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funded under the Excellence Initiative program
Call for PhD application happens each spring.
New topics will be added soon: stay tuned and visit this page regularly !
Arthur-César Le Bras (IRMA, Strasbourg)
Raphaël Beuzart-Plessis (I2M, Marseille)
IRMA, Strasbourg - Team "AGA"
In a recent spectacular preprint, Scholze proposed a geometric reformulation of the archimedean local Langlands correspondence, in the spirit of the geometric Langlands program. It is based on analytic geometry, a theory created by Clausen-Scholze via the formalism of condensed mathematics. This geometry allows the construction of new geometric objects, the existence of which Scholze crucially exploits. The goal of the thesis project is to revisit and generalize certain results from the representation theory of real groups, an ancient and very rich theory, in this new language.
Scheme theory
Homological algebra
Lie groups
Representation theory of compact groups, analytic rings and fields
Andrea Thomann (IRMA, Strasbourg)
IRMA, Strasbourg - Team MOCO
Weakly compressible, low-Mach number flows arise in many applications such as meteorology, astrophysics or aerospace engineering. They are difficult to simulate with standard numerical methods, such as the Finite Volume Methods (FVM) [1] or the Lattice-Boltzmann Method (LBM) [3]. The first reason is that using time-explicit schemes are only conditionally stable. Indeed, fast acoustic waves, which are characteristic for low Mach number flows, impose a very constraining CFL condition and thus vanishing time steps as the Mach number decreases. The second reason is that the numerical viscosity, which is important to stabilize the numerical methods, is imposed by the fastest wave speed and is generally too large in the FVM or too low in the LBM for resolving accurately the slow material waves. The third reason is that resolving small unsteady scales requires very fine meshes. This also impedes the computational cost which means that the simulation software must be carefully optimized. The optimization process is complex, time consuming and error-prone. The traditional approach for solving the first difficulty is to adopt (at least partially) implicit schemes. However, this leads to the resolution of large non-linear systems and is computationally very costly. Solutions of the second challenge can be based either on the direct resolution of the low-Mach limit equations (incompressible equations) or on recent schemes, based on a careful analysis of the effect of the numerical viscosity on all the waves [2, 4]. Up to our knowledge, the third challenge is today mainly addressed by lengthy trial-and-error software development. The objective of this research project is to test a new explicit Lattice-Boltzmann method tailored to solve weakly compressible fluid flows. The IRMA-MOCO team has developed recently LBM schemes with very low dissipation, but with proved entropy stability [5, 6]. The LBM is known to behave well on incompressible flows thus it is promising for an extension to weakly compressible flows. In recent years, MOCO has developed explicit, but CFL-less, kinetic schemes for general hyperbolic systems of conservation [7]. The idea is to represent the equations by an equivalent set of kinetic equations coupled through a stiff relaxation term. The number of kinetic velocities remains small and the kinetic model is solved by very efficient CFL-less transport solvers. As is, the method can be applied to compressible flows. But the loss of accuracy at low-Mach is expected. We propose to extend the number of kinetic velocities in order to improve the accuracy of the slow wave approximation. The new scheme, with two families of kinetic velocities (slow and fast), will be analyzed both from the stability and accuracy point of view. The theory has been developed in the team [5, 7] and will be adapted to this case. A part of the work will be devoted to the optimization of the LBM with new tools developed in the OptiTrust ANR project of ICUBE-ICPS. This tools allow to test rapidly code transformation, that are Coq-verified, in order to ensure that the optimized code performs exactly the same computations as the non-optimized source code.
[1] Guillard, H., & Viozat, C. (1999). On the behaviour of upwind schemes in the low Mach number limit. Computers & fluids, 28(1), 63-86.
[2] Farag, G., Zhao, S., Coratger, T., Boivin, P., Chiavassa, G., & Sagaut, P. (2020). A pressure-based regularized lattice-Boltzmann method for the simulation of compressible flows. Physics of Fluids, 32(6).
[3] Frapolli, N., Chikatamarla, S. S., & Karlin, I. V. (2015). Entropic lattice Boltzmann model for compressible flows. Physical Review E, 92(6), 061301.
[4] Chalons, C., Girardin, M., & Kokh, S. (2017). An all-regime Lagrange- Projection like scheme for 2D homogeneous models for two-phase flows on unstructured meshes. Journal of Computational Physics, 335, 885-904.
[5] Lukáová-Medvidová, M., Puppo, G., & Thomann, A. (2023). An all Mach number finite volume method for isentropic two-phase flow. Journal of Numerical Mathematics, 31(3), 175-204.
[6] Bellotti, T., Helluy, P., & Navoret, L. (2024). Fourth-order entropystable lattice Boltzmann schemes for hyperbolic systems. arXiv preprint arXiv:2403.13406.
[7] Gerhard, P., Helluy, P., Michel-Dansac, V., & Weber, B. (2024). Parallel kinetic schemes for conservation laws, with large time steps. Journal of Scientific Computing, 99(1), 5.
Background in fluid mechanics with a focus on hyperbolic partial differential equations, kinetic models and numerical methods, in particular Lattice-Boltzmann method.
Knowledge in theoretical and applied computer science is favourable, in particular scientific computing.
Experience in interdisciplinary projects is welcome as well as an advanced level of English and French.
Raphaël Côte and Benjamin Mélinand (IRMA, Strasbourg)
IRMA, Strasbourg - Teams “MOCO" and "Analysis"
In this thesis we are interested in the mathematical study of wave propagation through different asymptotic models with high frequency Schrödinger behavior. In a first axis, we want to rigorously derive and study a unidirectional asymptotic model of wave equations. This equation is written in the form √(1-∂_x^2 ) ∂_t u+ ∂_x u+ ∂_x^3 u+u ∂_x u=0, (t,x)∈R ×R. At low frequency, the symbol of the linear operator is of the Korteweg-de Vries (KdV) type, and at high frequency, of the Schrodinger type. As fas as we are aware, this model is the first to show this mixed behavior associating two of the most intensively studied dispersive equations. As a first step we plan to show that this equation approaches the wave equation with the same precision as the KdV or BBM equation. We will use for this the methodology described in [Lan13]. In a second step, we will focus on the local and global well-posedness of this equation, with particular attention to smoothness required on the initial data. This point is close to the work done on the BBM equation (see for example [BT09]). Next, we are to study the existence and stability of solitons (waves that propagate while retaining their shape, and which thus achieve an equilibrium between the dispersive and nonlinear effects). We have in mind to apply the techniques used in [KLPS25] and the associated references. Let us insist on two points. First, the Schrödinger-type high-frequency behavior is an important point for obtaining the existence of such objects: in fact, it boils down to a non-local elliptic equation. Second, a low regularity existence result is very important for studying stability. Finally, we also plan to study the behavior of these solitons according to their speed. At low speed, they are expected to behave like KdV solitons, possibly with monotonicity/Kato smoothing properties; and at high speeds, like those of the Schrodinger equation with quadratic nonlinearity. The goal is to make this intuition rigorous. To complete our study we can rely on numerical simulations (as was done for example in [KLPS25] on other equations). We typically have in mind to study numerically the long time behavior. The second line of research is to construct an asymptotic model satisfying the same properties as our scalar equation, but without the unidirectional character. It is therefore a question of deriving a system of equations (typically on the surface and the velocity of the fluid). We also want this système to be globally well posed, for initial data with low regularity. For this, we will start from the abcd Boussinesq systems, which were derived for the first time in [BCS02] and which will be modified to satisfy the desired properties. Once this is done, we will also study in this new framework the existence and stability of solitons.
References
[BCS02] J. L. Bona, M. Chen, and J.-C. Saut. Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and linear theory. J. Nonlinear Sci., 12(4) :283–318, 2002.
[BT09] J. Bona and N/ Tzvetkov. Sharp well-posedness results for the BBM equation. Discrete Contin. Dyn. Syst., 23(4) :1241–1252, 2009.
[KLPS25] C. Klein, F. Linares, D. Pilod, and J.-C. Saut. On the Benjamin and related equations. Bull. Braz. Math. Soc. (N.S.), 56(1) :Paper No. 4, 27, 2025.
[Lan13] D. Lannes. The water waves problem, volume 188 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2013. Mathematical analysis and asymptotics.
M2 Analysis & PDEs
Dominique Aubert (ObAS Strasbourg)
ObAS, Strasbourg - Team "GALHECOS"
The Epoch of Reionization (EoR) happened during the first billion years of the history of the Universe and hosted the buildup of the first structures and their components such as galaxies, active galactic nuclei, the first stars and supermassive black holes. The radiation produced by these objects led to the warm and ionised Intergalactic Medium as we know it today. In an era expected to be stacked with new (JWST) or soon-to-be available (SKA) results, EoR studies provide an ideal scientific case to push for the development of numerical methods : code development, production of state-of-the-art simulations, post-processing of large datasets and dissemination of simulation products. In particular, EoR models present specific challenges as they require simultaneously large simulated volumes (typically 200 cMpc), sufficiently resolved at galactic scale (ideally<1 kpc) while dealing with intensive computations for e.g. radiative transfer. The upcoming generation of supercomputers, which surpasses the Exaflop processing power thanks notably to Graphics Processing units (GPUs), will provide the means to overcome these difficulties. In october 2024, members of the ObAS (D. Aubert & P. Ocvirk) started the ExaSKAle ANR 48 months project that aims at 1/ developing the next generation of EoR simulation code for Exascale machines 2/ produce EoR simulations on Exascale supercomputers over this timeline 3/ use these simulations to analyse jointly the IGM reionization and first galaxies in a unified framework. In particular we will build upon the current Dyablo project [1], led by the French atomic agency CEA to build a new AMR framework for astrophysical simulations and for which Strasbourg has been actively contributing for a few years. Dyablo is designed to be massively parallel (including GPUs), architecture-agnostic as well as an experience of code co-design between astrophysicists and computational scientists. It is also part of the demonstrators of the Exa-DI Numpex initiative. As of today, the code is able to produce EoR simulations with ad-hoc modeling of astrophysical sources of radiation. The goals of the PhD in this context would be twofold : 1/ contribute to the ExaSKAle project by developing the Dyablo code, notably to include astrophysical models of source formation 2/ study the topology of the reionization process in Dyablo simulations using concepts of Morse Theory. The first part could be based on the Dyablo star formation modules currently being developed by collaborators at the Institut d’astrophysique de Paris or on semi-analytical prescriptions for unresolved radiation production by halos at large (z>20) redshift [2], that we currently re-implement in the context of a Master internship. This part of the work requires code development of the Dyablo C++ code and being able to test and validate these implementations on HPC infrastructures. For the second part, the PhD will produce and analyse Dyablo simulations to re-assess the topology of the propagation of the reionization process, following a methodogy developed recently in Strasbourg [3] on semi-analytical models. In particular the PhD will aim at retesting the validity of the gaussian random field hypothesis for reionization times fields produced in simulations and see if predictions on peak statistics, isocontours and the skeleton still hold. From there, the PhD will aim at investigating how galaxy properties can be traced back to the topology of EoR, both observable in principle by experiments such as JWST and SKA [4].
[1] Delorme, M., Durocher, A., Aubert, D., Brun, A. S., & Marchal, O. (2024), SF2A-2024: Proceedings
[2] Meriot, R., & Semelin, B. (2024), A&A, 683, A24.
[3] Thélie, E., Aubert, D., Gillet, N., Hiegel, J., & Ocvirk, P. (2023), A&A, 672, A184.
[4] Hiegel, J., Thélie, É., Aubert, D., Chardin, J., Gillet, N. et al. (2023), A&A, 679, A125.
Python and C++ programming skills
Knowledge of collaborative IT development tools
Reasonable facility with a “formal” and mathematical approach to physics: Gaussian field theory, Press-Schechter model
These subjects are proposed by ITI IRMIA++ members for PhD contracts starting in september/october 2025.
In order to apply, please click on the button below to access the application form and select the PhD Position Call for projects / Candidate profile :
You will need to provide the following information and documents:
Deadline for application reception : April 14th, 2025
If you are interested in a subject which is not in the subjects list, please contact directly the researchers and team you want to work with.
If you start a PhD in an ITI IRMIA++ team, we can offer you a financial help for your installation !
More information on the dedicated page.
Candidates recruited as PhDs will benefit from IRMIA++ funding and will have to follow the Graduate Program "Mathematics and Applications: Research and Interactions".
