Publications

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.

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.

This is a survey on formality results relying on weight structures. A weight structure is a naturally occurring grading on certain differential graded algebras. If this weight satisfies a purity property, one can deduce formality. Algebraic geometry provides us with such weight structures as the cohomology of algebraic varieties tends to present additional structures including a Hodge structure or a Galois action.

We define a properad Y (n) ∞ that encodes n-pre-Calabi-Yau algebras with vanishing copairing. These algebras include chains on the based loop space of any space X endowed with a fundamental class [X] such that (X, [X]) satisfies Poincaré duality of degree n ⩾ 1 with local system coefficients, such as an oriented manifold. Extending the notion of coformality of spaces, we define coformality of such a pair (X, [X]) in terms of properadic formality of Y (n) ∞ -algebra structures on C * (ΩX). Using a refined version of properadic Kaledin classes, we establish the intrinsic coformality of all spheres in characteristic zero.

We develop an obstruction theory for the existence of gauge equivalences in complete differential graded Lie algebras. Specifically, this theory provides a characterization of homotopy equivalences between differential graded algebras governed by operads or properads, potentially colored in a groupoid. We apply this framework to establish new homotopy equivalence results in both algebraic topology and algebraic geometry, with a particular focus on the study of minimal models for highly connected varieties.

We study the cohomological classification of vector bundles on smooth real affine surfaces and threefolds. We show that, as was observed in joint work in A. Asok and J. Fasel and with S. Banerjee and J. Fasel, under suitable cohomological assumptions on the real locus of such varieties, this classification mirrors the one obtained on algebraically closed base fields by Mohan Kumar and Murthy and by Asok and Fasel. Using an argument due to Fasel, we also give an efficient proof of a theorem of Kucharz characterising the triples of algebraic cycles that can be realised as the Chern classes of a rank $3$ bundle on a smooth real affine threefold. We further answer the questions left open by Kucharz; to our knowledge, we give the first instance of a projective module over a smooth affine $\mathbb{R}$-algebra of dimension $3$ with trivial Chern classes which is not stably free.

In this paper, we explore the statistical subtleties of the nonideal Rayleigh gas, in a grand canonical mixture framework. This model allows to consider a large amount of tagged particles close to equilibrium, and their empirical measure, whose first-order convergence has been shown to converge to the solution of the linear Rayleigh-Boltzmann equation [5]. Thanks to the study of the cumulants of the system, we analyze the asymptotic behaviour of the fluctuations and large deviations of this empirical measure, hence refining the previous statistical results in the same vein as [7]. This way, we exhibit the trivial limit behaviour of the fluctuations in any overdilute regime, proving the exact relevance at any statistical scale of the low density limit. In the case of large deviations, we present the linear Boltzmann-Hamilton-Jacobi system driving their asymptotic behaviour. Eventually, we optimize the geometrical estimates on the billiards dynamics [6] to finally achieve a full convergence rate for the cumulants.

This paper introduces a grand canonical mixture model to generalize the nonideal Rayleigh gas [5] to an asymptotically infinite amount of perturbed tagged particles. This model relies precisely on grand canonical tags, to preserve symmetry in the system, contrary to [2]. We hence define and study the convergence of the correlation functions of this system in large times, linking it to the expectancy of the empirical measure of tagged and non-tagged particles, to eventually prove a law of large numbers for this dynamics. We extend the quantitative study to all the correlation functions, and not only the first one, exhibiting the resultant additional factors, and we also generalize the perturbation to the whole phase space, instead of considering a space-only initial perturbation. Eventually, we fit our adaptive time cutting [12] to the mixture system, even improving it to get better convergence rates.

Solving optimal transport (OT) on random minibatches is a common surrogate for exact OT in large-scale learning. In flow matching (FM), this surrogate is used to obtain OT-like couplings that can straighten probability paths and reduce numerical integration cost. Yet, the population-level coupling induced by repeated minibatch OT remains only partially understood. We formalize this coupling as the expected batch OT plan $\overline{\pi}_{k}$, obtained by averaging empirical OT plans over independent minibatches of size $k$. We then establish its large-batch consistency and, in the semidiscrete case relevant to generative modeling, derive rates for both the transport-cost bias and the convergence of $\overline{\pi}_{k}$ to the OT plan. For FM, this yields a population coupling whose induced velocity field is regular enough to define a unique flow from the source to the discrete target. We finally quantify how OT batch size interacts with numerical integration in a tractable two-atom model and in synthetic and image experiments.

A surprising phenomenon in the training of neural networks is the ability of gradient descent to find global minimizers of the training loss despite its non-convexity. Following earlier works, we investigate this behavior for wide shallow networks. Existing results essentially cover the case of ReLU activations and the case of sigmoid activations with scalar output weights. We study a large class of models that includes multi-head attention layers and two-layer sigmoid networks with vector output weights. Building upon [Chizat and Bach, 2018], we prove that all non-global minimizers of the training loss are unstable under gradient descent dynamics. Thus, when the initial distribution of the parameters has full support (which includes the popular Gaussian case), and in the many hidden neurons or attention heads limit, continuoustime gradient descent can only converge to global minimizers. Establishing the instability of non-global minimizers corresponds to the construction of an "escaping active set" - we complete the proof of [Chizat and Bach, 2018] to construct this set for models with bounded nonlinearities and scalar output weights. We also extend this construction to new cases for models with vector output weights. Finally, we show the well-posedness and the stability with respect to discretization of the mean field training dynamic for sub-Gaussian initializations.

A surprising phenomenon in the training of neural networks is the ability of gradient descent to find global minimizers of the training loss despite its non-convexity. Following earlier works, we investigate this behavior for wide shallow networks. Existing results essentially cover the case of ReLU activations and the case of sigmoid activations with scalar output weights. We study a large class of models that includes multi-head attention layers and two-layer sigmoid networks with vector output weights. Building upon [Chizat and Bach, 2018], we prove that all non-global minimizers of the training loss are unstable under gradient descent dynamics. Thus, when the initial distribution of the parameters has full support (which includes the popular Gaussian case), and in the many hidden neurons or attention heads limit, continuoustime gradient descent can only converge to global minimizers. Establishing the instability of non-global minimizers corresponds to the construction of an "escaping active set" - we complete the proof of [Chizat and Bach, 2018] to construct this set for models with bounded nonlinearities and scalar output weights. We also extend this construction to new cases for models with vector output weights. Finally, we show the well-posedness and the stability with respect to discretization of the mean field training dynamic for sub-Gaussian initializations.

This note is motivated by the problem of "uniqueness of supercuspidal support" in the modular representation theory of p-adic groups. We show that any counterexample to the same property for a finite reductive group lifts to a counterexample for the corresponding unramified p-adic group. To this end, we need to prove the following natural property : any simple subquotient of a parabolically induced representation is isomorphic to a subquotient of the parabolic induction of some simple subquotient of the original representation. The point is that we put no finiteness assumption on the original representation.

This work is concerned with the generation of decay estimates in the velocity variable for solutions of the space-inhomogeneous Boltzmann equation without cutoff on a bounded spatial domain for hard and moderately soft potentials. We work with suitable weak solutions, provided that mass, energy and entropy density functions are under control. The following boundary conditions are treated: in-flow, bounce-back, specular reflection, diffuse reflection and Maxwell reflection. The notion of weak solutions relies on a family of Truncated Convex Inequalities that is inspired by the one recently introduced through F.~Golse, L.~Silvestre and the first author (2023) in the spatially homogeneous case. We show that the solutions generate some amount (up to $d+1$) of pointwise polynomial velocity decay. In case of moderately soft potentials, we show that it is not possible to generate a decay higher than $d+2$ if the energy is bounded.

Our goal is to highlight some deep connections between numerical splitting methods and control theory. We consider evolution equations of the form $\dot{x} = f_0(x) + f_1(x)$, where $f_0$ encodes non-reversible dynamics, motivating schemes that involve only forward flows of $f_0$. In this context, a splitting method can be interpreted as a trajectory of the control-affine system $\dot{x}(t)=f_0(x(t))+u(t)f_1(x(t))$, associated with a control $u$ that is a finite sum of Dirac masses. The goal is then to find a control such that the flow generated by $f_0 + u(t)f_1$ is as close as possible to the flow of $f_0+f_1$. Using this interpretation and classical tools from control theory, we revisit well-known results on numerical splitting methods and prove several new ones. First, we show that there exist numerical schemes of arbitrary order involving only forward flows of $f_0$, provided one allows complex coefficients for $f_1$. Equivalently, for complex-valued controls, we prove that the Lie algebra rank condition is equivalent to small-time local controllability. Second, for real-valued coefficients, we show that the well-known order restrictions are linked to so-called "bad" Lie brackets from control theory, which are known to obstruct small-time local controllability. We investigate the conditions under which high-order methods exist, thanks to a basis of the free Lie algebra that we recently constructed.

Our goal is to highlight some deep connections between numerical splitting methods and control theory. We consider evolution equations of the form $\dot{x} = f_0(x) + f_1(x)$, where $f_0$ encodes non-reversible dynamics, motivating schemes that involve only forward flows of $f_0$. In this context, a splitting method can be interpreted as a trajectory of the control-affine system $\dot{x}(t)=f_0(x(t))+u(t)f_1(x(t))$, associated with a control $u$ that is a finite sum of Dirac masses. The goal is then to find a control such that the flow generated by $f_0 + u(t)f_1$ is as close as possible to the flow of $f_0+f_1$. Using this interpretation and classical tools from control theory, we revisit well-known results on numerical splitting methods and prove several new ones. First, we show that there exist numerical schemes of arbitrary order involving only forward flows of $f_0$, provided one allows complex coefficients for $f_1$. Equivalently, for complex-valued controls, we prove that the Lie algebra rank condition is equivalent to small-time local controllability. Second, for real-valued coefficients, we show that the well-known order restrictions are linked to so-called "bad" Lie brackets from control theory, which are known to obstruct small-time local controllability. We investigate the conditions under which high-order methods exist, thanks to a basis of the free Lie algebra that we recently constructed.

Our goal is to highlight some deep connections between numerical splitting methods and control theory. We consider evolution equations of the form $\dot{x} = f_0(x) + f_1(x)$, where $f_0$ encodes non-reversible dynamics, motivating schemes that involve only forward flows of $f_0$. In this context, a splitting method can be interpreted as a trajectory of the control-affine system $\dot{x}(t)=f_0(x(t))+u(t)f_1(x(t))$, associated with a control $u$ that is a finite sum of Dirac masses. The goal is then to find a control such that the flow generated by $f_0 + u(t)f_1$ is as close as possible to the flow of $f_0+f_1$. Using this interpretation and classical tools from control theory, we revisit well-known results on numerical splitting methods and prove several new ones. First, we show that there exist numerical schemes of arbitrary order involving only forward flows of $f_0$, provided one allows complex coefficients for $f_1$. Equivalently, for complex-valued controls, we prove that the Lie algebra rank condition is equivalent to small-time local controllability. Second, for real-valued coefficients, we show that the well-known order restrictions are linked to so-called "bad" Lie brackets from control theory, which are known to obstruct small-time local controllability. We investigate the conditions under which high-order methods exist, thanks to a basis of the free Lie algebra that we recently constructed.

In this paper, we establish the low-temperature asymptotics of the Poincaré inequality constant for a class of convex potentials satisfying a Łojasiewicz inequality. In addition, we disprove a conjecture previously posed by Chewi and Stromme on the low-temperature asymptotics of the log-Sobolev constant and determine the correct asymptotic behavior in dimension one.

Let $\Delta$ be a finite set. We adapt the techniques of Carter-Kedlaya-Zábrádi to obtain a multivariable Fontaine equivalence which relates continuous finite dimensional $\mathbb{F}_q$-representations of $\prod_{\alpha\in \Delta} \mathcal{G}_{\mathbb{F}_q(\!(X)\!)}$ to multivariable $\varphi$-modules over a $\mathbb{F}_q$-algebra which is a domain. Building on this, we construct a multivariable Lubin-Tate period ring and deduce a multivariable Lubin-Tate Fontaine equivalence for continuous finite type $\mathcal{O}_K$-representations of $\prod_{\alpha\in \Delta} \mathcal{G}_K$, where $K|\mathbb{Q}_p$ is a finite extension. We also obtain a plectic Fontaine equivalence and two equivalences for the subgroup $\mathcal{G}_{K,\mathrm{glec}}$ of the plectic Galois group.

Functors involved in Fontaine equivalences decompose as extension of scalars and taking of invariants between full subcategories of modules over a topological ring equipped with semi-linear continuous action of a topological monoid. We give a general framework for these categories and the functors between them. We define the categories of étale projective $\mathcal{S}$-modules over $R$ to englobe categories that will correspond by Fontaine-type equivalences to finite free representations of a group. We study their preservation by base change, taking of invariants by a normal submonoid of $\mathcal{S}$ and coinduction to a bigger monoid. We define and study categories corresponding to finite type continuous representations over $\mathbb{Z}_p$ through the notions of finite projective $(r,μ)$-dévissage and of topological étale $\mathcal{S}$-modules over $R$.