PhD positions

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. 

Let K be a number field, one of the most celebrated theorems on the arithmetic of varieties over it is the Faltings Theorem (former Mordell Conjecture): One expects that the following conjecture is true: (Uniform Bound Conjecture) Let K be a number field and g≥2. There exists a number N(K,g) depending only on K and g such that, for every smooth projective curve X of genus g and defined over K, we have that Card(X(K))≤ N(K,g). In higher dimension a well established conjecture is the Lang conjecture: (Lang Conjecture) Let K be a number field, then for every smooth projective variety X defined over K and of general type, the set X(K) of the K--rational points of X is not Zariski dense. The Uniform bound conjecture is implied by the Lang Conjecture: (Caporaso, Harris, Mazur) Lang Conjecture implies the Uniform Bound Conjecture. The aim of this thesis is to study some instances of the implication in the function fields context, by a deeper analysis of the strategy used by Caporaso, Harris and Mazur which takes in account the presence of isotrivial varieties. Some facility with basics of birational geometry will certainly be very useful in this investitation.

The final product of stellar evolution, neutron stars remain enigmatic today, both in terms of their internal structure and their electrodynamic environment. Most of these compact objects appear as pulsars, emitting pulsed electromagnetic radiation in a broad spectral band, the origin of which is still poorly understood. There are currently around 4,000 pulsars. With the advent of a new generation of radio telescopes such as the SKA (Square Kilometre Array), the number of pulsars detected will increase by an order of magnitude, enabling a precise census of neutron star sub-populations, whether isolated or part of a binary system. Ultimately, we hope to gain a better understanding of the radiative processes taking place in their magnetospheres, and better constrain the stellar evolution of binary systems leading to compact objects (Tauris & van den Heuvel, 2023). The aim of this thesis is first to test scenarios for the evolution of binary pulsar geometry, and in particular the alignment between the stellar rotation axis and the binary's orbital angular momentum. Although theory predicts a very short alignment timescale compared with the pulsar's age, some systems deviate significantly from the assumption of near-perfect alignment. Joint modeling of the radio and gamma-ray pulsation will enable us to assess the deviation from alignment (Benli et al., 2021) (Pétri & Mitra, 2021). In a second step, we will extract the orientation of the magnetic field and the observer with respect to the axis of rotation, for all isolated pulsars, visible in radio and gamma. Thirdly, we will model specific systems, notably the millisecond pulsars observed by the NICER (Neutron Star Interior Composition ExploreR) collaboration, to deduce the geometry of the surface magnetic field (Pétri et al., 2023). Finally, we'll be looking at the statistics of the detection of different pulsar sub-populations, young or millisecond, visible in radio, X-rays and/or gamma rays. The work will be divided into three parts. The first part will involve theoretical modeling of multi-wavelength radio emission at very high energies at GeV/TeV, including in-depth knowledge of curvature, synchrotron and inverse Compton radiation processes. A second part will compare the models with observations gathered in radio (ATNF pulsar catalog) and gamma-ray (Fermi/LAT third pulsar catalog, 3PC) catalogs. The third part will involve implementing modules for calculating light curves in radio, X-rays and gamma rays, in order to compare them directly with data from recent observing campaigns (in X-rays with NICER, in gamma rays with Fermi/LAT and in radio with the Nançay, NRT and NenuFAR radio telescopes). This thesis represents preparatory work for the revolution brought about by the detection of a large population of pulsars in the radio domain thanks to SKA, while maintaining a multi-wavelength view of neutron star emission.

This project develops a hybrid approach combining classical numerical methods with neural networks to efficiently simulate radiative transfer in astrophysical simulations, particularly for the Epoch of Reionization. Current models like M1 compromise between precision and computational cost but have physical fidelity limitations. The project extends the Micro-Macro methodology developed for the M1 model using Physics-Informed Neural Networks (PINNs). It aims to design an optimal neural network architecture that captures radiative transfer physics by exploiting symmetries and physical constraints. The approach will be systematically evaluated against traditional methods and applied to classical tests from the literature. The three-year plan includes: (1) mathematical formulation of the micro-macro problem and implementation in 2D using the SCIMBA framework; (2) extension to 3D and development of physical regularization terms; (3) exploration of coupling strategies with Dyablo (https://github.com/Dyablo-HPC/Dyablo) and parallelization for massive simulations. The project emphasizes computational efficiency, documentation, and usability to ensure adoption by the scientific community.

This thesis will investigate bar and spiral arm instabilities in disk galaxies. While modern cosmological simulations reproduce many galaxy properties, they struggle with accurately reproducing bars. The Milky Way, a barred disk galaxy, has recently had its bar and spiral arm properties precisely determined (Khalil et al. 2024, arXiv:2411.12800). This thesis aims to reconstruct the Milky Way's past stellar disk state using the Vlasov-Poisson equation system to model its 4D (2D2V) phase-space evolution, initially excluding gas. Linear perturbation theory and backward integration will track the growth of instabilities, while Physics Informed Neural Networks (PINNs), validated against N-body simulations, will allow us to efficiently explore parameter space. A new patch for the RAMSES hydrodynamics code will simulate gas accretion, and the findings will allow us to refine our models.

This thesis proposes to develop a novel framework to automatically discover interpretable physical laws from observational data. The approach combines two methodologies: a specialized Levenberg-Marquardt optimization algorithm with analytic derivatives for fitting physical models, and a Class-Symbolic-Regression (CSR) method that enforces physical unit consistency across multiple datasets. The key innovation lies in fitting flexible neural networks with accurate derivatives across different physical scenarios, and then rendering the fits interpretable with CSR to discover underlying mathematical laws, conservation principles, and equations of motion. This will represent a significant advance toward automated scientific discovery. The methodology will be applied to study stellar dynamics in the outer Milky Way to map the dark matter distribution, utilizing two major brand new datasets: the 4MOST spectroscopic survey (providing data for over 100,000 RR Lyrae stars) and the LSST. The high-precision 6D phase-space data will enable mapping the gravitational potential and potentially discovering subtle dynamical laws or deviations from standard models.

Dark matter density profiles are key to address some of the challenges of the current standard model of cosmology and of our understanding of physical processes within galaxies. The aim of this thesis is, on the one hand, to test the physical hypotheses of the methods used to infer dark matter density profiles using simulations of isolated galaxies, carried out within the framework of the thesis, and existing cosmological simulations, and, on the other hand, to optimize the methods so that they are applicable to future observational surveys. This will involve both accelerating the Monte Carlo methods by optimally sampling the probability distribution, and exploring new methods such as Bayesian neural networks. This thesis will be carried out within the framework of the Interdisciplinary Thematic Institute IRMIA++, which brings together mathematicians, computer scientists, and astrophysicists. It will notably benefit from a collaboration within the statistics team of IRMA with Augustin Chevallier.

Large-scale structure (LSS) of the Universe is defined as inhomogeneity present on scales larger than that of individual galaxies. On these scales, from a few to hundreds of megaparsecs, matter is organized into a complex network of filaments, emanating from dense compact peaks, and sheet-like walls, forming the boundaries of large underdense voids in between. Such a distribution of matter is today understood in the framework of cosmic web theory and confirmed by all large N-body simulations of structure formation in a ΛCDM Universe. This anisotropic large-scale matter distribution depends on cosmological parameters, but it also naturally defines the environment in which galaxies form and evolve, hence it can be used to constrain cosmological and galaxy formation models. Observationally, 3D LSS is not directly accessible. However, indirect measurements, using various tracers of the matter distribution, can be used. In the nearby Universe (at redshifts z ≲ 1), the 3D LSS can be reconstructed using the positions of galaxies, luminous, yet sparse and biased tracers of the matter density field, available through the large spectroscopic redshift surveys. At higher redshifts (z > 2), the cosmic web is observationally accessible through an alternative probe, the Lyman-α forest absorption in the sight line of distant bright background sources. The Lyman-α forest traces the neutral hydrogen permeating the cosmic web, which is, in turn, correlated with the underlying distribution of dark matter. However, the main limiting factor is the need for a large number of sight lines in order to reconstruct the 3D density field with great precision. Reconstruction of the 3D density field from either sparse galaxy catalogs, or from one-dimensional information along multiple sight lines, can be formulated as an inverse modeling problem. The development of deep learning-based methods that propose a solution to this problem will be at the heart of this PhD project. Of particular interest are neural networks. Among them, Generative Adversarial Networks (GANs) allowing models to be flexible in the generalization of results by including two neural network architectures mutually training each other. GANs are known for their ability to reconstruct missing information since they do not necessarily require an input to generate an answer, and the conditional input can be of any type or dimension. The novelty and challenge in this area lies in the ability to transform low-dimensional information into a latent space representation capable of providing relevant information to the generative model. The deep learning models using this approach usually contain convolutional neural networks (CNNs), using specialized operations capable of retrieving spatial features, such as one-, two- or three-dimensional features present in the distribution of matter on large scales. Additional improvement can be obtained by considering physics-informed neural networks (PINNs), which implement physical formalisms to either guide the training of deep learning models through the loss function or determine the parameters of physical models such as in the inversion of the Lyman-α forest. This allows for a more theory-consistent prediction that is, at the same time, more interpretable in terms of what is driving the model to make decisions. All of these approaches will be combined and contrasted within the GAN framework. The flexibility provided by such an approach allows multiple solutions and ideas to be compared for the same task. More specifically, this PhD project will make use of multiple probes for the reconstruction of the large-scale structure of the Universe, namely, i/ galaxy positions in the redshift space tracing the underlying density field and ii/ Lyman-α forest tracing the underlying distribution of neutral hydrogen. The candidate will be required to develop these models from scratch, going through a learning process that will provide insight on how each of these work internally. The implementation of methods and their testing will be performed on existing large-scale cosmological simulations that are either publicly available or accessible through the existing collaborations of the PhD supervisor. Observational biases associated with each of the LSS probes will be modeled specifically in the context of ongoing and upcoming large international Euclid and PFS surveys, which the PhD candidate will have access to. The quality of the reconstruction will be assessed through the detailed analysis of the morphology of the LSS, which is of great interest for cosmology, but also for galaxy formation studies. Furthermore, given that the explainability of the models (XAI) is nowadays among the biggest challenges in the deep learning community, as well as a concern for the end users, an effort will be made to apply and develop XAI techniques (e.g., GradCam) to interpret the decisions made by the models.

Abstract: A previous result of the IGG team is that cartesian planes over pythagorean ordered fields are mutually interpretable with Tarski's axioms for Euclidean geometry without the continuity axiom. This result can be extended by linking cartesian planes over real closed fields and the full Tarski system of geometry, understanding the continuity axiom as an implementation of Dedekind cuts. This would allow to use the quantifier elimination procedure previously implemented and formally verified by Cyril Cohen to obtain a proof of completeness of Tarski's system of geometry. Stakes: Formally verifying the completeness of Tarski's axioms for Euclidean geometry will not only add to the already verified metatheorems of this theory, it will also paves the way for formally verified computational real algebraic geometry. Keywords: Proof assistant, formalization, completeness, Tarski's axioms, Euclidean geometry, real algebraic geometry.

The eye is a uniquely accessible organ that offers an invaluable window into both ocular and cerebral physiology. Our earlier work, encapsulated in the Ocular Mathematical Virtual Simulator (OMVS), has successfully modeled ocular hemodynamics and biomechanics by coupling heat transfer with the fluid dynamics of aqueous humor (AH) flow. This validated framework, applied to healthy eyes, serves as a robust foundation for our next research phase: extending the model to therapeutic applications.

In this project, we will build upon our core model—primarily based on heat transfer coupled with AH flow—and extend it to simulate advanced therapeutic scenarios. Our primary applications include:

- Laser Surgery: In this scenario, energy is rapidly deposited within the eye (on the order of milliseconds). To simulate such events, we will integrate a radiative transfer module that accurately captures light–tissue interactions and transient thermal effects.

- Cell Injection Therapy: Used for corneal repair, this application requires modeling the fluid dynamics of a cell suspension and the resulting tissue deformation. We will incorporate fluid–structure interaction (FSI) to simulate potential eye inflation during cell injections.

Our work will also enhance the model’s parameterization capabilities. While previous efforts have addressed physical and environmental parameters, we will extend this to include geometrical parameterization. By employing free-form deformation and PDE-based transformations, we can capture variations due to individual eye shapes, aging, and pathological changes.

To ensure our digital twin is patient-specific and can be used in clinical settings, the framework will be further enhanced with inverse problem methods and data assimilation techniques (e.g., ensemble Kalman filters) for real-time estimation of optical and biomechanical parameters.

Our objectives include developing and integrating these advanced modules, designing efficientnumerical schemes and reduced order models (ROM) for real-time simulation, and formulatingrobust strategies for inverse problems. The expected outcome is an integrated multiphysicssimulation framework that not only supports laser surgery and cell injection therapy but is alsoversatile enough to extend to other ocular therapeutic and diagnostic applications.

This project aligns with the ITI IRMIA++ objectives by supporting high-level interdisciplinaryresearch, developing advanced numerical methods and software tools, and promoting international collaboration in applied mathematics and its biomedical applications.