Ce séminaire est organisé par les doctorants, stagiaires et post-doctorants du CAMS.
Vendredi 15 juin à 11h, salle A4-47
The Dyson brownian motion
This talk will focus on a very known result from random matrix theory : the eigenvalue distribution for the Gaussian orthogonal ensemble (GOE), the space of real symmetric matrices with random iid gaussian entries.
Random matrices are a common tool in modelling complex systems, from the hamiltonian of heavy nuclei to correlation matrices in financial markets or even adjacency matrices in random graphs. In particular, the eigenvalues of such matrices are strongly correlated random variables, and it is quite remarkable that one can derive analytical results for their statistics.
The case of the GOE is the simplest, and one way of solving it uses a very elegant method. We will thus introduce a few simple tools from statistical physics – the Langevin equation and perturbation theory – to map the problem of computing the joint distribution of the eigenvalues of an NxN GOE matrix into that of the random motion of N repulsive particles confined in a potential. We will stress how this method helps one gain intuition into the problem at hand, and show how to obtain the joint distribution in the large N limit.
If time allows, we will also show how to derive the density of eigenvalues, obtaining thus the celebrated Wigner semi-circle law.
Vendredi 25 mai à 11h, salle A4-47
The moving plane method
In this talk, I will talk about PDEs while trying to be understood by non-specialists. Rather than presenting some celebrated results, I will focus on a technic, called the "Moving Plane Method", introduced by Alexandroff in the 40s for geometrical matters, and then widely used in PDEs as combined with the "Maximum Principle". This method, apart from being very elegant, is very efficient, robust, and give a grasp of the deep links between geometry and PDEs.
jeudi 7 décembre, 14h, salle A4-47
Comparison of two models for the perception of speech rhythm
Speech rhythm has been a controversial subject among linguists and psychologists for decades. The classical theory of isochrony tells that each language falls into one of three rhythmical categories: stress-timed languages, syllable-timed languages and mora-timed languages. Although the theory has been challenged several times, this typology is still used by many linguists. The topic has known a gain of interest in the field of language development as several pieces of evidence show that very young infants can discriminate between two languages when they belong to different families of speech rhythm.
In this presentation, I will compare two theoretical models for the perception of speech rhythm. The first model looks for correlates in durations of vocalic and intervocalic intervals (Ramus, 1999). The second makes use of recurrent neural networks (RNN) to find regular patterns in temporal measures of energy and entropy in the acoustic signal. The talk will be an opportunity to discuss the shift of paradigm from explicit modeling to machine learning algorithms in computational neuroscience.
Vendredi 27 novembre 2017 à 11h, A4-47, EHESS, 54 boulevard Raspail, 75006 – Paris
PhD Student, CAMS/CFM
A new mechanism for power-law distributions
Empirical power-law distributions are often found to be an excellent fit for very unevenly distributed quantities, such as the wealth of individuals, the size of firms or cities and the frequency use of words in a language. These distributions are strikingly different from gaussian, or normal, distributions, in that they may not have a mean or variance and that they do not possess a characteristic scale. Understanding the emergence of these distributions from a mechanistic standpoint is key to understanding the origins of inequality and the scale-free structure of the economy, and its implications in policy and regulation.
The mechanism leading to a given power-law greatly constrains the value of its exponent, and many quantities sharing the same exponent may in fact fall within the same universality class. In this talk, we will demonstrate this by showing a simple and intuitive mechanism that leads to a bounded continuum of exponents and that can be exactly solved using tools and premises from statistical physics.
Vendredi 27 octobre 2017 à 11h, A4-47, EHESS, 54 boulevard Raspail, 75006 – Paris
Gabrielle Saller Nornberg
PhD student, Puc-Rio, Brazil
Multiplicity and Regularity results for fully nonlinear elliptic equations with quadratic growth in the gradient
The study of quasi-linear elliptic equations with quadratic dependence in the gradient started in the 80s, with the works of Boccardo, Murat and Puel and has been an object of research until now. The multiplicity of solutions phenomenon in the coercive case, for a particular example with the laplacian, was first observed by Sirakov, in 2010. Further improvements were done in the last years, specially by Jeanjean et al. in order to give a more clear picture of the set of solutions, still for the laplacian and using divergence tools. In this talk, we will discuss some recent results obtained for the fully nonlinear case in the context of Lp-viscosity solutions connected to purely nondivergence techniques, together with a generalization of the regularity results of Swiech-Winter, to our equations, that naturally appears in the midway.
Vendredi 29 septembre 2017 à 11h, A4-47, EHESS, 54 boulevard Raspail, 75006 – Paris
Post-doctoral reasearcher, CAMS, ReaDi
Propagation in multi-dimensional Fisher-KPP equations
Fisher-KPP equations are a type of reaction-diffusion equations that model population dynamics. Their behavior is characterized by the invasion of an unstable state by a stable one, which leads to the phenomenon of spreading. In one dimension, it is known that localized data give rise to solutions that lag behind the slowest traveling front by a logarithmic term. In this talk, I will first discuss these past results, then focus as some recent ones about the asymptotic rates of spreading along each direction for the Cauchy problem in periodic media in R^n.