Publications

Across mathematics, formality results are usually established through general criteria rather than by constructing explicit equivalences. These criteria include domination, puncturing, highly-connected formality, intrinsic formality, formality descent, formality in families, purity, etc. In this article, we prove that all these criteria are instances of a single theorem, formality transmission. This unifying framework allows us to extend these criteria in several directions, beyond characteristic zero and to differential graded algebras over operads or properads, leading to diverse new formality results.

We show, under a smoothness assumption, that probability measures whose exponential tilts have a uniformly bounded variance are given by a strongly convex potential, up to a bounded perturbation, yielding a partial converse to the Bakry-Émery theorem combined with Holley-Stroock's perturbation principle. A stronger version enforcing uniform convexity of the potential is obtained by considering the low-temperature regime.

We show, under a smoothness assumption, that probability measures whose exponential tilts have a uniformly bounded variance are given by a strongly convex potential, up to a bounded perturbation, yielding a partial converse to the Bakry-Émery theorem combined with Holley-Stroock's perturbation principle. A stronger version enforcing uniform convexity of the potential is obtained by considering the low-temperature regime.

Since the early days of quantum information science, diverse strategies have been proposed to encode qubits and implement quantum gates across various physical platforms. Modes and bosonic states offer a flexible framework for quantum information processing, from single-photon optics to superconducting circuits, combining massless and massive particles. Universal gate sets have been established for specific encodings—such as those based on Fock states or the field’s quadratures. However, a general framework clarifying how different bosonic resources—such as mode structure, particle-number statistics, Gaussian and non-Gaussian states and operations—combine to enable universality and potential quantum advantage irrespectively of the encoding remains lacking. A main difficulty lies in the coexistence of distinct paradigms that appear to combine loosely in bosonic quantum information: discrete-variable schemes—where information is encoded in qubits or effective finite-dimensional systems—and continuous-variable schemes—which rely on infinite-dimensional Hilbert spaces associated with the field’s quadratures. Despite the success of each approach, they remain seemingly incompatible in many aspects—in particular in terms of physical resource analysis for universality—due to their underlying mathematical differences. In this work, we apply a superselection-rule-compliant framework for bosonic quantum information that unifies these paradigms. This representation explicitly incorporates the phase reference and enforces total particle number conservation, allowing for a consistent treatment of bosonic quantum information protocols and resources across both discrete and continuous regimes. We derive necessary and sufficient conditions expressed in terms of Gaussian and non-Gaussian resources for universality that apply to any encoding, without resorting to specific mappings. Our results show that continuous variable encodings are not fundamentally distinct from discrete ones, but rather emerge as subspaces within a general bosonic structure. Importantly, in such a widely studied framework, non-Gaussian properties of bosonic states and operations can be washed out, providing evidence that the continuous variables regime is not the most suitable to fully assess the quantum properties of bosons.

We prove a sharp quantitative version of the Talenti rearrangement inequality.

We survey recent results on soliton resolution for the energy critical nonlinear wave equation.

We consider the F -equivalence problem for parabolic systems: under which conditions a control system, governed by a parabolic operator A and a control operator B, can be made equivalent to an arbitrarily exponentially stable evolution system through an appropriate control feedback law? While this problem has been resolved for finite-dimensional systems fifty years ago, good conditions for infinite-dimensional systems remain a challenge, especially for systems in spatial dimension larger than one. Our main result establishes optimal conditions for the existence of an F -equivalence pair (T, K) for a given parabolic control system (A, B). We introduce an extended framework for Fequivalence of parabolic operators, addressing key limitations of existing approaches, and we prove that the pair (T, K) is unique if and only if (A, B) is approximately controllable. As a consequence, this provides a method to construct feedback operators for the rapid stabilization of nonlinear parabolic systems, possibly multidimensional in space. We provide several illustrative examples, including the rapid stabilization of the heat equation, the Kuramoto-Sivashinsky equation, the Navier-Stokes equations and the quasilinear heat equation.

Let $X$ be a smooth algebraic variety over $\mathbb{R}$. We study the separation of the connected components of $X(\mathbb{R})$ by equivariant cohomology, unramified cohomology, and signatures of unramified quadratic forms. Our results highlight how the geometry of $X$ influences these questions. We apply them to cubic hypersurfaces, complete intersections of two quadrics, abelian varieties, and Enriques surfaces. We also obtain new bounds on the exponent of the cokernel of higher signature morphisms in $\mathbf{I}$-cohomology and investigate the optimality of these bounds for smooth projective varieties.

This PhD thesis presents contributions to the field of deep learning. From convolutional ResNets to Transformers, residual connections are ubiquitous in state-of-the-art deep learning models. The continuous depth analogues of residual networks, neural ODEs, have been widely adopted, but the connection between the discrete and continuous models still lacks a solid mathematical foundation. In this manuscript, we will show that for a formal correspondence between residual networks and neural ODEs to hold, the residual functions must be smooth with depth, and we will present an implicit regularization result of deep residual networks towards neural ODEs. We will then present two applications of this analogy to the design and study of new architectures. First, we will introduce a drop-in replacement for any residual network that can be trained with the same accuracy, but with much less memory. Second, by viewing the attention mechanism as an interacting particle system, where the particles are the tokens, we will study the impact of attention map normalization on the Transformer model. Finally, we will present some other contributions to Transformers: how Transformers perform in-context autoregressive learning and how to differentiably route tokens to experts in Sparse Mixture of Experts Transformers.

Fully capturing cellular identity requires integrating multiple molecular layers. Bridge integration, i.e. aligning unimodal datasets using a paired multi-omic reference, has emerged as a practical solution, yet existing methods offer limited interpretability and use paired information without regularization, making them sensitive to limited size and coverage. We introduce CHAMPOLLION, which uses regularized optimal transport to learn an interpretable cross-modal metric that drives the alignment of unpaired cells while capturing relationships between molecular features. Benchmarks on RNA-protein and RNA-ATAC datasets show that CHAMPOLLION outperforms existing approaches, remaining accurate with few paired cells and even generalizing to unseen cell types. Beyond alignment, CHAMPOLLION reveals biologically meaningful cross-modal relationships, highlighting in scRNA-protein data a potential role for CD18 across multiple cancers, and, in a human tonsil atlas combining scRNA-seq and scATAC-seq, suggesting that MEF2C may regulate inflammatory responses beyond the brain, notably in plasmacytoid dendritic cells.

Single-cell RNA sequencing provides high-resolution snapshots of cellular states but lacks direct information about transcriptional dynamics. Metabolic RNA labeling addresses this limitation by distinguishing newly synthesized RNA, offering insight into the direction of cell state changes, and providing valuable information when attempting to recover the underlying continuous dynamics from static snapshots of cell distributions. However, existing trajectory inference methods do not fully exploit this additional signal. Here, we propose FLOWSATATE, a framework for single-cell trajectory inference that leverages time-resolved RNA labeling within an Optimal Transport setting. We model cell dynamics as a gradient flow in an inferred potential landscape parameterized by a neural network, integrating both total and labeled RNA across time points. The learned potential enables identification of key genes and transcription factors driving cell fate decisions and supports prediction of future cellular states. We benchmark our approach on its ability to generalize unseen data and recover coherent trajectories. We also apply it to study colorectal cancer response to demethylation treatment as well as neuronal differentiation of embryonic stem cells.

Cell biology has been booming with foundation models trained on large single-cell RNA-seq databases, but benchmarks and capabilities remain unclear. We propose an additive benchmark across a gymnasium of tasks to discover which features improve performance. From these findings, we present scPRINT-2, a single-cell Foundation Model pre-trained across 350 million cells and 16 organisms. Our contributions in pre-training tasks, tokenization, and losses made scPRINT-2 state-of-the-art in expression denoising, cell embedding, and cell type prediction. Furthermore, with our cell-level architecture, scPRINT-2 becomes generative, as demonstrated by our expression imputation and counterfactual reasoning results. Finally, thanks to our pre-training database, we uncover generalization to unseen modalities and organisms. These studies, together with improved abilities in gene embeddings and gene network inference, place scPRINT-2 as a next-generation cell foundation model.

In dynamic biological processes such as development, spatial transcriptomics is revolutionizing the study of the mechanisms underlying spatial organization within tissues. Inferring cell fate trajectories from spatial transcriptomics profiled at several time points has thus emerged as a critical goal, requiring novel computational methods. Wasserstein gradient flow learning is a promising framework for analyzing sequencing data across time, built around a neural network representing the differentiation potential. However, existing gradient flow learning methods face challenges in analyzing spatially resolved transcriptomic data. Here, we propose STORIES, a method that uses an extension of Optimal Transport to learn a spatially informed potential. We benchmark our approach using three large Stereo-seq spatiotemporal atlases and demonstrate superior spatial coherence compared to existing approaches. Finally, we provide an in-depth analysis of axolotl neural regeneration and mouse gliogenesis, recovering gene trends for known markers such as Nptx1 in neuron regeneration and Aldh1l1 in gliogenesis and additional putative drivers.

27 pages, 3 figures, major revision: a simplification in an intermediate formula shortens the proofs of the main theorem

We introduce a class of Hall sets, which we call factor-parity Hall sets, whose elements split into good and bad brackets. We prove that the good brackets can be steered simultaneously and arbitrarily in small time, which yields sufficient conditions for the small-time local controllability of control-affine systems. This positive result draws on constructions of Kawski, Agrachev-Gamkrelidze and Krastanov. Conversely, we prove that each bad bracket generates an obstruction to controllability, hence a family of necessary conditions.

We investigate the role of quartic terms in the small-time local controllability of scalar-input systems. First, we prove a new sufficient condition for controllability, which exploits simultaneously more good quartic Lie brackets than previous results. Second, we identify a family of quartic obstructions to controllability, relying on Lie brackets whose coordinates of the second kind are positive-definite functionals of the control. Third, we show that the complementarity of these results can be seen as an answer to Kawski's 1987 open problem concerning the construction of a Hall basis which somehow separates good and bad quartic brackets. We give examples and remarks illustrating some of the intricacies of these notions.

We derive a weak-strong uniqueness and stability principle for the Landau equation in the soft potentials case (including Coulomb interactions). The distance between two solutions is measured by their relative entropy, which to our knowledge was never used before in stability estimates. The logarithm of the strong solution is required to have polynomial growth while the weak solution can be any H-solution with sufficiently many moments at initial time. Since we require a substantial amount of regularity on the strong solution, we also provide an example of sufficient conditions on the initial data that ensure this regularity in the Coulomb (and very soft potentials) case.

Several problems in machine learning are naturally expressed as the design and analysis of time-evolving probability distributions. This includes sampling via diffusion methods, optimizing the weights of neural networks, and analyzing the evolution of token distributions across layers of large language models. While the targeted applications differ (samples, weights, tokens), their mathematical descriptions share a common structure. A key idea is to switch from the Eulerian representation of densities to their Lagrangian counterpart through vector fields that advect particles. This dual view introduces challenges, notably the non-uniqueness of Lagrangian vector fields, but also opportunities to craft density evolutions and flows with favorable properties in terms of regularity, stability, and computational tractability. This survey presents an overview of these methods, with emphasis on two complementary approaches: diffusion methods, which rely on stochastic interpolation processes and underpin modern generative AI, and optimal transport, which defines interpolation by minimizing displacement cost. We illustrate how both approaches appear in applications ranging from sampling, neural network optimization, to modeling the dynamics of transformers for large language models.

Low-Rank Adaptation (LoRA) is the most widely adopted method for fine-tuning large language models. Notably, LoRA is inherently overparameterized: multiple pairs of low-rank factors can yield the same adapted weight matrix. We show--both theoretically and empirically--that these pairs exhibit significantly different condition numbers. As a result, converging to different loss minimizers directly impacts the convergence rate of LoRA. Building on this observation, we introduce Balanced Low-Rank Adaptation (BaLoRA), a variant of LoRA that projects iterates onto a balanced manifold. This manifold improves the conditioning of the loss landscape while preserving the adapted matrix. The projection step is computationally lightweight and integrates seamlessly into existing fine-tuning pipelines. Empirically, BaLoRA converges faster than standard LoRA and achieves superior performance across a range of fine-tuning tasks.

We study the convergence of gradient flow for the training of deep neural networks. While residual neural networks (ResNet) are a popular example of very deep architectures, their training constitutes a challenging optimization problem, notably due to the non‐convexity and the non‐coercivity of the objective. Yet, in applications, such tasks are successfully solved by simple optimization algorithms such as gradient descent. To better understand this phenomenon, we focus here on a “mean‐field” model of an infinitely deep and arbitrarily wide ResNet, parameterized by probability measures on the product set of layers and parameters, and with constant marginal on the set of layers. Indeed, in the case of shallow neural networks, mean field models have been proven to benefit from simplified loss landscapes and good theoretical guarantees when trained with gradient flow w.r.t. the Wasserstein metric on the set of probability measures. Motivated by this approach, we propose to train our model with gradient flow w.r.t. the conditional optimal transport (COT) distance: a restriction of the classical Wasserstein distance which enforces our marginal condition. Relying on the theory of gradient flows in metric spaces, we first show the well‐posedness of the gradient flow equation and its consistency with the training of ResNets at finite width. Performing a local Polyak–Łojasiewicz analysis, we then show convergence of the gradient flow for well‐chosen initializations: if the number of features is finite but sufficiently large and the risk is sufficiently small at initialization, the gradient flow converges to a global minimizer. This is the first result of this type for infinitely deep and arbitrarily wide ResNets. In addition, this work is an opportunity to study in more detail the COT metric, particularly its dynamic formulation. Some of our results in this direction might be interesting on their own.

Transformers, which are state-of-the-art in most machine learning tasks, represent the data as sequences of vectors called tokens. This representation is then exploited by the attention function, which learns dependencies between tokens and is key to the success of Transformers. However, the iterative application of attention across layers induces complex dynamics that remain to be fully understood. To analyze these dynamics, we identify each input sequence with a probability measure and model its evolution as a Vlasov equation called Transformer PDE, whose velocity field is non-linear in the probability measure. Our first set of contributions focuses on compactly supported initial data. We show the Transformer PDE is well-posed and is the mean-field limit of an interacting particle system, thus generalizing and extending previous analysis to several variants of self-attention: multi-head attention, L2 attention, Sinkhorn attention, Sigmoid attention, and masked attention--leveraging a conditional Wasserstein framework. In a second set of contributions, we are the first to study non-compactly supported initial conditions, by focusing on Gaussian initial data. Again for different types of attention, we show that the Transformer PDE preserves the space of Gaussian measures, which allows us to analyze the Gaussian case theoretically and numerically to identify typical behaviors. This Gaussian analysis captures the evolution of data anisotropy through a deep Transformer. In particular, we highlight a clustering phenomenon that parallels previous results in the non-normalized discrete case.

Drifting models offer a promising route to faster generative AI: they perform gradual transport during training, while generating new samples in a single step. This paper asks whether the underlying drifting process can converge rapidly to a target distribution under ideal conditions, before finite-data or optimization effects are introduced. We show that its convergence rate depends critically on how it handles spatial scale. With a single fixed resolution, fine-scale features of the target can become nearly invisible, leading to extremely slow convergence. We introduce a multihead approach that combines scale-normalized information across a continuum of resolutions. We prove that this multihead approach restores exponential convergence near standard reference distributions. These results identify fixed resolution as a key bottleneck and provide a simple route to faster one-step generative models.

We establish a general Poisson approximation for rare local patterns in critical Bienaymé--Galton--Watson trees with offspring distribution $μ$ in the domain of attraction of a stable law, conditioned to have a large number of vertices. A pattern is specified by a sequence-dependent mark on fringe subtrees. If marked fringe subtrees remain microscopic and nearby marked occurrences have negligible clustering, then their count is asymptotically Poisson in total variation whenever its mean remains bounded; when the mean diverges, the count satisfies a law of large numbers. The main difficulty is the global dependence created by size conditioning. We overcome it by combining the cyclic-shift representation with a refined form of the Chen--Stein bound and a bridge-removal estimate controlling the interaction between a local mark and the remainder of the conditioned random walk. For non-fringe patterns, overlapping occurrences may form clusters and the raw count need not be asymptotically Poisson. We introduce declumped indicators which select boundary witnesses of these clusters and prove a general Poisson approximation for their count. As applications, we obtain sharp asymptotics for the maximum leaf-height, equivalently the maximum protection number, and for the height of the largest complete $r$-ary tree appearing as a non-fringe subtree. Unary-chain maxima, and the maximum leaf-height when $μ_1>0$, exhibit lattice-modulated Gumbel behavior. Complete $r$-ary patterns for $r\ge2$, and the maximum leaf-height when $μ_1=0$, are localized on one or two consecutive integers. The results require no exponential moment and include offspring distributions with infinite variance.

Flow-matching schedules affect both sampling dynamics and the variance of the regression target. For centered commuting Gaussians, we show that a direction-dependent schedule decomposes into two independent design choices: a variance path, which fully determines the intermediate laws and probability flow, and a factorization, which leaves this flow unchanged while controlling irreducible regression variance. On the sampling side, we analyze finite-step Euler accuracy and derive a necessary drift bound for exact N -step sampling, connecting the geodesic and the logarithmic path. On the training side, for any fixed path, we derive closed-form factorizations that either minimize time-averaged regression variance or make it constant along the path.

Abstract Conifers, which comprise nearly two-thirds of extant gymnosperm species, are ecologically and economically important but remain genomically understudied because of their exceptionally large, repeat-rich genomes. Here, we report a chromosome-level assembly of the haploid genome of Cupressus sempervirens generated using PacBio HiFi reads and scaffolded with optical and genetic maps. The 10 Gb assembly shows exceptional contiguity for a conifer genome (contig N50 = 29.8 Mb) and was organized into 11 pseudomolecules. Iso-Seq-supported annotation identified 42,980 protein-coding genes. Repetitive elements account for over 80% of the genome, with LTR retrotransposons alone representing 52.5%. Transposable elements (TE) are pervasive in both intergenic and genic regions and have a major impact on gene architecture: TE insertions within introns generate ultra-long introns, often exceeding 100 kb, and drive gene size expansion. Analyses of LTR retrotransposon dynamics indicate that genome enlargement in C. sempervirens was driven not by recent transpositional bursts, but by the long-term accumulation and incomplete removal of ancient LTR retrotransposons. Consistent with this pattern, paleogenomic reconstruction across representative gymnosperms found no evidence of whole-genome duplication in the Cupressus lineage. This reference genome provides a valuable resource for studying conifer genome evolution, gene structure, and traits of agronomic and ecological interest, including cypress pollinosis.

We study Flow Matching in a semi-discrete setting where a Gaussian source is transported toward a discrete target supported on finitely many points. This semi-discrete regime is the theoretical setting behind the use of Flow Matching for generative modeling, where the target distribution is represented by a finite dataset. In this semi-discrete regime, the exact Flow Matching velocity field is available in closed form, which makes it possible to analyze the geometry induced by the terminal flow map independently of optimization and approximation effects. We investigate the terminal assignment regions, namely the preimages of the target atoms under the terminal flow. We show that these regions are open, simply connected and, under an additional assumption, homeomorphic to the unit ball. At the same time, a planar four-point example shows that these cells can differ sharply from Laguerre cells arising in semi-discrete optimal transport: they may be non-convex, have curved boundaries, and exhibit different boundedness and adjacency patterns. These results clarify the geometry intrinsically induced by the exact semi-discrete Flow Matching objective before neural approximation enters the picture.

Practical diffusion sampling is a numerical approximation problem: under a fixed inference budget, one must simulate a reverse-time ODE or SDE using only a limited number of denoising steps, so discretization error is often the dominant source of error. Existing non-asymptotic analyses provide convergence guarantees, but are typically too loose and too insensitive to diffusion parameters to guide practical design: broad families of schedules receive the same rates, which depend on coarse worst-case quantities such as the dimension or the drift Lipschitz constant. We take a less ambitious but more informative route. In the exact-score setting, we derive first-order asymptotic expansions of the Euler-Maruyama weak and Fréchet discretization errors. These formulas hold for general smooth reverse diffusions and become fully explicit under Gaussian data. They show how discretization error adapts to the geometry of the data through the covariance spectrum, and how this geometry interacts with key diffusion parameters, including the diffusion schedules and the diffusion-term coefficient. This yields tractable objectives for geometry-aware parameter optimization. Finally, we show that the qualitative predictions of the Gaussian formulas remain robust across diffusion sampling problems with different geometries, including image generation on different datasets and image posterior sampling.

These notes survey the question of embedding Wasserstein spaces into normed spaces, with applications to data science as a primary motivation. While quantile functions provide a remarkably simple isometric embedding of ($P_p(R)$, $W_p$) into $L^p([0, 1])$, strong obstructions prevent a comparable picture in higher dimension. We discuss and compare two alternative constructions, both rooted in the notion of quantiles. The sliced-Wasserstein embeddings is built from the quantile functions of the one-dimensional projections of a measure, whereas the linearized optimal transport embedding represents a measure by the Brenier map from a fixed reference measure, which may be regarded as a higher-dimensional analogue of the quantile function. Along the way, we review global non-embeddability results, recent progress on reverse comparisons between Wasserstein and sliced-Wasserstein distances, and quantitative stability estimates for Brenier maps.

Rectified flows, also called flow matching or stochastic interpolants, are generative models that learn a time-dependent vector field steering a probability curve between two probability distributions, usually referred to as latent and target distributions. Reflow accelerates inference by iteratively straightening the trajectories induced by this vector field. We study the asymptotic behavior of this iteration and characterize its limit points. First, we define weak rectified couplings which always exist. Next, when rectified flow updates are alternated with minibatch optimal transport steps of fixed batch size, we show that any limit is N -cyclically monotone, where N is the batch size. Such N -cyclically monotone couplings enjoy favorable structural and stability properties such as rectifiability and straightness. Finally, restricting velocities to gradient fields and assuming additional support conditions, we prove that reflow limits coincide with the optimal transport map between the endpoint distributions.

This collective introduction presents the historical, institutional, and mathematical context of the forty-year correspondence between Gösta Mittag-Leffler and Vito Volterra. It traces their relationship, the development of Acta Mathematica, Volterra’s international career, the emergence of functional analysis, and the effects of the First World War.

12 pages, 2 graphics

We study the $\mathbf{I}^*$-cohomology of a smooth real algebraic curve in terms of its real locus and its geometric genus. We notably extend results of Monnier to the twisted case, which is crucial to the understanding of proper pushforwards of Witt groups. We also perform some computations related to transfers along the finite étale extension $\mathbb{C}/\mathbb{R}$. We further describe how to compute twisted Witt groups of surfaces, extending work of Sujatha, and the image of the global signature homomorphism following Monnier. As an application of the main methods of the paper, we describe the shifted and twisted Witt groups of smooth anisotropic quadrics over $\mathbb{R}$ of dimension $\leq 3$.

We study Suslin's cancellation conjecture on smooth real affine varieties whose real locus is empty or, more generally, of small cohomological dimension.

In this paper, we show how the recent analysis of (sharp) strong convexity properties of the Kantorovich functional by the third author can be used to establish new global quantitative stability estimates for Wasserstein barycenters and their entropic regularizations with respect to perturbations of the population distribution. For classical Wasserstein barycenters, we improve the state-of-the-art Hölder continuity exponent with respect to the W1 metric from 1/6 to 1/4, while drastically weakening the assumptions on the population distribution. Motivated by computational considerations, we also analyze regularizations of Wasserstein barycenters with outer and inner entropic penalizations. For outer entropic barycenters, we establish novel higher-order stability results and in particular give a quantitative bound on relative Fisher information. Another contribution of our paper is a strong convexity estimate for an entropic version of the Kantorovich functional via soft Legendre transforms, enabling us to extend our stability results to doubly entropic barycenters, as introduced by Chizat.

We prove that the singular cohomology with finite coefficients of a finite-dimensional Stein space S is isomorphic to the étale cohomology of the Stein algebra O(S). We deduce that any class in H^k(S,Z) comes from an algebraic variety by pullback by a holomorphic map (if k≥1), and vanishes on the complement of a nowhere dense closed analytic subset of S (if k≥2).

We investigate the Brauer group of the ring O(S) of holomorphic functions on a finite-dimensional Stein space S. We provide a purely topological computation of this group and deduce a comparison theorem between the étale cohomology of Spec(O(S)) and the singular cohomology of S in degree 2. Furthermore, we prove a purity theorem when S is nonsingular and study the index of classes in the Brauer group of O(S).

Fix two integers $1\leq d\leq e$. We study the birational geometry of a parameter space for pairs of homogeneous polynomials of degrees $d$ and $e$ in two variables (in which the higher degree polynomial is well defined only up to a multiple of the lower degree polynomial). We show that one can run the MMP on this space, and that it eventually contracts the resultant divisor.

We establish quantitative stability bounds for the quadratic optimal transport map $T_\mu$ between a fixed probability density $\rho$ and a probability measure $\mu$ on $\mathbb{R}^d$. Under general assumptions on $\rho$, we prove that the map $\mu\mapsto T_\mu$ is bi-Hölder continuous, with dimension-free Hölder exponents. The linearized optimal transport metric $W_{2,\rho}(\mu,\nu)=\|T_\mu-T_\nu\|_{L^2(\rho)}$ is therefore bi-Hölder equivalent to the $2$-Wasserstein distance, which justifies its use in applications. We show this property in the following cases: (i) for any log-concave density $\rho$ with full support in $\mathbb{R}^d$, and any log-bounded perturbation thereof; (ii) for $\rho$ bounded away from $0$ and $+\infty$ on a John domain (e.g., on a bounded Lipschitz domain), while the only previously known result of this type assumed convexity of the domain; (iii) for some important families of probability densities on bounded domains which decay or blow-up polynomially near the boundary. Concerning the sharpness of point (ii), we also provide examples of non-John domains for which the Brenier potentials do not satisfy any Hölder stability estimate. Our proofs rely on local variance inequalities for the Brenier potentials in small convex subsets of the support of $\rho$, which are glued together to deduce a global variance inequality. This gluing argument is based on two different strategies of independent interest: one of them leverages the properties of the Whitney decomposition in bounded domains, the other one relies on spectral graph theory.

It is known since the seminal work of Guillemin and Sternberg that Lie subalgebras of finite codimension can be realized as subalgebras of formal vector fields over formal power series. In this note, we characterize the Lie subalgebras which admit a convergent realization in the sense of locally analytic vector fields. We give generalizations of these properties for the problem of output realization. We give reformulations and applications of these algebraic results in the context of control theory. In particular, we recover and clarify previous results on the realization of Chen-Fliess series for control-affine systems, the equivalence of control systems, the existence of embedded or canonical systems.

In 1986, Sussmann proposed an expansion of the Chen series as an infinite product of exponentials of Hall basis elements multiplied by coefficients defined through a simple induction. He proved that this product converges locally in time for bilinear control systems associated with matrices or bounded operators. In this note, we prove that Sussmann's product also converges locally in time for control-affine systems driven by analytic vector fields. The difficulty is that one must keep track of the structure of the involved Lie brackets, as the natural decay of the explicit coefficients is not enough to counterbalance the natural growth of the iterated Lie brackets.

Étant donné un arbre et un groupe d’automorphismes de , nous étudions les propriétés markoviennes du flot géodésique sur le quotient de l’espace des géodésiques de par . Par exemple, quand est l’arbre de Bruhat-Tits d’un groupe algébrique linéaire connexe semi-simple de rang 1 sur un corps local non archimédien et si est un réseau (éventuellement non uniforme) dans , nous montrons que l’action des puissances paires de la transformation géodésique est Bernoulli d’entropie finie sur chacune des deux composantes ergodiques. Sous des hypothèses générales bénignes, nous montrons que si le flot géodésique est mélangeant pour une mesure de probabilité de Patterson-Sullivan-Bowen-Margulis, alors il est lâchement Bernoulli

Let M be a complete simply connected Riemannian manifold, with sectional curvature K ≤ −1. Under certain assumptions on the geometry of ∂M, which are satisfied for instance if M is a symmetric space, or has dimension 2, we prove that given any family of horoballs in M, and any point x₀ outside these horoballs, it is possible to shrink uniformly, by a finite amount depending only on M, these horoballs so that some geodesic ray starting from x₀ avoids the shrunk horoballs. As an application, we give a uniform upper bound on the infimum of the heights of the closed geodesics in the finite volume quotients of M.

We prove that scaling limits of random planar maps which are uniformly distributed over the set of all rooted 2k-angulations are a.s. homeomorphic to the two-dimensional sphere. Our methods rely on the study of certain random geodesic laminations of the disk.

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

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.

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.