Institut de recherche mathématique avancée

Résumé : Missing values in measurements for carbon dioxide emissions on drained peatlands remains an open challenge for training forecasting techniques to achieve net zero. At the field scale, existing methods struggle to model CO_2 emissions to fill gaps, especially in nighttime measurements. We propose robust Physics-Informed Autoencoders (PIAEs), which combine the generative capabilities of Autoencoders with the reliability of physical models of Net Ecosystem Exchange (NEE) that quantify CO_2 exchanges between the atmosphere and major carbon pools. Our method integrates equations describing the physical processes and associated uncertainties to fill gaps in NEE measurements from eddy covariance (EC) flux towers. In the PIAE, various sensor measurements are encoded into the latent space, and a set of decoders is then used to approximate the ecosystem parameters and the optimal NEE forecast, directed by dynamics described by a stochastic differential equation. These decoders utilize nighttime and daytime NEE models that describe carbon transfer as a Wiener process. Finally, we use a two-phased training routine with two loss functions describing each phase: Mean Squared Error (MSE) and Maximum Mean Discrepancy (MMD) between the measurements and the reconstructed samples. PIAE outperforms the current state-of-the-art Random Forest Robust on the prediction of nighttime NEE measurements on various distribution-based and data-fitting metrics. We present significant improvement in capturing temporal trends in the NEE at daily, weekly, monthly and quarterly scales.

Résumé : Résumé : Les arrangements de droites dans le plan projectif complexe sont des objets fascinants qui interrogent les liens entre topologie, géométrie, et combinatoire. Dans cet exposé je revisiterai un invariant topologique des arrangements de droites, initialement défini par Artal, Florens, et Guerville-Ballé par analogie avec la théorie des nœuds, qui permet parfois de distinguer des paires de Zariski (deux arrangements de droites avec la même combinatoire mais des topologies différentes). On verra que cet invariant est de nature homologique, et on l'étudiera grâce aux outils, classiques et moins classiques, de l'homologie des variétés algébriques. Il s'agit d'un travail en commun avec Benoît Guerville-Ballé.

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.