La semaine de l’IMB

Quantum mechanics is well approximated by classical physics when Planck's constant is considered small, i.e., in the semi-classical limit. Typically, one can study an observable associated with a particle, such as its momentum or its position, and show that its dynamics is given by classical dynamics at first order, with corrections of the order of Planck's constant. In this talk, I will present more precisely the concept of semi-classical limits, the standard mathematical results known for non-relativistic quantum mechanics, and my work that concerns the semi-classical limit in the context of relativistic quantum mechanics. Concretely, I will show how to adapt the modulated energy method to the Klein-Gordon and Klein-Gordon-Maxwell equations and how to recover relativistic mechanics (instead of classical mechanics) at the semi-classical limit.

The determinant of the Hessian matrix of a homogeneous polynomial f defines a hypersurface H(f)=0. When f is either a ternary cubic or a binary quartic, H descends to a rational self-map on the moduli space X(1). This talk explores the resulting dynamical systems, showing how they arise from the dynamics of suitable group homomorphisms, and how this dictates their remarkably symmetric structure. 

Joint work with E. Broggini, M. Houben, D. Lazzarini, R. Lolato, F. Pintore, and D. Taufer.

L'ordre du jour sera le suivant :

1. Approbation du compte rendu de la réunion du conseil scientifique du 21/10.

2. Examen et classement des demandes de gratifications de stages de M2 fléchés (les propositions reçues sont déposées ici au fur et à mesure de la réception).

3. Validation des critères à adopter par le CS pour l'examen et le classement des propositions de contrats doctoraux fléchés.

4. Informations de la direction.

5. Questions diverses.

The Doughnut Factory is in danger. A few weeks ago, the recipes of the famous bakery have been stolen by some dudes in yellow jackets, and with them, hundreds of years of hidden geometrical secrets that have made the reputation of their wide collection of torical doughnuts. In this presentation, we will go back to the history of this bakery and try to understand the value of their collection as well as some glances at their new projects. We might also speak about pizzas, so no excuses !

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.

In this presentation, we discuss recent results in the mathematical modeling of Alzheimer’s disease, based on the amyloid cascade hypothesis related to the polymerization process of the Aβ protein. We first present the analytical and numerical study of a spatial model describing the propagation of Aβ in interaction with neuronal activity. We then discuss a second model that incorporates the immune response, in particular the inflammation mediated by microglial cells. Finally, we present qualitative results related to the estimation of the nucleation rate, a critical parameter in the progression of the disease, for which no experimental measurements are currently available.

Dans ce séminaire, je parlerai de différents schémas d’optimisation stochastique, tels que la descente de gradient stochastique et le schéma de Heavy-Ball stochastique. Nous établirons des estimations d’erreur faible uniformes en temps pour l’erreur entre la solution du schéma numérique et celle d’équations différentielles continues modifiées (ou à haute résolution) aux premier et deuxième ordres, par rapport à la taille du pas de temps. Enfin, nous illustrerons ces résultats par des présentations de simulations numériques.

Aujourd'hui, l'Europe et la France en particulier sont face à des défis de très grande envergure : adaptation aux impacts du changement climatique, positionnement face aux superpuissances (USA, Chine), désindustrialisation, perte de souveraineté, d'attractivité et de compétitivité, etc. Le numérique n'échappe pas à cette situation. Alors, face aux GAFAM et aux BATX, développer des communs numériques ne serait-il pas une réponse adaptée ?

Given a measure on the real line, one can consider the corresponding orthogonal polynomials that satisfy a three-term recurrence that defines the Jacobi matrix. The classical Szegö theorem gives a condition on the measure for which these polynomials satisfy the strong asymptotics. We will discuss the generalization to the case of multiple orthogonality and show that the corresponding polynomials give rise to a Jacobi matrix on the tree. Finally, we will explain how the classical fixed-point theorems from functional analysis can be used to obtain their strong asymptotics. Based on joint work with A. Aptekarev and M. Yattselev.

Starting from the Abel-Jacobi Theorem we will motivate the interest in moduli spaces of vector bundles on algebraic varieties. 

We study a robust extensible bin packing problem with budgeted uncertainty, under a budgeted uncertainty model where item sizes are defined to lie in the intersection of a box with a one-norm ball. We propose a scenario generation algorithm for this problem, which alternates between solving a master robust bin-packing problem with a finite uncertainty set and solving a separation problem. We first show that the separation is strongly NP-hard given solutions to the continuous relaxation of the master problem. Then, focusing on the separation problem for the integer master problem, we show that this problem becomes a special case of the continuous convex knapsack problem, which is known to be weakly NP-hard. Next, we prove that our special case when each of the functions is piecewise linear, having only two pieces, remains NP-hard. We develop a pseudo-polynomial dynamic program (DP) and a fully polynomial-time approximation scheme (FPTAS) for our special case whose running times match those of a binary knapsack FPTAS. Finally, our computational study shows that the DP can be significantly more efficient in practice compared with solving the problem with specially ordered set (SOS) constraints using advanced mixed-integer (MIP) solvers. Our experiments also demonstrate the application of our separation problem method to solving the robust extensible bin packing problem, including the evaluation of deferring the exact solution of the master problem, separating based on approximate master solutions in intermediate iterations. Finally, a case-study, based on real elective surgery data, demonstrates the potential advantage of our model compared with the actual schedule and optimal nominal schedules.

We define a determinant on the automorphisms of non-trivial Severi-Brauer surfaces. Using the generators and relations, we extend this determinant to birational maps between Severi-Brauer surfaces. Using this determinant and a group homomorphism found in [BSY23] we can determine the abelianisation of the Cremona group of a non-trivial Severi-Brauer surface. This is the first example of an abelianization of the Cremona group of a geometrically rational surface where the automorphism group is not trivial.