front into an unstable state. It was first introduced, in 1937, in a biological context, but this equation (or similar ones) has since been used in many fields of physics, such as reaction-diffusion models, directed polymers or, more recently, QCD. In this seminar, I will try to explain, from a phenomenological point of view, how microscopic noise influences the shape and position of the front, and I will present a set of unproved but plausible hypothesis that allow to make a prediction for the velocity and diffusion constant of the front.

David A. Kessler (Ramat-Gan): Running for your life: Optimal Disperal Rates

We investigate a model of two competing species, identical except for their dispersal rate (i.e. diffusion constant). A standard reaction-diffusion treatment indicates that the slower of the two species will always emerge victorious in any inhomogeneous environment. We show that for sufficiently low population levels (which however number hundreds of individuals per lattice site) the fast species always wins. The critical population level for fast victory is shown to increase for very homonogeneous or alternatively, very inhomogenous environments. These effects are explained and the beginnings of a quantitative theory outlined.

Jean-François Le Gall (Paris): Branching processes and the Bolthausen-Sznitman coalescent

We discuss connections between the coalescent process introduced by Bothausen and Sznitman and a certain branching process with continuous state space. These probabilistic objects play an important role in the analysis of Derrida's GREM model.

Esteban Moro (Madrid): Modeling and simulation of branching problems using continuum models

Understanding when and whether fluctuations may play a major role is or paramount importance in several areas of research. In my talk I will present some recent results about the use and simulation of stochastic continuum models to understand the behavior of branching processes and how fluctuations may be so important to completely determine the dynamics. New algorithms will be presented to integrate stochastic differential equations associated with some reaction- diffusion problems. In particular the talk will cover a) the effect of microscopic fluctuations in the travelling waves present in the Fisher equation and its importance in high dimensions, b) the dynamical properties of the compact support of the super-Brownian motion and c) the use and interesting properties of the subcritical Bellman-Harris process to understand the dynamics of information spreading in social networks and its application to viral marketing.

Carl Mueller (Rochester): The speed of a random traveling wave for the KPP equation

We describe joint work with Leonid Mytnik and Jeremy Quastel. The KPP equation is a standard model for the study of traveling waves. A large class of initial conditions yield solutions which converge to a limiting shape, which moves with constant velocity. Adding noise to the equation may give a stationary ensemble of shapes, with an average speed which is different than the speed of the deterministic wave. Brunet and Derrida have conjectured some surprising results about the speed of the wave in the random case, when the noise is small. Conlon and Doering have given an inequality which partially verifies half of the conjecture. We completely prove both halves of the conjecture. In addition, we give some further error terms which partially confirm a more recent conjecture of Brunet and Derrida.

Stephane Munier (Palaiseau) Effect of selection on ancestry

We consider a peculiar class of models of population evolution under selection whose dynamics may be associated to the propagation of noisy traveling waves. From an exact calculation in one particular model in this class, we observe that the coalescence times scale like a definite power of the log of the population size, and that the genealogical trees have the same statistics as the trees of replicas in the Parisi theory of spin glasses. We show how these features are related to properties of the propagation of traveling waves of the Fisher-Kolmogorov-Petrovsky-Piscounov type.

Feng Yu (Oxford): Limit to the Rate of Adaptation and Overcoming Muller's Ratchet

Asexual populations accumulate beneficial and deleterious mutations over time. The rate of adaptation is affected by the mutation rate, the proportion of beneficial mutations, and the population size. If all mutations are deleterious, then the population inexorably grows less fit, which is known as Muller's ratchet. We view the distribution of individuals over mutation numbers as a travelling wave, and study the stationary measure of the centred wave. We show that regardless of mutation rates and proportion of beneficial mutations (as long as it is nonzero), the adaptation rate is at least $O(\log^{1-\delta} (population size))$. Thus Muller's ratchet can be overcome as long as a certain proportion of the mutations are beneficial and the population size is sufficiently large.