Évènements

In this talk we will discuss Self-Interacting diffusions (SID), its basic properties and applications. We also discuss what constitutes the Exit-time problem, why it is important, and for which dynamical systems it was already established. We present the recent results of exit-time problem for a specific case (convex confinement and convex interaction) of SID and how they were established. In the end of the talk we discuss some ideas to generalize this result

Après avoir présenté les défis numériques actuels pour atteindre des tailles de réseaux de l’ordre du cerveau humain, j’expliquerai pourquoi les processus de Hawkes peuvent être un bon modèle pour passer à l’échelle. J’expliquerai comment les algorithmes classiques peuvent être renouvelés pour y arriver. Une des approches les plus innovantes est basée sur un résultat probabiliste fort : la décomposition de Kalikow, qui permet de tirer au hasard les dépendances spatiales et dans le passé et que nous avons redéfini en temps continu. Elle permet en particulier de simuler un neurone immergé dans un réseau infini sans avoir à simuler le réseau infini. Ce travail est effectué en collaboration avec Eva Löcherbach (mathématicienne, Paris I), Alexandre Muzy (informaticien, Université Côte d’Azur) ainsi que nos étudiants : Cyrille Mascart, Tien Cuong Phi et Paul Gresland.

Considering partial character sums as defining a family of random processes (by choosing the characters randomly from some set), it becomes natural to ask questions about the limiting distribution. I’ll describe this in some contexts and give examples of what we can find in the support. This is largely work of Ayesha Hussain, but also some joint work.

The quantum gauge theory is the current basis for explaining the interactions of elementary particles. Among the four fundamental interactions: electromagnetic, weak, strong, and gravitational, for the last two, a proper mathematical construction does not yet exist. Quantum gauge theory is based on three different branches of mathematics: 1) geometry, 2) group theory, and 3) functional analysis.

The strong interactions are studied in the SU(3) quantum Yang-Mills theory but an axiomatic QFT with the mass gap and color confinement does not yet exist. In the classical field theory model, the gluon^1 is a massless particle but one can not see the color-charge^2 of a particle in nature, and then the gluons must be in the bound states, forming massive particles. This is the mass gap problem. The mass gap is the difference in energy between the vacuum and the next lowest energy state. The energy of the vacuum is zero by definition, and the mass gap is the mass of the lightest particle. However, the color confinement postulated permits only bound states of gluons, it is an obvious contradiction.

In this presentation, I would like to discuss the fact that if we replace: 1)  Minkowskian geometry with the de Sitterian geometry, 2) Poincaré group with de Sitter group and 3) Hilbert space quantization with the Krein space quantization, an axiomatic quantum Yang-Mills theory with a mass gap and the color confinement can be constructed. It is important to note that the ambient space formalism allows us to make this construction possible.

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1 . A gluon is an elementary particle that acts as the exchange particle (or gauge boson) for the strong force between quarks.

2 . Color charge is a property of quarks and gluons that is related to the particles’ strong interactions in the theory of quantum chromodynamics