Institut de recherche mathématique avancée

Résumé : We work over a closed field of positive characteristic. A classic result of Serre-Tate says that the deformation space of an ordinary abelian variety has the structure of a formal torus, and one can consider the closed subvarieties which are given by formal subtori. Tate-linear formal varieties play the role of formal subtori in the deformation space of abelian varieties of arbitrary Newton polygon. Recent work of Chai-Oort established an important link between Tate-linear subvarieties and the Hecke orbit conjecture for \mathcal{A}_g, which then led to a full solution for Shimura varieties of Hodge type by D'Addezio and van Hoften. We will explain the role of Tate-linear varieties in the Hecke orbit conjecture, their conjectural link with special subvarieties of \mathcal{A}_g, and show how p-adic monodromy techniques can help shed light on their structure.

Résumé : The goal of this talk is to discuss the motivic Galois group of a double zeta value. To do so, I will recall some elements of Galois theory. Then, I will introduce periods and more precisely multiple zeta values. We will see that the Galois philosophy can be used to study periods. At the end, I will express the motivic Galois group of a single zeta value and I will compare it to the case of a double zeta value.

Résumé : State-Space Models (SSMs) are deterministic or stochastic dynamical systems defined by two processes. The state process, which is not observed directly, models the transformation of the system states over time, while the observation process produces the observables on which model fitting and prediction are based. Ecology frequently uses stochastic SSMs to represent the imperfectly observed dynamics of population sizes or animal movement. However, several simulation-based evaluations of model performance suggest broad identifiability issues in ecological SSMs. Formal SSM identifiability is typically investigated using exhaustive summaries, which are simplified representations of the model. The theory on exhaustive summaries is largely based on continuous-time deterministic modelling and those for discrete-time stochastic SSMs have developed by analogy. While the discreteness of time does not constitute a challenge, finding a good exhaustive summary for a stochastic SSM is more difficult. The strategy adopted so far has been to create exhaustive summaries based on a transfer function of the expectations of the stochastic process. However, this evaluation of identifiability does not allow to take into account the possible dependency between the variance parameters and the process parameters. We show that the output spectral density plays a key role in stochastic SSM identifiability assessment. This allows us to define a new suitable exhaustive summary. Using several ecological examples, we show that usual ecological models are often theoretically identifiable, suggesting that most SSM estimation problems are due to practical rather than theoretical identifiability issues.

Résumé : Natural systems are open to their environment and thus subjected to external perturbations. Some of such systems may be described as large randomly connected networks. The present work investigates how external additive random perturbations affect the state and stability of such finite-size random networks. We observe analytically an additive noise-induced system evolution (ANISE), whose stationary state depends on the additive noise level. In a first pilot study, the principal analytical approach is demonstrated by application to a nonlinear Erdos-Renyi network. In a second more detailed study, a nonlinear random network of excitatory and inhibitory sub-networks describes successfully Event-Related Desynchronization and Synchronization (ERD/ERS) observed in experimental brain signals. In sum, we find that additive noise impacts on the system's stationary state and in turn also affects the system's stability and hence its spectral properties.