Thesis subjects proposals

The aim of the thesis is to derive reduced models for water waves on a rough bottom in the regime when the size of the roughness and the height of water are small and of the same order. The main challenge is to handle possible resonances, typically via Anderson localization (of a linearized operator).

The aim of this thesis is to study a wide class of partially hyperbolic dynamical systems of geometric origin such as the frame flow over a negatively-curved Riemannian manifold. The main statistical properties of these flows will be investigated (e.g. ergodicity, mixing or exponential mixing). Other connected questions such as the study of the Wilson loop operator from conformal field theory will be explored. This topic is at the interplay between dynamical systems, algebraic topology and microlocal analysis.

The aim of this thesis is to establish quantitative convergence bounds for numerical schemes of the generalized Langevin diffusions used for sampling high-dimensional probability measures, and to study the use of such stochastic processes for the description of the dynamics of low-dimensional representations of complex systems (in particular in atomistic simulations). The Phd would be co-supervised by Urbain Vaes (INRIA) and Pierre Monmarché (Sorbonne Université).

The aim of this thesis is to work on the development of efficient computational approaches in the context of multiscale materials. Examples of applications include materials with microstructures (the characteristic lengthscale of these being much smaller than the characteristic size of the computational domain). It is well-known that, for such problems, standard discretization methods yield a poorly accurate approximation, unless a prohibitively expensive fine mesh/grid is employed. Dedicated multiscale approaches have thus been introduced, which provide a reasonably accurate approximation of the problem for a limited computational cost. Our research team has a long-term experience in the development of those approaches, which are often based on finite elements that are adapted to the precise microstructure of the media, in order to appropriately encode the fine-scale features (at the fine lengthscale). The case of purely elliptic equations is now well-understood. However, many questions remain open in this overall area (for equations other than purely elliptic equations, for parameterized problems, for questions in relation to the construction of coarse approximations of operators with oscillatory-in-space coefficients, ...), and this is the aim of the thesis to push forward the development of such multiscale methods.

The purpose of this thesis is to understand how the insertion of localized defects in otherwise periodic structures affect homogenization processes for partial differential equations. Our research team has a long-term experience in homogenization theory for a large class of PDEs, ranging from linear elliptic equations in divergence form to fully nonlinear equations such as Hamilton-Jacobi equations. The thesis will be focused on the latter class of equations, but the consideration of several settings concurrently will be the opportunity to identify specific mathematical phenomena. The work is based on our previous works, some devoted to elliptic, linear or quasilinear equations, some other devoted to Hamilton-Jacobi equations, all works performed in close collaboration with P.-L. Lions and other colleagues (N. Tchou, etc). The connections with the Calderon Zygmund theory for linear operators, and the theory of viscosity solutions will be explored and exploited in depth. Depending upon the background and aspirations of the PhD candidate, some numerical simulations could also be performed, in particular in order to illustrate important phenomena.

The list of different subjects proposed at IRIF is detailed in the web page indicated above.

Project background: Crohn's disease (CD) is an inflammatory bowel disease with an estimated life-long incidence rate close to 2/1.000 in developed countries. About 120.000 people are affected in France and several millions around the world. Aim of the project: The objective of this project is to develop an algorithm able to identify Crohn Disease (CD) patients with the highest risk of severe disease and treatment failure based on integrated datasets with clinical, histological and molecular parameters recorded at the time of diagnosis. This study is based on: - A retrospective cohort of CD pediatric cases followed in the same hospital and with records of clinical, biological, endoscopic and radiologic data from the initial diagnosis to at least one year of follow-up. - A newly developed transcriptomic method able to analyze a large panel of genes with a limited quantity of intestinal tissue and with limited biases (the Biopred panel). - A newly developed machine learning algorithm able to analyze histological slides. Originality and expected impact of the project: add a renewed histological information to the clinical and molecular information. The development of new biomarkers able to predict the disease course and response to treatments would be a key advance in the management of CD. Working plan: Patients: 436 patients under the age of 18 years at diagnosis are the studied population. The inclusion criteria are i) a diagnosis of CD made according to ESPGHAN criteria (17) at Robert Debré Hospital and ii) a first biopsy available and iii) at least one year of follow up in this hospital. This population is extremely useful for this work for the two following reasons. 1) The available biopsies were performed at the time of diagnosis, most often before any treatment. They reflect data not modified by the various therapies. Such an approach is usually impossible for cohorts from adult hospitals where access to inaugural biopsies is rare. 2) Cumulative clinical data are available, because children and adolescents are usually followed at Robert Debré's hospital until adulthood.

The purpose of this project is to investigate the breakdown of ergodicity in piecewise expanding systems of coupled maps, as it emerges when the coupling strength increases, in particular due to symmetry breaking. The main objective is to prove the existence of such phenomena for an arbitrary number of maps/units.

In the framework of the ANR project MecaFlux, we are working together with experimentalists at Sorbonne University who are building a quantum device coupling a mechanical oscillator (very slow decay) to a superconducting circuit (most prominent platformr for qcontrol: it is the one used by Google, IBM Amazon,...). The goal of this thesis is to precisely analyze the dynamics of this coupled system and study how to operate it in an optimal manner to robustly manipulate quantum information, proposing it as a building block for quantum technology.

A more detailed description of the subject can be found at the following link: https://cermics.enpc.fr/~ehrlachv/sujet_these2.pdf

A more detailed description of the subject can be found at the following link: https://cermics.enpc.fr/~ehrlachv/sujet_these1.pdf

Dynamic games feature several players interacting in an evolving environment. They have found numerous applications in Economics, and their study has motivated a prolific literature in mathematical and algorithmic game theory. Two prominent classes are discrete-time stochastic games, where the environment (the state) is modeled as a controlled Markov chain, and continuous-time differential games, where the state follows a controlled differential equation. The thesis concerns the asymptotic analysis in large time horizon of both models, and their relation. Connections with homogenization theory of Hamilton-Jacobi equations and first-passage percolation will notably be considered. This topic is part of the ANR project ''CONVERGENCE'' (2022-2026).