Publications

We prove a formula for the number of occurrences of certain labels and local configurations in two-step puzzles introduced by Buch, Kresch, Purbhoo and Tamvakis building on Knutson’s original conjecture for the cohomology of partial flag varieties. These puzzles are tilings of the triangular lattice by edge-labeled tiles and compute the Schubert structure constants of the cohomology of two-step flag varieties. Our formula depends only on the boundary conditions of the puzzle. The proof relies on the study of color maps, which are tilings of the triangular lattice derived from puzzles.

We provide a manifestly positive expression for the volume of moduli spaces of flat unitary connections on punctured compact oriented surfaces. This volume is obtained by summing volumes of explicit polytopes describing coloured honeycombs on a polygon, in the spirit of the work of Knutson and Tao describing the spectrum of the sum of two hermitian matrices. As a corollary, we also provide a positive formula for marginals of the unitary Yang-Mills measure on a compact oriented surface in terms of the probability distribution of an explicit path process.

En 1887 Volterra est lancé dans la recherche d’une vision générale pour l’analyse qui va prendre plusieurs formes. Ses travaux les plus connus le mènent ainsi à définir les fonctionnelles, ou plus précisément à développer un calcul différentiel et intégral pour des « fonctions qui dépendent d’autres fonctions » ou des « fonctions de lignes ». Pourtant les efforts de Volterra pour définir un contexte général pour certains problèmes d’analyse vont aussi l’amener à étendre les notions de dérivation et d’intégration aux substitutions—matrices dont les coefficients sont des fonctions—qui ont un rôle important dans l’étude des équations différentielles linéaires. Dans un mémoire intitulé Sui fondamenti della teoria delle equazioni differenziali lineari Volterra établit un calcul différentiel et intégral pour les substitutions. Ce travail, qui permet de penser les équations différentielles linéaires grâce à deux opérations sur les substitutions—dérivation et intégration, permet aussi d’analyser la stratégie de progression mise en œuvre par le mathématicien italien dans sa recherche d’une analyse généralisée dès le début de sa carrière. Nous examinons les processus de sélection et de réorganisation qui ont permis à Volterra de transposer une théorie bien établie pour les fonctions ordinaires à un cadre adapté aux substitutions. Nous mettons ainsi au jour une dynamique de progression vers le général révélant les éléments sur lesquels s’appuie sa pensée et les motifs qui l’animent. Loin d’être anecdotique, ce texte qui ne résout pourtant aucune conjecture, permet de voir une cohérence dans la manière de progresser de Volterra, et éclaire son rôle dans la recherche d’une analyse générale qui deviendra petit-à-petit l’analyse fonctionnelle du 20ᵉ siècle.

Given a compact Riemannian surface $M$, with Laplace-Beltrami operator $\Delta$, for $\lambda > 0$, let $P_{\lambda,\lambda^{-\frac{1}{3}}}$ be the spectral projector on the bandwidth $[\lambda-\lambda^{-\frac{1}{3}}, \lambda + \lambda^{\frac{1}{3}}]$ associated to $\sqrt{-\Delta}$. We prove a polynomial improvement on the $L^2 \to L^{\infty}$ norm of $P_{\lambda,\lambda^{-\frac{1}{3}}}$ for generic simple spheres of revolution (away from the poles and the equator) and for the Euclidean disk away from its center but up to the boundary. We use the Quantum Integrability of those surfaces to express the norm in terms of a joint basis of eigenfunctions for $\left(\sqrt{-\Delta}, \frac{1}{i}\frac{\partial}{\partial \theta}\right)$. Then, we use that those eigenfunctions are asymptotically Lagrangian oscillatory functions, each supported on a Lagrangian torus with fold-type caustic. Thus, studying the distribution of the caustics, and using BKW decay away from the caustics, we are able to reduce the problem to counting estimates.

Benchmarks are important tools for tracking the rapid advancements in large language model (LLM) capabilities. However, benchmarks are not keeping pace in difficulty: LLMs now achieve more than 90% accuracy on popular benchmarks such as Measuring Massive Multitask Language Understanding1, limiting informed measurement of state-of-the-art LLM capabilities. Here, in response, we introduce Humanity’s Last Exam (HLE), a multi-modal benchmark at the frontier of human knowledge, designed to be an expert-level closed-ended academic benchmark with broad subject coverage. HLE consists of 2,500 questions across dozens of subjects, including mathematics, humanities and the natural sciences. HLE is developed globally by subject-matter experts and consists of multiple-choice and short-answer questions suitable for automated grading. Each question has a known solution that is unambiguous and easily verifiable but cannot be quickly answered by internet retrieval. State-of-the-art LLMs demonstrate low accuracy and calibration on HLE, highlighting a marked gap between current LLM capabilities and the expert human frontier on closed-ended academic questions. To inform research and policymaking upon a clear understanding of model capabilities, we publicly release HLE at https://lastexam.ai.

Benchmarks are important tools for tracking the rapid advancements in large language model (LLM) capabilities. However, benchmarks are not keeping pace in difficulty: LLMs now achieve more than 90% accuracy on popular benchmarks such as Measuring Massive Multitask Language Understanding1, limiting informed measurement of state-of-the-art LLM capabilities. Here, in response, we introduce Humanity’s Last Exam (HLE), a multi-modal benchmark at the frontier of human knowledge, designed to be an expert-level closed-ended academic benchmark with broad subject coverage. HLE consists of 2,500 questions across dozens of subjects, including mathematics, humanities and the natural sciences. HLE is developed globally by subject-matter experts and consists of multiple-choice and short-answer questions suitable for automated grading. Each question has a known solution that is unambiguous and easily verifiable but cannot be quickly answered by internet retrieval. State-of-the-art LLMs demonstrate low accuracy and calibration on HLE, highlighting a marked gap between current LLM capabilities and the expert human frontier on closed-ended academic questions. To inform research and policymaking upon a clear understanding of model capabilities, we publicly release HLE at https://lastexam.ai.

We consider ergodic translation-invariant Gibbs measures for the dimer model (i.e. perfect matchings) on the hexagonal lattice. The complement to a dimer configuration is a fully-packed loop configuration: each vertex has degree two. This is also known as the loop $O(1)$ model at $x=\infty$. We show that, if the measure is non-frozen, then it exhibits either infinitely many loops around every face or a unique bi-infinite path. Our main tool is the flip (or XOR) operation: if a hexagon contains exactly three dimers, one can replace them by the other three edges. Classical results in the dimer theory imply that such hexagons appear with a positive density. Up to some extent, this replaces the finite-energy property and allows to make use of tools from the percolation theory, in particular the Burton--Keane argument, to exclude existence of more than one bi-infinite path.

Flow Matching is a recent framework for learning continuous transformations between probability measures. The method constructs a time-dependent velocity field whose flow transports a source distribution to a target distribution, and whose training reduces to a simple regression problem on paired samples. This simulation-free objective makes Flow Matching an attractive alternative to continuous normalizing flows and diffusion models. This tutorial provides a self-contained and mathematically rigorous introduction to Flow Matching, aimed at applied mathematicians. Starting from the continuity equation, we establish the theoretical foundations linking velocity fields, probability paths, and flows, and explain how Flow Matching arises from a particular construction based on couplings of probability measures. We carefully state the assumptions under which the induced ordinary differential equation defines a unique flow and yields a valid pushforward between distributions, and we illustrate the limitations of the theory through explicit counterexamples. We derive closed-form velocity fields in several important settings, including one-dimensional distributions, Gaussian and Gaussian mixture models, and semi-discrete targets, and we clarify the connections with score matching, diffusion models, and optimal transport. Throughout the paper, theoretical results are complemented by reproducible numerical experiments designed to build intuition and illustrate practical behavior. Our goal is to provide readers with both a solid mathematical understanding of Flow Matching and concrete tools for its application.

Abstract This study investigates the occurrence of surface elevation oscillations, known as seiches, at the atoll scale. We show that the innovative Ka-band Radar Interferometer (KaRIn) onboard the Surface Water Ocean Topography (SWOT) satellite makes it possible to visualize seiche-like structures within lagoons of French Polynesia. This ability to capture two-dimensional sea surface undulations of low amplitude (on the order of a few centimeters) in relatively small water bodies (less than 80 km in length) is unprecedented in satellite altimetry, and opens new avenues of research into these events and their contribution to coastal erosion and flooding. Our study combines sea surface height observations from SWOT with in situ measurements conducted in the Raroia lagoon and theoretical calculations based on an eigenvalue model. These three complementary approaches-satellite remote sensing, in situ data, and theoretical modeling-allow us to investigate both the spatial and spectral properties of seiches in atoll islands. Plain Language Summary In closed or semi-enclosed water bodies, a resonant phenomenon can happen in which the water surface performs standing oscillations. These oscillations, known as seiches, have well-defined periods and geometries at the basin scale. Here we investigate their presence in the lagoon of atoll islands using in situ measurements, modeling, and the Surface Water Ocean Topography (SWOT) mission, a new satellite that provides for the first time two-dimensional images of the ocean surface elevation with an unprecedented precision and resolution. We show that SWOT does capture seiche-like structures, and our in situ measurements confirm that such resonant modes do occur in the lagoon of Raroia. This is a first step toward understanding and assessing the importance of these modes in Pacific atoll islands, where they have received relatively little attention though they appear to play a role in sediment transport and submersion events.

These notes are based on a short course delivered at the Summer School EUR MINT 2025 "Control, Inverse Problems and Spectral Theory", held in June 2025 in Toulouse, France. The course presents three important strategies in control theory, formulated as time-iteration methods, where each time step brings the state of the system closer to the desired target. For linear PDEs, we survey the classical Lebeau-Robbiano method and its more recent developments. This method combines spectral inequalities and dissipation estimates to prove null controllability of a dissipative linear system. For nonlinear PDEs, we reinterpret the Liu-Takahashi-Tucsnak method, which establishes local controllability of a nonlinear system by analyzing the control cost of its linearization. We provide an easily applicable black-box formulation of their method. Finally, for nonlinear ODEs, we present the tangent vectors method, which establishes local exact controllability starting from approximately reachable directions.

As context windows in large language models continue to expand, it is essential to characterize how attention behaves at extreme sequence lengths. We introduce token-sample complexity: the rate at which attention computed on $n$ tokens converges to its infinite-token limit. We estimate finite-$n$ convergence bounds at two levels: pointwise uniform convergence of the attention map, and convergence of moments for the transformed token distribution. For compactly supported (and more generally sub-Gaussian) distributions, our first result shows that the attention map converges uniformly on a ball of radius $R$ at rate $C(R)/\sqrt{n}$, where $C(R)$ grows exponentially with $R$. For large $R$, this estimate loses practical value, and our second result addresses this issue by establishing convergence rates for the moments of the transformed distribution (the token output of the attention layer). In this case, the rate is $C'(R)/n^β$ with $β<\tfrac{1}{2}$, and $C'(R)$ depends polynomially on the size of the support of the distribution. The exponent $β$ depends on the attention geometry and the spectral properties of the tokens distribution. We also examine the regime in which the attention parameter tends to infinity and the softmax approaches a hardmax, and in this setting, we establish a logarithmic rate of convergence. Experiments on synthetic Gaussian data and real BERT models on Wikipedia text confirm our predictions.

The work presented here is concerned with breaking water waves, a well-known phenomenon arising as an oceanic wave approaches the shore: its crest starts to move faster than the trough up front, which ultimately leads to the appearance of an overhanging region that quickly curls over while falling down until it collides with the water lying below. An important contemporary issue concerns the incorporation of the viscous dissipation associated with the breaking into the many models that have been introduced to describe the ocean. This is mostly done empirically. In the present work, we follow a different path: we aim at modelling wave breaking up to the free surface self-intersection (the splash singularity), relying thus on a more geometrical approach to the subject. The first part of this thesis will be devoted to the motivation of a set of equations that describes overhanging waves in the inviscid irrotational regime, with either a one-dimensional or a two-dimensional free surface. This is done by setting aside the commonly used Eulerian framework and working in (pseudo)Lagrangian coordinates instead. This should be seen as an extension of the Zakharov-Craig-Sulem formulation of the Water Waves problem. The non-canonical Hamiltonian structure of these partial differential equations is investigated and it is shown that in the absence of breaking, they can be reduced to the usual set of equations. Emphasis is put on the various physical assumptions that are made along the way. In a second moment, we come back to these very hypotheses and put them to the test. This is done numerically using a Navier-Stokes based computational framework based on the Finite-Element Method (FEM). The major novelty compared to other studies lies in the use of the Arbitrary Lagrangian-Eulerian method (ALE), which diminishes the interpolation error greatly. The viscosity can therefore be decreased to values that allow the comparison with the inviscid solution (computed using another method, based on potential theory in the complex plane) to be carried out. Over a flat topography, it is found that both the free-surface and bed boundary layers are sufficiently well-behaved as to not perturb the bulk irrotational flow. Water being characterised by a relatively small viscosity, the consequence is that, in this regime the inviscid models accurately describe the oceanic flow. We do not prove this assertion rigorously, however. Difficulties seem to arise, however, when a non-flat topography is considered. Indeed, the typical velocities associated with the wave are high enough to eventually trigger boundary layer separation near curved-enough portions of the bed, resulting in vorticity being shed in the initially irrotational flow, far from the topography. The convergence to the inviscid solution is therefore compromised.

We classify the unimodular Euclidean integral lattices of rank 29 by developing an elementary, yet very efficient, inductive method. As an application, we determine the isometry classes of even lattices of rank at most 28 and prime (half-)determinant at most 7. We also provide new isometry invariants allowing for independent verification of the completeness of our lists, and we give conceptual explanations of some unique orbit phenomena discovered during our computations. Some of the genera classified here are orders of magnitude larger than any genus previously classified. In a forthcoming companion paper, we use these computations to study the cohomology of GL_n(Z).

We classify the unimodular Euclidean integral lattices of rank 29 by developing an elementary, yet very efficient, inductive method. As an application, we determine the isometry classes of even lattices of rank at most 28 and prime (half-)determinant at most 7. We also provide new isometry invariants allowing for independent verification of the completeness of our lists, and we give conceptual explanations of some unique orbit phenomena discovered during our computations. Some of the genera classified here are orders of magnitude larger than any genus previously classified. In a forthcoming companion paper, we use these computations to study the cohomology of GL_n(Z).

Lectures grothendieckiennes rassemble les textes qui font suite à un séminaire qui s’est tenu au département de mathématiques de l’École Normale Supérieure de 2017 à 2018. Le livre présente une pensée complexe à l’œuvre, celle de l’un des mathématiciens les plus influents et énigmatiques du 20e siècle : Alexander Grothendieck. Les auteurs, Pierre Cartier, Olivia Caramello, Alain Connes, Laurent Laforgue, Colin McLarty, Gilles Pisier, Jean-Jacques Szczeciniarz et Fernando Zalamea, dévoilent à leur façon les conséquences mathématiques ou philosophiques que l’on peut tirer d’une œuvre monumentale qui a transformé le paysage mathématique du 20e siècle et qui a probablement ouvert une nouvelle ère mathématique que nous avons seulement commencé à explorer. Préface de Peter Scholze.

We study a class of parabolic quasilinear systems, in which the diffusion matrix is not uniformly elliptic, but satisfies a Petrovskii condition of positivity of the real part of the eigenvalues. Local wellposedness is known since the work of Amann in the 90s, by a semi-group method. We first revisit these results in the context of Sobolev spaces modelled on L^2 and then explore the endpoint Besov case B_{p,1}^{d/p}. We also exemplify our method on the SKT system, showing the existence of local, non-negative, strong solutions.

We prove a bumpy metric theorem in the sense of Ma\~{n}e for non-convex Hamiltonians that are satisfying a certain geometric property.

We construct self-similar solutions to the 2D Navier--Stokes equations evolving from arbitrarily large $-1$--homogeneous initial data and present numerical evidence for their non-uniqueness.

A fundamental challenge in the theory of deep learning is to understand whether gradient-based training in high-dimensional parameter spaces can be captured by simpler, lower-dimensional structures, leading to so-called implicit bias. As a stepping stone, we study when a gradient flow on a high-dimensional variable $θ$ implies an intrinsic gradient flow on a lower-dimensional variable $z = ϕ(θ)$, for an architecture-related function $ϕ$. We express a so-called intrinsic dynamic property and show how it is related to the study of conservation laws associated with the factorization $ϕ$. This leads to a simple criterion based on the inclusion of kernels of linear maps which yields a necessary condition for this property to hold. We then apply our theory to general ReLU networks of arbitrary depth and show that, for any initialization, it is possible to rewrite the flow as an intrinsic dynamic in a lower dimension that depends only on $z$ and the initialization, when $ϕ$ is the so-called path-lifting. In the case of linear networks with $ϕ$ the product of weight matrices, so-called balanced initializations are also known to enable such a dimensionality reduction; we generalize this result to a broader class of {\em relaxed balanced} initializations, showing that, in certain configurations, these are the \emph{only} initializations that ensure the intrinsic dynamic property. Finally, for the linear neural ODE associated with the limit of infinitely deep linear networks, with relaxed balanced initialization, we explicitly express the corresponding intrinsic dynamics.

189 pages, comments welcome. Figures added. Numerous typos corrected. Contents of Sections 4.4, 6.2, 9.1 and 9.2 of Version 1 corrected and completed

This paper is devoted to the asymptotic analysis of strongly rotating and stratified fluids, under a $\beta$-plane approximation, and within a three-dimensional spatial domain with strong topography. Our purpose is to propose a linear idealized model, which is able to capture one of the key features of western boundary currents, in spite of its simplicity: the separation of the currents from the coast. Our simplified framework allows us to perform explicit computations, and to highlight the intricate links between rotation, stratification and bathymetry. In fact, we are able to construct approximate solutions at any order for our system, and to justify their validity. Each term in the asymptotic expansion is the sum of an interior part and of two boundary layer parts: a ``Munk'' type boundary layer, which is quasi-geostrophic, and an ``Ekman part'', which is not. Even though the Munk part of the approximation bears some similarity with previously studied 2D models, the analysis of the Ekman part is completely new, and several of its properties differ strongly from the ones of classical Ekman layers. Our theoretical analysis is supplemented with numerical illustrations, which exhibit the desired separation behavior.

We show the convergence of the characteristic polynomial for random permutation matrices sampled from the generalized Ewens distribution. Under this distribution, the measure of a given permutation depends only on its cycle structure, according to certain weights assigned to each cycle length. The proof is based on uniform control of the characteristic polynomial using results from the singularity analysis of generating functions, together with the convergence of traces to explicit random variables expressed via a Poisson family. The limit function is the exponential of a Poisson series which has already appeared in the case of uniform permutation matrices. It is the Poisson analog of the Gaussian Holomorphic Chaos, related to the limit of characteristic polynomials for other matrix models such as Circular Ensembles, i.i.d. matrices, and Gaussian elliptic matrices.

Relative entropy, as a divergence metric between two distributions, can be used for offline change-point detection and extends classical methods that mainly rely on moment-based discrepancies. To build a statistical test suitable for this context, we study the distribution of empirical relative entropy and derive several types of approximations: concentration inequalities for finite samples, asymptotic distributions, and Berry-Esseen bounds in a pre-asymptotic regime. For the latter, we introduce a new approach to obtain Berry-Esseen inequalities for nonlinear functions of sum statistics under some convexity assumptions. Our theoretical contributions cover both one-and two-sample empirical relative entropies. We then detail a change-point detection procedure built on relative entropy and compare it, through extensive simulations, with classical methods based on moments or on information criteria. Finally, we illustrate its practical relevance on two real datasets involving temperature series and volatility of stock indices.

In this thesis, we study interpretable groups and fields in various theories of enriched fields, using tools from geometric model theory. The work is divided into the following three parts. The group configuration theorem for generically stable types. Following the proof of the group configuration theorem in the usual stable setting, we generalize it to the case of generically stable group configurations in arbitrary theories. More explicitly, we show how one can construct a type-definable group (action) from a generically stable sextuple of points satisfying the usual algebraicity and independence properties of a group configuration. On groups and fields interpretable in NTP2 fields. We show that, in NTP2 theories of enriched fields, under mild model-theoretic and algebraic assumptions, any definably amenable interpretable group admits a definable morphism to an algebraic group with purely imaginary kernel, i.e. that does not admit definable maps to the field sort with infinite image. We deduce a structure theorem for interpretable fields, which we instantiate for henselian valued fields of characteristic 0. We also extend these results to NIP (possibly enriched) differential fields, and prove a full classification of interpretable fields for differentially closed valued fields. In passing, we prove that in arbitrary theories, if K and F are definable fields such that the group of affine transformations F+ ⋊ F × can be definably embedded into an algebraic group over K, then F admits a definable field embedding into a finite extension of the field K. On groups and fields definable in D-henselian fields. Finally, we focus on the theory of D-henselian valued fields with differentially closed residue field and divisible value group, studied by Scanlon and Rideau-Kikuchi. Adapting the proof of Hrushovski’s p-configuration theorem, we prove that groups definable in the valued field sort with generically stable generics orthogonal to all differentially algebraic types, admit definable group homomorphisms to alge- braic groups, with kernels of finite rank. We then show that any definable field, in the valued field sort, with a generating subring admitting such a generic, is definably isomorphic to the valued field itself, assuming its Kolchin closure is of infinite rank.

In this thesis, we study interpretable groups and fields in various theories of enriched fields, using tools from geometric model theory. The work is divided into the following three parts. The group configuration theorem for generically stable types. Following the proof of the group configuration theorem in the usual stable setting, we generalize it to the case of generically stable group configurations in arbitrary theories. More explicitly, we show how one can construct a type-definable group (action) from a generically stable sextuple of points satisfying the usual algebraicity and independence properties of a group configuration. On groups and fields interpretable in NTP2 fields. We show that, in NTP2 theories of enriched fields, under mild model-theoretic and algebraic assumptions, any definably amenable interpretable group admits a definable morphism to an algebraic group with purely imaginary kernel, i.e. that does not admit definable maps to the field sort with infinite image. We deduce a structure theorem for interpretable fields, which we instantiate for henselian valued fields of characteristic 0. We also extend these results to NIP (possibly enriched) differential fields, and prove a full classification of interpretable fields for differentially closed valued fields. In passing, we prove that in arbitrary theories, if K and F are definable fields such that the group of affine transformations F+ ⋊ F × can be definably embedded into an algebraic group over K, then F admits a definable field embedding into a finite extension of the field K. On groups and fields definable in D-henselian fields. Finally, we focus on the theory of D-henselian valued fields with differentially closed residue field and divisible value group, studied by Scanlon and Rideau-Kikuchi. Adapting the proof of Hrushovski’s p-configuration theorem, we prove that groups definable in the valued field sort with generically stable generics orthogonal to all differentially algebraic types, admit definable group homomorphisms to alge- braic groups, with kernels of finite rank. We then show that any definable field, in the valued field sort, with a generating subring admitting such a generic, is definably isomorphic to the valued field itself, assuming its Kolchin closure is of infinite rank.

We prove convergence of solutions of Dirichlet problems and Green's functions on Tutte's harmonic embeddings to those of the linearized Monge-Ampère equation $\mathcal{L}_φh=0$. The potential $φ$ appears as the limit of piecewise linear potentials associated with the embeddings and the only assumption that we use is the uniform convexity of $φ$. Even if $φ$ is quadratic, this setup significantly generalizes known results for discrete harmonic functions on orthodiagonal tilings. Motivated by potential applications to the analysis of 2d lattice models on irregular graphs, we also study the situation in which the limits are harmonic in a different complex structure.

We prove convergence of solutions of Dirichlet problems and Green's functions on Tutte's harmonic embeddings to those of the linearized Monge-Ampère equation $\mathcal{L}_φh=0$. The potential $φ$ appears as the limit of piecewise linear potentials associated with the embeddings and the only assumption that we use is the uniform convexity of $φ$. Even if $φ$ is quadratic, this setup significantly generalizes known results for discrete harmonic functions on orthodiagonal tilings. Motivated by potential applications to the analysis of 2d lattice models on irregular graphs, we also study the situation in which the limits are harmonic in a different complex structure.

Numerical experiments of dynamo action designed to understand the generation of Earth's magnetic field produce different regime branches identified within bifurcation diagrams. Notable are distinct branches where the resultant magnetic field is either weak or strong. Weak‐field solutions are identified by the prominent role of viscosity (and/or inertia) on the motion, whereas the magnetic field has a leading‐order effect on the flow in strong‐field solutions. We demonstrate the persistence of the strong‐field branch, preserving the expected force balance of Earth's core, and provide scaling laws governing its onset as parameters move toward values appropriate for the Geodynamo. We introduce a new output parameter, based on dynamically important parts of rotational and magnetic forces, that captures expected values of strong‐field solutions throughout input parameter space. This new measure of the field strength and our bounds on scaling laws can guide future studies in locating strong‐field dynamos in parameter space.

Recent numerical experiments of dynamo action relevant to the generation of the geomagnetic field have produced different regime branches identified within bifurcation diagrams. Notable are separate branches in which the resultant magnetic field is either weak or strong. Weak-field solutions can be identified by the prominent role of viscosity on the motion whereas the magnetic field has a leading order effect on the flow in strong-field solutions. For a given Ekman number, E (measuring the ratio of viscosity to rotational effects), the existence of these branches and bistability between them is reliant on a small enough magnetic Ekman number, E m (measuring the ratio of magnetic diffusion to rotational effects, so E / E m = P m , the magnetic Prandtl number). Both branches are known to produce large scale dipolar magnetic fields but do not exhibit an expected scale separation between the flow and magnetic field. In this work, by reducing E m , we identify a variety of dynamo states on the weak-field branch beyond the known dipolar solutions. Specifically, hemispherical and nondipolar dynamos were found, in addition to the usual dipolar solutions. Some solutions exhibit clear scale separation between small-scale flow and large-scale magnetic field, despite the large ratio of viscosity to magnetic diffusion. Numerical solutions in this regime have not been observed before and they offer a first connection with earlier theoretical work based on mean-field theory.

Convection is the main heat transport mechanism in the Earth's liquid core and is thought to power the dynamo that generates the geomagnetic field. Core convection is strongly constrained by rotation while being turbulent. Given the difficulty in modeling these conditions, some key properties of core convection are still debated, including the dominant energy‐carrying lengthscale. Different regimes of rapidly rotating, unmagnetized, turbulent convection exist depending on the importance of viscous and inertial forces in the dynamics, and hence different theoretical predictions for the dominant flow lengthscale have been proposed. Here we study the transition from viscously dominated to inertia‐dominated regimes using numerical simulations in spherical and planar geometries. We find that the cross‐over occurs when the inertial lengthscale approximately equals the viscous lengthscale. This suggests that core convection in the absence of magnetic fields is dominated by the inertial scale, which is hundred times larger than the viscous scale.

Nous proposons une exploration de la notion de distance, sous les regards croisés d'un mathématicien et d'un géographe.

Asymptotic solutions are investigated for the travelling wave consisting of infectives I ( x − c t ) propagating at speed c into a region of uninfected susceptibles S = S + , on the basis that S + is large. In the moving frame, three domains are identified. In the narrow leading frontal region, the infectives terminate relatively abruptly. Conditions ahead (increasing x ) of the front control the speed c of the front advance. In the trailing region (decreasing x ), the number of infectives decay relatively slowly. Our asymptotic development focuses on the dependence of I on S in the central region. Then, the apparently simple problem is complicated by the presence of both algebraic and logarithmic dependencies. Still, we can construct an asymptotic expansion to a high order of accuracy that embeds the trailing region solution. A proper solution in the frontal region is numerical, but here the central region solution works well too. We also investigated numerically the evolution from an initial state to a travelling wave. Following the decay of transients, the speed adopted by the wave is fast, though the slowest of those admissible. The asymptotic solutions are compared with the numerical solutions and display excellent agreement.

A classical approach to the Calderón problem is to estimate the unknown conductivity by solving a nonlinear least-squares problem. It leads to a nonconvex optimization problem which is generally believed to be riddled with bad local minimums. We revisit this issue in the case of piecewise constant radial conductivities and prove that, contrary to previous claims, there are no spurious critical points in the case of two scalar unknowns with no measurement noise. We also provide a partial proof of this result in the general setting which holds under a numerically verifiable assumption. Finally, we investigate whether a recently proposed approach based on convexification yields better reconstructions. For the first time, we propose a way to implement it in practice and show that it is consistently outperformed by some least squares solvers, which are also faster and require less measurements.

We prove existence, uniqueness and regularity of weak solutions of Kolmogorov--Fokker--Planck equations with either local or non-local diffusion in the velocity variable and rough diffusion coefficients or kernels. Our results cover the Cauchy problem and allow a broad class of source terms under minimal assumptions. The core of the analysis is a set of sharp kinetic embeddings \`a la Lions and transfer-of-regularity results \`a la Bouchut--H\"ormander. We formulate these tools in a homogeneous, scale-invariant form, available for a large range of regularity parameters.

In this paper, we are interested in the $\beta$-ensembles (or 1D log-gas) with Freud weights, namely with a potential of the form $|x|^{p}$ with $p \geq 2$. Since this potential is not of class $\mathcal{C}^{3}$ when $p \in (2,3]$, most of the literature does not apply. In this singular setting, we prove the central limit theorem for linear statistics with general test-functions and compute the subleading correction to the free energy. Our strategy relies on establishing an optimal local law in the spirit of [Bourgade, Mody, Pain 22']. Our results allow us to give a large $N$ expansion up to $o(N)$ of the log-volume of the unit balls of $N\times N$ self-adjoint matrices for the $p$-Schatten norms and to give a consistency check of the KLS conjecture. For the latter, we consider the functions $f(X)=\mathrm{Tr}(X^r)^q$ and the uniform distributions on these same Schatten balls for $N$ large enough. While the case $p>3$, $q=1, r=2$, was proven in [Dadoun, Fradelizi, Guédon, Zitt 23'], we address in the present paper the case $p\geq2$, $q\geq1$ and $r\geq2$ an even integer. The proofs are based on a link between the moments of norms of uniform laws on $p$-Schatten balls and the $\beta$-ensembles with Freud weights.

We propose a model theoretic interpretation of the theorems about the equivalence between mixed characteristic perfectoid spaces and their tilts.

We study groups definable in existentially closed geometric fields with commuting derivations. Our main result is that such a group can be definably embedded in a group interpretable in the underlying geometric field. Compared to earlier work of the first two authors toguether with K. Peterzil, the novelty is that we also deal with infinite dimensional groups.

In this thesis, we investigate combinatorial, geometric, and probabilistic properties of wreath products and other group extensions. The work is divided into the following two parts. [1] Non-extendable geodesics in Cayley graphs. We study the property of having unbounded depth in Cayley graphs of wreath products. That is, whether there exist elements at arbitrarily large distance from other elements of larger word length. We prove that for any finite group A and any finitely generated group B, the wreath product A ≀ B admits a standard generating set with unbounded depth. If B is abelian, then the above is true for every standard generating set. This generalizes the case B = ℤ, due to Cleary and Taback. When B = H ∗ K for two finite groups H and K, we characterize which standard generators of A ≀ B have unbounded depth in terms of a geometrical constant related to the Cayley graphs of H and K. [2] Random walks and Poisson boundaries of groups. First, we study random walks on the lampshuffler group FSym(H) ⋊ H, where H is a finitely generated group and FSym(H) is the group of finitary permutations of H. We show that for any step distribution µ with a finite first moment that induces a transient random walk on H, the permutation coordinate of the random walk almost surely stabilizes pointwise to a limit function. Our main result states that for H = ℤ, the Poisson boundary of the random walk (FSym(ℤ)⋊ℤ, μ) is equal to the space of limit functions endowed with the hitting measure. Our result provides new examples of completely described non-trivial Poisson boundaries of elementary amenable groups. Next, in collaboration with Joshua Frisch, we completely describe the Poisson boundary of the wreath product A ≀ B of countable groups A and B, for all probability measures µ with finite entropy and such that the lamp configurations stabilize almost surely along sample paths. If in addition the projection of µ to B is Liouville, we prove that the Poisson boundary of (A ≀ B, µ) coincides with the space of limit lamp configurations, endowed with the corresponding hitting measure. This improves earlier results by Lyons-Peres and, in particular, we answer an open question asked by Kaimanovich and Lyons-Peres for B = ℤᵈ, d ≥ 3, and measures µ with a finite first moment.

We prove that the Fisher information is monotone decreasing in time along solutions of the space-homogeneous Boltzmann equation for a large class of collision kernels covering all classical interactions derived from systems of particles. For general collision kernels, a sufficient condition for the monotonicity of the Fisher information along the flow is related to the best constant for an integro-differential inequality for functions on the sphere, which belongs in the family of the Log-Sobolev inequalities. As a consequence, we establish the existence of global smooth solutions to the space-homogeneous Boltzmann equation in the main situation of interest where this was not known, namely the regime of very soft potentials. This is opening the path to the completion of both the classical program of qualitative study of space-homogeneous Boltzmann equation, initiated by Carleman, and the program of using the Fisher information in the study of the Boltzmann equation, initiated by McKean. From the proofs and discussion emerges a strengthened picture of the links between kinetic theory, information theory and log-Sobolev inequalities.

Most existing learning-based methods for solving imaging inverse problems can be roughly divided into two classes: iterative algorithms, such as plug-and-play and diffusion methods leveraging pretrained denoisers, and unrolled architectures that are trained end-to-end for specific imaging problems. Iterative methods in the first class are computationally costly and often yield suboptimal reconstruction performance, whereas unrolled architectures are generally problem-specific and require expensive training. In this work, we propose a novel non-iterative, lightweight architecture that incorporates knowledge about the forward operator (acquisition physics and noise parameters) without relying on unrolling. Our model is trained to solve a wide range of inverse problems, such as deblurring, magnetic resonance imaging, computed tomography, inpainting, and super-resolution, and handles arbitrary image sizes and channels, such as grayscale, complex, and color data. The proposed model can be easily adapted to unseen inverse problems or datasets with a few fine-tuning steps (up to a few images) in a self-supervised way, without ground-truth references. Throughout a series of experiments, we demonstrate state-of-the-art performance from medical imaging to low-photon imaging and microscopy. Our code is available at https://github.com/matthieutrs/ram.

DeepInverse is an open-source PyTorch-based library for imaging inverse problems. DeepInverse implements all steps for image reconstruction, including efficient forward operators, defining and solving variational problems and designing and training advanced neural networks, for a wide set of domains (medical imaging, astronomical imaging, remote sensing, computational photography, compressed sensing and more).