Séminaires du LAREMA

Séminaire de topologie et géométrie algébriques

Séminaire des doctorant.es
In this presentation, I focus on the semiclassical Schrödinger equation, a fundamental equation in quantum mechanics that describes the evolution of quantum particles over time. Since exact solutions to this equation are rarely explicit and conventional numerical methods are often impractical, my goal is to develop approximate solutions that are both easier to compute and accurate. To achieve this, I study special functions called wave packets, which represent localized quantum states. First, I will present how, starting from initial data defined by a wave packet, we can construct a good approximate solution using a wave packet, for the scalar semiclassical Schrödinger equation. Then, I will explain how this approach can be extended to more complex vector-valued systems, where new phenomena arise.

Séminaire des doctorant.es
In this talk, we will explore the behavior of a random walk when conditioned to remain within a cone. We will begin by introducing the problem of conditioning a random walk to never escape a cone. To address this, we will examine the concept of the Doob h-transform and its role in shaping the walk's behavior. Next, we will delve into the set of harmonic functions associated with random walks and discuss their significance in the context of conditioning the walk. Finally, we will derive the conditions under which a unique discrete harmonic function exists for a Dirichlet problem posed within a cone under certain assumptions about the transition kernel of the random walk and the cone itself.

Séminaire des doctorant.es
Inference of the tail parameters of a distribution is a question of interest. Indeed, some extreme events can have disastrous consequences and being able to estimate their probability of appearance allows us to prevent them. It is however a difficult question because usual statistics theory does not work well in that case. Extreme value theory has been developed for this purpose. In particular, the Conditional Tail Moments (CTMs) are useful tools in risk quantification. For instance, the Expected Shortfall (ES), a particular case of CTM, is a risk measure widely used in finance. The estimation of CTMs and the demonstration of convergence results on these estimators have been the purpose of my first year of PhD. In this talk, I will start with an introduction and motivation to Extreme Value Theory. I will then define the Conditional Tail Moment and give the mathematical framework in which estimation of extreme CTM is manageable. Finally, if time permits, I would like to present some of the convergence results me and my PhD supervisors have been able to produce so far.