Publications

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).

We prove that fields of meromorphic functions on Stein surfaces have cohomological dimension 2, and solve the period-index problem and Serre's conjecture II for these fields. We obtain analogous results for fields of real meromorphic functions on Stein surfaces equipped with an antiholomorphic involution. We deduce an optimal quantitative solution to Hilbert's 17th problem on analytic surfaces.

We show that any sum of squares in a field of transcendence degree 1 over Q is a sum of squares, answering a question of Pop and Pfister. We deduce this result from representation theorem, in k(C), for quadratic forms of rank at least 5 with coefficients in k, where C is a curve over a number field k.

We prove that holomorphic maps from an open subset of a complex smooth projective curve to a complex smooth projective rationally simply connected variety can be approximated by algebraic maps for the compact-open topology. This theorem can be applied in particular when the target is a smooth hypersurface of degree d in P^n with n greater than or equal to d^2-1. We deduce it from a more general result: the tight approximation property holds for rationally simply connected varieties over function fields of complex curves.

We study the rationality of some geometrically rational three-dimensional conic and quadric surface bundles, defined over the reals and more general real closed fields, for which the real locus is connected and the intermediate Jacobian obstructions to rationality vanish. We obtain both negative and positive results, using unramified cohomology and birational rigidity techniques, as well as concrete rationality constructions.

The problem of the geodynamo is simple to formulate (Why does the Earth possess a magnetic field?), yet it proves surprisingly hard to address. As with most geophysical flows, the fluid flow of molten iron in the Earth's core is strongly influenced by the Coriolis effect. Because the liquid is electrically conducting, it is also strongly influenced by the Lorentz force. The balance is unusual in that, whereas each of these effects considered separately tends to impede the flow, the magnetic field in the Earth's core relaxes the effect of the rapid rotation and allows the development of a large-scale flow in the core that in turn regenerates the field. This review covers some recent developments regarding the interplay between rotation and magnetic fields and how it affects the flow in the Earth's core.

Transformers are deep architectures that define "in-context mappings" which enable predicting new tokens based on a given set of tokens (such as a prompt in NLP applications or a set of patches for a vision transformer). In this work, we study in particular the ability of these architectures to handle an arbitrarily large number of context tokens. To mathematically, uniformly address their expressivity, we consider the case that the mappings are conditioned on a context represented by a probability distribution of tokens which becomes discrete for a finite number of these. The relevant notion of smoothness then corresponds to continuity in terms of the Wasserstein distance between these contexts. We demonstrate that deep transformers are universal and can approximate continuous in-context mappings to arbitrary precision, uniformly over compact token domains. A key aspect of our results, compared to existing findings, is that for a fixed precision, a single transformer can operate on an arbitrary (even infinite) number of tokens. Additionally, it operates with a fixed embedding dimension of tokens (this dimension does not increase with precision) and a fixed number of heads (proportional to the dimension). The use of MLPs between multi-head attention layers is also explicitly controlled. We consider both unmasked attentions (as used for the vision transformer) and masked causal attentions (as used for NLP and time series applications). We tackle the causal setting leveraging a space-time lifting to analyze causal attention as a mapping over probability distributions of tokens.

Causal Transformers are trained to predict the next token for a given context. While it is widely accepted that self-attention is crucial for encoding the causal structure of sequences, the precise underlying mechanism behind this in-context autoregressive learning ability remains unclear. In this paper, we take a step towards understanding this phenomenon by studying the approximation ability of Transformers for next-token prediction. Specifically, we explore the capacity of causal Transformers to predict the next token xt+1 given an autoregressive sequence (x1,…,xt) as a prompt, where xt+1=f(xt), and f is a context-dependent function that varies with each sequence. On the theoretical side, we focus on specific instances, namely when f is linear or when (xt)t≥1 is periodic. We explicitly construct a Transformer (with linear, exponential, or softmax attention) that learns the mapping f in-context through a causal kernel descent method. The causal kernel descent method we propose provably estimates xt+1 based solely on past and current observations (x1,…,xt), with connections to the Kaczmarz algorithm in Hilbert spaces. We present experimental results that validate our theoretical findings and suggest their applicability to more general mappings f.

Super-resolution of pointwise sources is of utmost importance in various areas of imaging sciences. Specific instances of this problem arise in single molecule fluorescence, spike sorting in neuroscience, astrophysical imaging, radar imaging, and nuclear resonance imaging. In all these applications, the Lasso method (also known as Basis Pursuit or \( \ell^1 \)-regularization) is the de facto baseline method for recovering sparse vectors from low-resolution measurements. This approach requires discretization of the domain, which leads to quantization artifacts and consequently, an overestimation of the number of sources. While grid-less methods, such as Prony-type methods or non-convex optimization over the source position, can mitigate this, the Lasso remains a strong baseline due to its versatility and simplicity. In this work, we introduce a simple extension of the Lasso, termed ``super-resolved Lasso" (SR-Lasso). Inspired by the Continuous Basis Pursuit (C-BP) method, our approach introduces an extra parameter to account for the shift of the sources between grid locations. Our method is more comprehensive than C-BP, accommodating both arbitrary real-valued or complex-valued sources. Furthermore, it can be solved similarly to the Lasso as it boils down to solving a group-Lasso problem. A notable advantage of SR-Lasso is its theoretical properties, akin to grid-less methods. Given a separation condition on the sources and a restriction on the shift magnitude outside the grid, SR-Lasso precisely estimates the correct number of sources.

46 pages

We have reached a point where many bio foundation models exist across 4 different scales, from molecules to molecular chains, cells, and tissues. However, while related in many ways, these models do not yet bridge these scales. We present a framework and architecture called Xpressor that enables cross-scale learning by (1) using a novel cross-attention mechanism to compress high-dimensional gene representations into lower-dimensional cell-state vectors, and (2) implementing a multi-scale fine-tuning approach that allows cell models to leverage and adapt protein-level representations. Using a cell Foundation Model as an example, we demonstrate that our architecture improves model performance across multiple tasks, including cell-type prediction (+12%) and embedding quality (+8%). Together, these advances represent first steps toward models that can understad and bridge different scales of biological organization.

A cell is governed by the interaction of myriads of macromolecules. Such a network of interaction has remained an elusive milestone in cellular biology. Building on recent advances in large foundation models and their ability to learn without supervision, we present scPRINT, a large cell model for the inference of gene networks pre-trained on more than 50M cells from the cellxgene database. Using novel pretraining methods and model architecture, scPRINT pushes large transformer models towards more interpretability and usability in uncovering the complex biology of the cell. Based on our atlas-level benchmarks, scPRINT demonstrates superior performance in gene network inference to the state of the art, as well as competitive zero-shot abilities in denoising, batch effect correction, and cell label prediction. On an atlas of benign prostatic hyperplasia, scPRINT highlights the profound connections between ion exchange, senescence, and chronic inflammation.

Uniform attachment with freezing is an extension of the classical model of random recursive trees, in which trees are recursively built by attaching new vertices to old ones. In the model of uniform attachment with freezing, vertices are allowed to freeze, in the sense that new vertices cannot be attached to already frozen ones. We study the impact of removing attachment and/or freezing steps on the height of the trees. We show in particular that removing an attachment step can increase the expected height, and that freezing cannot substantially decrease the height of random recursive trees. Our methods are based on coupling arguments.

We are interested in the geometry of the ``infection tree'' in a stochastic SIR (Susceptible-Infectious-Recovered) model, starting with a single infectious individual. This tree is constructed by drawing an edge between two individuals when one infects the other. We focus on the regime where the infectious period before recovery follows an exponential distribution with rate 1, and infections occur at a rate λn∼λ/n where n is the initial number of healthy individuals with λ>1. We show that provided that the infection does not quickly die out, the height of the infection tree is asymptotically κ(λ)logn as n→∞, where κ(λ) is a continuous function in λ that undergoes a second-order phase transition at λc≃1.8038. Our main tools include a connection with the model of uniform attachment trees with freezing and the application of martingale techniques to control profiles of random trees.

We are interested in the geometry of the ``infection tree'' in a stochastic SIR (Susceptible-Infectious-Recovered) model, starting with a single infectious individual. This tree is constructed by drawing an edge between two individuals when one infects the other. We focus on the regime where the infectious period before recovery follows an exponential distribution with rate 1, and infections occur at a rate λn∼λ/n where n is the initial number of healthy individuals with λ>1. We show that provided that the infection does not quickly die out, the height of the infection tree is asymptotically κ(λ)logn as n→∞, where κ(λ) is a continuous function in λ that undergoes a second-order phase transition at λc≃1.8038. Our main tools include a connection with the model of uniform attachment trees with freezing and the application of martingale techniques to control profiles of random trees.

Girko matrices have independent and identically distributed entries of mean zero and unit variance. In this note, we consider the random matrix model formed by the ratio of two independent Girko matrices, its entries are dependent and heavy-tailed. Our main message is that divided by the square root of the dimension, the spectral radius of the ratio converges in distribution, when the dimension tends to infinity, to a universal heavy-tailed distribution. We provide a mathematical proof of this high-dimensional phenomenon, under a fourth moment matching with a Gaussian case known as the complex Ginibre ensemble. In this Gaussian case, the model is known as the spherical ensemble, and its spectrum is a determinantal planar Coulomb gas. Its image by the inverse stereographic projection is a rotationally invariant gas on the two-sphere. A crucial observation is the invariance in law of the model under inversion, related to its spherical symmetry, and that makes, in a sense, edge and bulk equivalent. Our approach involves Girko Hermitization, local law estimates for Wigner matrices, lower bound estimates on the smallest singular value, and convergence of kernels of determinantal point processes. The universality of the high-dimensional fluctuation of the spectral radius of the ratio of Girko matrices turns out to be remarkably more accessible mathematically than for a single Girko matrix!

We prove that, for any Morse function on a compact manifold and any adapted gradient satisfying the Morse-Smale condition, there is a homotopically unique complex-valued symplectic Lefschetz fibration on the cotangent bundle whose restriction to the zero-section is the given function, whose imaginary part is the evaluation of covectors on the gradient, and which is equivariant under the actions of the fiberwise antipodal involution and the complex conjugation. Then we study the topology and symplectic geometry of the regular fibers of this fibration, which are well-defined Weinstein manifolds.

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.

We consider a model for a gas of N confined particles subject to a two-body repulsive interaction, namely the one-dimensional log or Riesz gas. We are interested in the so-called high temperature regime, ie where the inverse temperature β_N scales as Nβ_N → 2P>0. We establish, in the log case, a large deviation (LD) principle and moderate deviations estimates for the largest particle x_max when appropriately rescaled . Our result is an extension of [BADG01, Pak20] where such estimates were shown for the largest particle of the β-ensemble respectively at fixed β_N=β>0 and β_N≫N -1 . We show that the corresponding rate function is the same as in the case of iid particles. We also provide LD estimates in the Riesz case. Additionally, we consider related models of symmetric tridiagonal random matrices with independent entries having Gaussian tails; for which we establish the LD principle for the top eigenvalue. In a certain specialization of the entries, we recover the result for the largest particle of the log-gas. We show that LD are created by a few entries taking abnormally large values.

We prove the existence and uniqueness of strong solutions to the equation $u u_x - u_{yy} = f$ in the vicinity of the linear shear flow, subject to perturbations of the source term and lateral boundary conditions. Since the solutions we consider have opposite signs in the lower and upper half of the domain, this is a quasilinear forward-backward parabolic problem, which changes type across a critical curved line within the domain. In particular, lateral boundary conditions can be imposed only where the characteristics are inwards. There are several difficulties associated with this problem. First, the forward-backward geometry depends on the solution itself. This requires to be quite careful with the approximation procedure used to construct solutions. Second, and more importantly, the linearized equations solved at each step of the iterative scheme admit a finite number of singular solutions, of which we provide an explicit construction. This is similar to well-known phenomena in elliptic problems in nonsmooth domains. Hence, the solutions to the equation are regular if and only if the source terms satisfy a finite number of orthogonality conditions. A key difficulty of this work is to cope with these orthogonality conditions during the nonlinear fixed-point scheme. In particular, we are led to prove their stability with respect to the underlying base flow. To tackle this deceivingly simple problem, we develop a methodology which we believe to be both quite natural and adaptable to other situations in which one wishes to prove the existence of regular solutions to a nonlinear problem for suitable data despite the existence of singular solutions at the linear level. This paper is a shorter version of [3].

We introduce twisted triple crossing diagram maps, collections of points in projective space associated to bipartite graphs on the cylinder, and use them to provide geometric realizations of the cluster integrable systems of Goncharov and Kenyon constructed from toric dimer models. Using this notion, we provide geometric proofs that the pentagram map and the cross-ratio dynamics integrable systems are cluster integrable systems. We show that in appropriate coordinates, cross-ratio dynamics is described by geometric R-matrices, which solves the open question of finding a cluster algebra structure describing cross-ratio dynamics.

Here are seemingly unrelated problems: computing rational homotopy groups of spheres in rational homotopy theory, purity in algebraic geometry, Koszul duality for the category of a reductive group in representation theory, splitting Drinfeld space's de Rham complex in the p-adic Langlands program, deformation quantization of Poisson manifolds in mathematical physics. And yet, all of them boil down to the same question: formality. A differential graded algebraic structure A (e.g. an associative algebra, a Lie algebra, a Pre-Calabi-Yau algebra, etc.) is formal if it is related to its homology H(A) by a zig-zag of quasi-isomorphisms preserving the algebraic structure. This thesis develops obstruction classes allowing to prove formality results. On the one hand, it incorporates aforementioned results into a single theory. On the other hand, it provides tools to study these questions in cases little studied hitherto: over any coefficient ring and for algebraic structures with several outputs: algebras encoded by properads.

We develop a general obstruction theory to the formality of algebraic structures over any commutative ground ring. It relies on the construction of Kaledin obstruction classes that faithfully detect the formality of differential graded algebras over operads or properads, possibly colored in groupoids. The present treatment generalizes the previous obstruction classes in two directions: beyond characteristic zero and including a wider range of algebraic structures. This enables us to establish novel formality criteria, including formality descent with torsion coefficients, formality in families, intrinsic formality, and criteria in terms of chain-level lifts of homology automorphism.

We develop a general obstruction theory to the formality of algebraic structures over any commutative ground ring. It relies on the construction of Kaledin obstruction classes that faithfully detect the formality of differential graded algebras over operads or properads, possibly colored in groupoids. The present treatment generalizes the previous obstruction classes in two directions: beyond characteristic zero and including a wider range of algebraic structures. This enables us to establish novel formality criteria, including formality descent with torsion coefficients, formality in families, intrinsic formality, and criteria in terms of chain-level lifts of homology automorphism.

We study the small-time local controllability (STLC) of a bilinear Schrödinger equation with Neumann boundary conditions near its ground state. We focus on the degenerate case where the linearized system is not controllable, necessitating a second-order analysis. We prove two complementary results. The negative result provides a new PDE instance of Sussmann's classical quadratic obstruction, corresponding to a non-vanishing Lie bracket. The positive result appears to be the first to establish STLC at the quadratic order for a physical PDE with a single scalar control. Both proofs rely on a Fourier-based approach, which is crucial because the integral kernel of the second-order term lacks the regularity required by standard integration-by-parts arguments. Along the way, we develop tools valid in a more general setting to analyze such quadratic forms. In particular, we prove results that allow for the multiplication of a kernel by a modulation function

We study a multi--particle model including a kinetic energy and a non linear local self-interaction, both in the bosonic and fermionic cases. In both cases, we prove that the model is well-posed if the number of particles is large enough. In particular, we show that there is a nonlinearity for which the model with $N=2$ particles is well-posed, while the model with $N=1$ is not.

En continuation des travaux de Hrushovski sur les corps pseudo-finis avec un caractère additif, nous étudions la théorie des modèles des corps aux différences (en caractéristique 0) avec un caractère additif sur le corps fixe ajouté comme prédicat au sens de la logique continue. Leur théorie possède un modèle-compagnon, la théorie ACFA+. Elle admet l’élimination des quantificateurs aux quantificateurs algébriquement bornés prés. Nous montrons que ACFA+ est la théorie asymptotique en caractéristique 0 de la clôture algébrique des corps finis munis du Frobenius et d’un caractère additif sur le corps fixe. ACFA+ est simple, mais en général on n’a pas la 3-amalgamation sur des ensembles aclσ-clos. En suivant une direction de recherche suggérée par Hrushovski, nous donnons une caractérisation complète de ce phénomène. Cela nous permet de déterminer la composante connexe du groupe de Kim-Pillay en tant que groupe topologique. En particulier, nous pouvons en déduire, comme attendu par Hrushovski, que le groupe est abélien. De plus, nous obtenons une caractérisation des imaginaires (en logique continue) dans ACFA+. Nous étudions ensuite l’amalgamation en dimension supérieure. Contrairement aux théories ACFA et PF+, nous pouvons construire un contre-exemple à la 4-amalgamation sur un ensemble pour lequel on a la 3-amalgamation. Néanmoins, nous montrons qu’on a la n-amalgamation sur tous les modèles pour tout n ∈ N. Dans le dernier chapitre, nous généralisons les résultats de Hrushovski dans une direction différente. Motivés par des exemples naturels provenant de la théorie des nombres, nous introduisons la théorie PF+,× des corps pseudo-finis avec un caractère additif et un caractère multiplicatif (d’ordre infini). Nous montrons l’élimination des quantificateurs dans une extension naturelle du langage. Ensuite nous obtenons que PF+,× est la théorie asymptotique en caractéristique 0 des corps finis avec un caractère additif non-trivial et un caractère multiplicatif suffisamment générique. Ultérieurement nous montrons que l’intégration des prédicats définissables par rapport à la mesure de comptage de Chatzidakis-Macintyre-van den Dries est uniformément définissable en fonction des paramètres.

This note reviews a recent contribution about the Fisher information for the space-homogeneous Boltzmann equation by L. Silvestre, C. Villani and the author (arXiv, 2024 ). This classical functional from information theory is shown to be nonincreasing along the flow of the non-linear PDE for all physically relevant particle interactions. The proof consists in establishing a new functional inequality on the sphere of Log-Sobolev type. This new a priori estimate on solutions yields global-in-time well posedness of the equation, in particular in the case of very singular interactions, a left open question up to this work.

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 study the geodesic convexity of various energy and entropy functionals restricted to (non-geodesically convex) submanifolds of Wasserstein spaces with their induced geometry. We prove a variety of convexity results by means of a simple general principle, which holds in the metric space setting, and which crucially requires no knowledge of the structure of geodesics in the submanifold: If the EVI gradient flow of a functional exists and leaves the submanifold invariant, then the restriction of the functional to the submanifold is geodesically convex. This leads to short new proofs of several known results, such as one of Carlen and Gangbo on strong convexity of entropy on sphere-like submanifolds, and several new results, such as the λ-convexity of entropy on the space of couplings of λ-log-concave marginals. Along the way, we develop sufficient conditions for existence of geodesics in Wasserstein submanifolds. Submanifold convexity results lead systematically to improvements of Talagrand and HWI inequalities which we speculate to be closely related to concentration of measure estimates for conditioned empirical measures, and we prove one rigorous result in this direction in the Carlen-Gangbo setting.

This article studies large and local large deviations for sums of i.i.d. real-valued random variables in the domain of attraction of an $\alpha$-stable law, $\alpha\in (0,2]$, with emphasis on the case $\alpha=2$. There are two different scenarios: either the deviation is realised via a collective behaviour with all summands contributing to the deviation (a Gaussian scenario), or a single summand is atypically large and contributes to the deviation (a one-big-jump scenario). Such results are known when $\alpha \in (0,2)$ (large deviations always follow a one big-jump scenario) or when the random variables admit a moment of order $2+\delta$ for some $\delta>0$. We extend these results, including in particular the case where the right tail is regularly varying with index $-2$ (treating cases with infinite variance in the domain of attraction of the normal law). We identify the threshold for the transition between the Gaussian and the one-big-jump regimes; it is slightly larger when considering local large deviations compared to integral large deviations. Additionally, we complement our results by describing the behaviour of the sum and of the largest summand conditionally on a (local) large deviation, for any $\alpha\in (0,2]$, both in the Gaussian and in the one-big-jump regimes. As an application, we show how our results can be used in the study of condensation phenomenon in the zero-range process at the critical density, extending the range of parameters previously considered in the literature.

We introduce a new method of symmetrization of mappings on the $n$-sphere ($n\geq 2$). They are applied to estimate solutions of quasilinear elliptic partial differential equations of $p$-Laplacian type, with combinations of Dirac measures on the right-hand side. The case $p=n$ is reduced to a problem on the sphere, using a conformal transformation. The cases when $1n$ are considered more briefly, full details being available in other papers of the author.

We establish minimax convergence rates for score-based generative models (SGMs) under the $1$-Wasserstein distance. Assuming the target density $p^\star$ lies in a nonparametric $\beta$-smooth H\"older class with either compact support or subGaussian tails on $\mathbb{R}^d$, we prove that neural network-based score estimators trained via denoising score matching yield generative models achieving rate $n^{-(\beta+1)/(2\beta+d)}$ up to polylogarithmic factors. Our unified analysis handles arbitrary smoothness $\beta > 0$, supports both deterministic and stochastic samplers, and leverages shape constraints on $p^\star$ to induce regularity of the score. The resulting proofs are more concise, and grounded in generic stability of diffusions and standard approximation theory.

Le perimètre d'une partie mesurable de $\mathbb R^N$ est la variation totale de sa fonction caractéristique. On en donne une généralisation au cas d'une partie $E$ d'une variété riemannienne compacte orientée. On montre que ce périmètre est la limite des variations totales des régularisées de $\chi_E$ par le noyau de la chaleur. On en déduit une inégalité isopérimétrique, et une formule de type Fleming-Rishel. On applique ensuite ces résultats à un problème quasilinéaire elliptique dans $\mathbb R^N$, pour lequel les méthodes usuelles de symétrisation dans $\mathbb R^N$ achoppent, mais que l'on pourra traiter en introduisant une méthode de symétrisation sur $S^N$.

A classical approach to the Calderón problem is to estimate the unknown conductivity by solving a nonlinear least-squares problem. It is generally believed that it leads to a nonconvex optimization problem which is riddled with bad local minimums. This has motivated the development of reconstruction methods based on convex optimization, one recent contribution being the nonlinear convex semidefinite programming approach of Harrach (2023). In this work, we investigate the computational viability of this convex approach in a simple setting where the conductivities are piecewise constant and radial. We implement this convex reconstruction method and compare it extensively to the least squares approach. Our experiments suggest that this convex programming approach only allows to accurately estimate the unknown for problems with a very small size. Moreover, surprisingly, it is consistently outperformed by Newton-type least squares solvers, which are also faster and require less measurements. We revisit the issue of nonconvexity in this piecewise constant radial setting and prove that, contrary to previous claims, there are no local minimums 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.

We give an expression for the minimum energy of a map between a three-dimensional manifold and a surface, with prescribed vortex-like singularities. We find that the topological type of the target needs to be restricted for the problem to be meaningful. We extend results of Brezis-Coron-Lieb for maps from 3-space to the standard 2-sphere. This is made possible by the use of Fermi-Walker transport.

The Calderón problem consists in recovering an unknown coefficient of a partial differential equation from boundary measurements of its solution. These measurements give rise to a highly nonlinear forward operator. As a consequence, the development of reconstruction methods for this inverse problem is challenging, as they usually suffer from the problem of local convergence. To circumvent this issue, we propose an alternative approach based on lifting and convex relaxation techniques, that have been successfully developed for solving finite-dimensional quadratic inverse problems. This leads to a convex optimization problem whose solution coincides with the sought-after coefficient, provided that a nondegenerate source condition holds. We demonstrate the validity of our approach on a toy model where the solution of the partial differential equation is known everywhere in the domain. In this simplified setting, we verify that the non-degenerate source condition holds under certain assumptions on the unknown coefficient. We leave the investigation of its validity in the Calderón setting for future works.

In the interest of reproducible research, this is exactly the version of the code used to generate the figures in the paper "Opening the Black Box: Reverse-Engineering of Sparse Neural Networks" by the same authors.

Dans le contexte de la compression de réseaux de neurones, la distillation consiste à entraîner un petit réseau élève (si possible parcimonieux) à partir d'échantillons générés par un réseau enseignant. Malgré le succès empirique de ces approches on connaît mal leurs conditions de succès. En guise de preuve de concept nous introduisons une nouvelle famille de réseaux parcimonieux structurés dits perceptrons multiarbre, et un algorithme d'estimation des paramètres de tels réseaux à partir d'échantillons de la fonction correspondante et de sa Jacobienne. L'algorithme en question n'est pas basé sur la descente de gradient.

We consider a drift-diffusion process of N stochastic particles and show that its empirical measure converges, as N → ∞, to the solution of the Landau equation. We work in the regime of very soft potentials, which had never been covered before, using a tightness/uniqueness method. To claim uniqueness, we need high integrability estimates that we obtain by crucially exploiting the dissipation of the Fisher information at the level of the particle system. To be able to exploit these estimates as N → ∞, we prove the affinity in infinite dimension of the entropy production and Fisher information dissipation (and general higher-order versions of the Fisher information through an abstract theorem), results which were up to now only known for the entropy and the Fisher information.

We study the discretization, convergence, and numerical implementation of recent reformulations of the quadratic porous medium equation (multidimensional and anisotropic) and Burgers' equation (one-dimensional, with optional viscosity), as forward in time variants of the Benamou-Brenier formulation of optimal transport. This approach turns those evolution problems into global optimization problems in time and space, of which we introduce a discretization, one of whose originalities lies in the harmonic interpolation of the densities involved. We prove that the resulting schemes are unconditionally stable w.r.t. the space and time steps, and we establish a quadratic convergence rate for the dual PDE solution, under suitable assumptions. We also show that the schemes can be efficiently solved numerically using a proximal splitting method and a global space-time fast Fourier transform, and we illustrate our results with numerical experiments.

The purpose of this paper is to investigate the well-posedness of several linear and nonlinear equations with a parabolic forward-backward structure, and to highlight the similarities and differences between them. The epitomal linear example will be the stationary Kolmogorov equation $y\partial_x u -\partial_{yy} u=f$ in a rectangle. We first prove that this equation admits a finite number of singular solutions, of which we provide an explicit construction. Hence, the solutions to the Kolmogorov equation associated with a smooth source term are regular if and only if $f$ satisfies a finite number of orthogonality conditions. This is similar to well-known phenomena in elliptic problems in polygonal domains. We then extend this theory to a Vlasov--Poisson--Fokker--Planck system, and to two quasilinear equations: the Burgers type equation $u \partial_x u - \partial_{yy} u = f$ in the vicinity of the linear shear flow, and the Prandtl system in the vicinity of a recirculating solution, close to the line where the horizontal velocity changes sign. We therefore revisit part of a recent work by Iyer and Masmoudi. For the two latter quasilinear equations, we introduce a geometric change of variables which simplifies the analysis. In these new variables, the linear differential operator is very close to the Kolmogorov operator $y\partial_x -\partial_{yy}$. Stepping on the linear theory, we prove existence and uniqueness of regular solutions for data within a manifold of finite codimension, corresponding to some nonlinear orthogonality conditions.

While conservation laws in gradient flow training dynamics are well understood for (mostly shallow) ReLU and linear networks, their study remains largely unexplored for more practical architectures. This paper bridges this gap by deriving and analyzing conservation laws for modern architectures, with a focus on convolutional ResNets and Transformer networks. For this, we first show that basic building blocks such as ReLU (or linear) shallow networks, with or without convolution, have easily expressed conservation laws, and no more than the known ones. In the case of a single attention layer, we also completely describe all conservation laws, and we show that residual blocks have the same conservation laws as the same block without skip connection. We then introduce the notion of conservation laws that depend only on a subset of parameters (corresponding e.g. to a pair of consecutive layers, to a residual block, or to an attention layer). We demonstrate that the characterization of such laws can be reduced to the analysis of the corresponding building block in isolation. Finally, we examine how these newly discovered conservation principles, initially established in the continuous gradient flow regime, persist under discrete optimization dynamics, particularly in the context of Stochastic Gradient Descent (SGD).