Publications

The training of deep residual neural networks (ResNets) with backpropagation has a memory cost that increases linearly with respect to the depth of the network. A simple way to circumvent this issue is to use reversible architectures. In this paper, we propose to change the forward rule of a ResNet by adding a momentum term. The resulting networks, momentum residual neural networks (MomentumNets), are invertible. Unlike previous invertible architectures, they can be used as a drop-in replacement for any existing ResNet block. We show that MomentumNets can be interpreted in the infinitesimal step size regime as second-order ordinary differential equations (ODEs) and exactly characterize how adding momentum progressively increases the representation capabilities of MomentumNets. Our analysis reveals that MomentumNets can learn any linear mapping up to a multiplicative factor, while ResNets cannot. In a learning to optimize setting, where convergence to a fixed point is required, we show theoretically and empirically that our method succeeds while existing invertible architectures fail. We show on CIFAR and ImageNet that MomentumNets have the same accuracy as ResNets, while having a much smaller memory footprint, and show that pre-trained MomentumNets are promising for fine-tuning models.

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 revisit these results in the context of Sobolev spaces modelled on L^2 and exemplify our method with the SKT system, showing the existence of local, non-negative, strong solutions.

We prove that the entropy solution to a scalar conservation law posed on the real line with a flux that is discontinuous at one point (in the space variable) coincides with the derivative of the solution to a Hamilton-Jacobi (HJ) equation whose Hamiltonian is discontinuous. Flux functions (Hamiltonians) are not assumed to be convex in the state (gradient) variable. The proof consists in proving the convergence of two numerical schemes. We rely on the theory developed by B.~Andreianov, K.~H.~Karlsen and N.~H.~Risebro (\textit{Arch. Ration. Mech. Anal.}, 2011) for such scalar conservation laws and on the viscosity solution theory developed by the authors (\textit{arxiv}, 2023) for the corresponding HJ equation. This study allows us to characterise certain germs introduced in the AKR theory (namely maximal and complete ones) and relaxation operators introduced in the viscosity solution framework.

In this article, we consider joint returns to zero of $n$ Bessel processes ($n\geq 2$): our main goal is to estimate the probability that they avoid having joint returns to zero for a long time. More precisely, considering $n$ independent Bessel processes $(X_t^{(i)})_{1\leq i \leq n}$ of dimension $\delta \in (0,1)$, we are interested in the first joint return to zero of any two of them: \[ H_n := \inf\big\{ t>0, \exists 1\leq i t) = t^{-\theta_n+o(1)}$ as $t\to\infty$, and we provide some non-trivial bounds on $\theta_n$. In particular, when $n=3$, we show that $2(1-\delta)\leq \theta_3 \leq 2 (1-\delta) + f(\delta)$ for some (explicit) function $f(\delta)$ with $\sup_{[0,1]} f(\delta) \approx 0.079$.

We study interpolation inequalities between Hölder Integral Probability Metrics (IPMs) in the case where the measures have densities on closed submanifolds. Precisely, it is shown that if two probability measures $\mu$ and $\mu^\star$ have $\beta$-smooth densities with respect to the volume measure of some submanifolds $\mathcal{M}$ and $\mathcal{M}^\star$ respectively, then the Hölder IPMs $d_{\mathcal{H}^\gamma_1}$ of smoothness $\gamma\geq 1$ and $d_{\mathcal{H}^\eta_1}$ of smoothness $\eta>\gamma$, satisfy $d_{ \mathcal{H}_1^{\gamma}}(\mu,\mu^\star)\lesssim d_{ \mathcal{H}_1^{\eta}}(\mu,\mu^\star)^\frac{\beta+\gamma}{\beta+\eta}$, up to logarithmic factors. We provide an application of this result to high-dimensional inference. These functional inequalities turn out to be a key tool for density estimation on unknown submanifold. In particular, it allows to build the first estimator attaining optimal rates of estimation for all the distances $d_{\mathcal{H}_1^\gamma}$, $\gamma \in [1,\infty)$ simultaneously.

In network analysis, a measure of node centrality provides a scale indicating how central a node is within a network. The coreness is a popular notion of centrality that accounts for the maximal smallest degree of a subgraph containing a given node. In this paper, we study the coreness of random geometric graphs and show that, with an increasing number of nodes and properly chosen connectivity radius, the coreness converges to a new object, that we call the continuum coreness. In the process, we show that other popular notions of centrality measures, namely the H-index and its iterates, also converge under the same setting to new limiting objects.

We investigate the mechanism for eye formation in hurricane-like vortices, using a formulation adapted from Oruba et al. (2017). Numerical simulations are performed using an axisymmetric model of dry rotating Rayleigh-Bénard convection under the Boussinesq approximation. The fluxes of heat and momentum at the sea surface are described using the bulk aerodynamic formula. A simplified model for radiative cooling is also implemented. We find that the mechanism for eye formation introduced in Oruba et al. (2017), relying on vorticity stripping from the boundary layer, is robust in dry hurricane-like vortices. Furthermore, with these boundary conditions the structure of the flow is closer to the flow of actual tropical cyclones. The applicability of this mechanism to the moist case however remains uncertain and deserves further study. Finally, energy budgets, obtained either by a heat engine approach, or by a direct estimation of the work of buoyancy forces, are investigated. They provide estimations of the surface wind speed as a function of the controlling parameters.

We wish to correct an error pointed out by the third author in the paper “Valued fields, Metastable groups” by the first two authors.

We wish to correct an error pointed out by the third author in the paper “Valued fields, Metastable groups” by the first two authors.

This thesis is devoted to the regularity theory of kinetic Fokker-Planck equations. Firstly, we study the interior regularization effect for the equations with general transport operators and rough coefficients, by revisiting the De Giorgi-Nash-Moser theory and velocity averaging lemmas. The second part addresses the Cauchy problem and the diffusion asymptotics for a kinetic model associated to a nonlinear Fokker-Planck operator. We derive the global well-posedness with instantaneous smoothness effect, and the global diffusion asymptotics quantitively. Finally, we study the existence, uniqueness, and boundary regularization mechanism for the equations in the presence of boundary conditions, including the inflow, diffuse reflection and specular reflection cases.

Conservation laws are well-established in the context of Euclidean gradient flow dynamics, notably for linear or ReLU neural network training. Yet, their existence and principles for non-Euclidean geometries and momentum-based dynamics remain largely unknown. In this paper, we characterize "all" conservation laws in this general setting. In stark contrast to the case of gradient flows, we prove that the conservation laws for momentum-based dynamics exhibit temporal dependence. Additionally, we often observe a "conservation loss" when transitioning from gradient flow to momentum dynamics. Specifically, for linear networks, our framework allows us to identify all momentum conservation laws, which are less numerous than in the gradient flow case except in sufficiently over-parameterized regimes. With ReLU networks, no conservation law remains. This phenomenon also manifests in non-Euclidean metrics, used e.g. for Nonnegative Matrix Factorization (NMF): all conservation laws can be determined in the gradient flow context, yet none persists in the momentum case.

We investigate the ocean wave eld under Hurricane SAM (2021). Whilst measurements of waves under Tropical Cyclones (TCs) are rare, an unusually large number of quality in situ and remote mea- surements are available in that case. First, we highlight the good consistency between the wave spectra provided by the Surface Waves Investigation and Monitoring (SWIM) instrument onboard the China- France Oceanography Satellite (CFOSAT), the in situ spectra mea- sured by National Data Buoy Center (NDBC) buoys, and a saildrone. The impact of strong rains on SWIM spectra is then further investi- gated. We show that whereas the rain de nitely a ects the normalized radar cross section, both the innovative technology (beam rotating scanning geometry) and the post-processing processes applied to re- trieve the 2D wave spectra ensure a good quality of the resulting wave spectra, even in heavy rain conditions. On this basis, the satellite, air- borne and in situ observations are confronted to the analytical model proposed by Kudryavtsev et al (2015). We show that a trapped wave mechanism may be invoked to explain the large signi cant wave height observed in the right front quadrant of Hurricane SAM.

In this paper, we continue some investigations on the periodic NLSE started by Lebowitz, Rose and Speer and by Bourgain with the addition of a distributional multiplicative potential. We prove that the equation is globally wellposed for a set of data of full normalized Gibbs measure, after suitable truncation in the focusing case. The set and the measure are invariant under the flow. The main ingredients used are Strichartz estimates on periodic NLS with distributional potential to obtain local well-posedness for low regularity initial data.

A recent study (Lee et al., 2021) has shown that contrails are the main contributor to aviationrelated radiative forcing. However, the same study shows that this contribution is highly imprecise due to numerous uncertainties. Among the most important are the numerous contingencies regarding the vertical and horizontal extent of ice plumes, as well as their altitude, which may differ from the flight level of the emitting aircraft, rising to hundreds of meters. This uncertainty is largely due to its interaction with the aircraft’s dynamic wake, which, very soon after the aircraft’s passage, is reduced to two counter-rotating vortices known as wingtip vortices. These two vortices descend by induction into the atmosphere, driving the plumes to lower altitudes. However, these dynamics are influenced by atmospheric stratification, as shown in Spalart (1996). In most cases, the two wake vortices continue their descent, but certain dynamic structures are created in their vicinity by the baroclinic torque due to buoyancy, and rise to flight altitude. The wake then splits into two parts: one descending into the atmosphere and the other rising back up to, or slightly above, flight altitude. A long, rising column of fluid joins the two wakes. The plume initially trapped around the two vortices can then evolve in three different ways. Either the plume remains with the vortices well below the flight altitude, or it rises to this altitude or even higher, entrained in the secondary wake, or it is distributed between the two wakes and the column uniting them, as shown by Saulgeot et al. (2023). Among the parameters influencing these dynamics is the relationship between atmospheric stratification, quantified by the Brunt-Väisälä frequency N, and the characteristic time τ0 of the vortex dipole τ0 = b0/ W0 where the natural motion of the vortices is a descent at constant speed W0 caused by mutual induction. This is the reference time scale, and the initial vortex separation b0 is the reference distance. In this scale framework, the effective stratification of the vortex flow is measured by the inverse of the Froude number Fr−1 = Nτ0. The intermediate vorticity column plays a fundamental role in the upwelling of the plume: it is the only link between the primary and secondary wakes and can therefore influence both the latter and the plume. At the end of the two-dimensional phase of wake evolution, before the onset of the Crow instability, this column can destabilize, isolating the two parts of the wake and preventing the plume from rising. This can be thought of as thermal plume jet instabilities. These are of two types: sinusoidal and varicose. In most cases, the two instabilities follow one another: the varicose instability appears first, then the sinusoidal instability takes over due to a higher growth rate. Nevertheless, the appearance of one or the other can be observed independently.

When considering statistical mechanics models on trees, such that the Ising model, percolation, or more generally the random cluster model, some concave tree recursions naturally emerge. Some of these recursions can be compared with non-linear conductances, or $p$-conductances, between the root and the leaves of the tree. In this article, we estimate the $p$-conductances of $T_n$, a supercritical Galton-Watson tree of depth $n$, for any $p>1$ (for a quenched realization of $T_n$). In particular, we find the sharp asymptotic behavior when $n$ goes to infinity, which depends on whether the offspring distribution admits a finite moment of order $q$, where $q=\frac{p}{p-1}$ is the conjugate exponent of $p$. We then apply our results to the random cluster model on~$T_n$ (with wired boundary condition) and provide sharp estimates on the probability that the root is connected to the leaves. As an example, for the Ising model on $T_n$ with plus boundary conditions on the leaves, we find that, at criticality, the quenched magnetization of the root decays like: (i) $n^{-1/2}$ times an explicit tree-dependent constant if the offspring distribution admits a finite moment of order $3$; (ii) $n^{-1/(\alpha-1)}$ if the offspring distribution has a heavy tail with exponent $\alpha \in (1,3)$.

We introduce a numerical strategy to study the evolution of two-dimensional water waves in the presence of a plunging jet. The free-surface Navier–Stokes solution is obtained with a finite, but small, viscosity. We observe the formation of a surface boundary layer where the vorticity is localised. We highlight convergence to the inviscid solution. The effects of dissipation on the development of a singularity at the tip of the wave is also investigated by characterising the vorticity boundary layer appearing near the interface.

In this article, we establish embeddings à la Lions and transfer of regularity à la Bouchut for a large scale of kinetic spaces. We use them to identify a notion of weak solutions to Kolmogorov-Fokker-Planck equations with (local or integral) diffusion and rough (measurable) coefficients under minimal requirements. We prove their existence and uniqueness for a large class of source terms, first in full space for the time, position and velocity variables and then for the kinetic Cauchy problem on infinite and finite time intervals.

This paper’s objective is to improve the existing proof of the derivation of the Rayleigh–Boltzmann equation from the nonideal Rayleigh gas [6], yielding a far faster convergence rate. This equation is a linear version of the Boltzmann equation, describing the behavior of a small fraction of tagged particles having been perturbed from thermodynamic equilibrium. This linear equation, derived from the microscopic Newton laws as suggested by the Hilbert’s sixth problem, is much better understood than the quadratic Boltzmann equation, and even enable results on long time scales for the kinetic description of gas dynamics. The present paper improves the physically poor convergence rate that had been previously proved, into a much more satisfactory rate which is more than exponentially better.

We show the existence and uniqueness of fundamental solution operators to Kolmo\-gorov-Fokker-Planck equations with rough (measurable) coefficients and local or integral diffusion on finite and infinite time strips. In the local case, that is to say when the diffusion operator is of differential type, we prove $\L^2$ decay using Davies' method and the conservation property. We also prove that the existence of a generalized fundamental solution with the expected pointwise Gaussian upper bound is equivalent to Moser's $\L^2-\L^\infty$ estimates for local weak solutions to the equation and its adjoint. When coefficients are real, this gives the existence and uniqueness of such a generalized fundamental solution and a new and natural way to obtain pointwise decay.

We identify the local limit of massive spanning forests on the complete graph. This generalizes a well-known theorem of Grimmett on the local limit of uniform spanning trees on the complete graph.

We study the Maximum Zero-Sum Partition problem (or MZSP), defined as follows: given a multiset S={a₁, a₂, ..., a_n} of integers a_i∈Z* such that Σ_i=1..n a_i=0, find a maximum cardinality partition {S₁, S₂, ... , S_k} of S such that, for every 1≤ i ≤ k, Σ_{aj ∈ Si} a_j=0. Solving MZSP is useful in genomics for computing evolutionary distances between pairs of species. Our contributions are a series of algorithmic results concerning MZSP, in terms of complexity, (in)approximability, with a particular focus on the fixed-parameter tractability of MZSP with respect to either (i) the size k of the solution, (ii) the number of negative (resp. positive) values in S and (iii) the largest integer in S.

This paper aims at developing model-theoretic tools to study interpretable fields and definably amenable groups, mainly in $\mathrm{NIP}$ or $\mathrm{NTP_2}$ settings. An abstract theorem constructing definable group homomorphisms from generic data is proved. It relies heavily on a stabilizer theorem of Montenegro, Onshuus and Simon. The main application is a structure theorem for definably amenable groups that are interpretable in algebraically bounded perfect $\mathrm{NTP_2}$ fields with bounded Galois group (under some mild assumption on the imaginaries involved), or in algebraically bounded theories of (differential) NIP fields. These imply a classification of the fields interpretable in differentially closed valued fields, and structure theorems for fields interpretable in finitely ramified henselian valued fields of characteristic $0$, or in NIP algebraically bounded differential fields.

We study one- and two-dimensional periodic tight-binding models under the presence of a potential that grows to infinity in one direction, hence preventing the particles to escape in this direction (the soft wall). We prove that a spectral flow appears in these corresponding edge models, as the wall is shifted. We identity this flow as a number of Bloch bands, and provide a lower bound for the number of edge states appearing in such models.

The so-called Faster is Slower (FIS) effect is observed in some particular real-life or experimental situations. In the context of an evacuation process, it expresses that increasing the speed (or, more generally, the competitive- ness) of individuals may induce a reduction of the flow through the exit door. We propose here a parameter-free model to reproduce and investigate this effect (more precisely its backward “Slower is Faster” equivalent). In spite of its non-smooth character, which makes it difficult to analyze, this gran- ular approach is based on very basic ingredients in terms of behavior. In its native, purely asocial version, individuals are represented by hard-discs, each of which has a desired velocity, and the actual velocity is built as the projection of this field on the set of admissible velocities (which respect the non-overlapping constraints). We implement the slower effect by introducing here an extra step to account for the fact that individuals refrain from pushing, and therefore tend to reduce their desired velocity accounting for the velocities of people upfront. The present paper has two objectives: estab- lish the relevance of this model by showing that it satisfactorily reproduces various empirical effects in highly crowded evacuations with various levels of competitiveness, and explore how it can be implemented to recover and explain the FIS effect. In this spirit, we confront this Inhibition-Based (IB) model to experimental data, focusing on the Faster is Slower effect. We show in particular that this approach makes it possible to accurately recover the effect of competitiveness upon power-law distributions of time lapses which have been experimentally observed. We also study the effect of mixed behaviors, by introducing a two-population model using both approaches. Weinvestigate in particular the effect upon evacuation efficiency of the ratio be- tween competitive agents and non-competitive ones. In a similar context, we investigate the role of an obstacle placed upstream the exit upon evacuation efficiency.

This article focuses on a large family of cross-diffusion systems of the form ∂ t U-∆A(U) = 0, in dimension d ∈ N * , and where U ∈ R 2. We show that under natural conditions on the nonlinearity A, those systems have a unique smooth (nonnegative for all components) solution when the initial data are small enough in a suitable norm.

We consider conservative cross-diffusion systems for two species where individual motion rates depend linearly on the local density of the other species. We develop duality estimates and obtain stability and approximation results. We first control the time evolution of the gap between two bounded solutions by means of its initial value. As a by product, we obtain a uniqueness result for bounded solutions valid for any space dimension, under a non-perturbative smallness assumption. Using a discrete counterpart of our duality estimates, we prove the convergence of random walks with local repulsion in one dimensional discrete space to cross-diffusion systems. More precisely, we prove quantitative estimates for the gap between the stochastic process and the cross-diffusion system. We give first rough but general estimates; then we use the duality approach to obtain fine estimates under less general conditions.

We consider the nonlinear Schrödinger equation with double power nonlinearity. We extend the scattering result in [17] for all L 2-supercritical powers, specially, our results adapt to the cases of energy-supercritical nonlinearity.

We give new examples of algebraic integral cohomology classes on smooth projective complex varieties that are not integral linear combinations of classes of smooth subvarieties. Some of our examples have dimension 6, the lowest possible. The classes that we consider are minimal cohomology classes on Jacobians of very general curves. Our main tool is complex cobordism.

We show that any smooth cubic hypersurface of dimension n defined over a finite field Fq contains a line defined over Fq in each of the following cases: • n = 3 and q ≥ 11; • n = 4 and q 6= 3; • n ≥ 5. For a smooth cubic threefold X, the variety of lines contained in X is a smooth projective surface F(X) for which the Tate conjecture holds, and we obtain information about the Picard number of F(X) and the 5-dimensional principally polarized Albanese variety A(F(X)

We show that any smooth cubic hypersurface of dimension n defined over a finite field Fq contains a line defined over Fq in each of the following cases: • n = 3 and q ≥ 11; • n = 4 and q 6= 3; • n ≥ 5. For a smooth cubic threefold X, the variety of lines contained in X is a smooth projective surface F(X) for which the Tate conjecture holds, and we obtain information about the Picard number of F(X) and the 5-dimensional principally polarized Albanese variety A(F(X)

We study a self-attractive random walk such that each trajectory of length $N$ is penalised by a factor proportional to $exp(-|R_N\left|\right)$, where $R_N$ is the set of sites visited by the walk. We show that the range of such a walk is close to a solid Euclidean ball of radius approximately ${\rho }_d{N}^{1/(d+2)}$, for some explicit constant ${\rho }_d>0$. This proves a conjecture of Bolthausen [Bol94] who obtained this result in the case $d=2$.

Understanding the mechanisms behind the remote triggering of landslides by seismic waves at micro-strain amplitude is essential for quantifying seismic hazards. Granular materials provide a relevant model system to investigate landslides within the unjamming transition framework, from solid to liquid states. Furthermore, recent laboratory experiments have revealed that ultrasound-induced granular avalanches can be related to a reduction in the interparticle friction through shear acoustic lubrication of contacts. However, investigating slip at the scale of grain contacts within an optically opaque granular medium remains a challenging issue. Here, we propose an original coupling model and numerically investigate 2D dense granular flows triggered by basal acoustic waves. We model the triggering dynamics at two separated time-scales—one for grain motion (milliseconds) and the other for ultrasound (10 microseconds)—relying the computation of vibrational modes with a discrete element method through the reduction of the local friction. We show that ultrasound predominantly propagates through the strong-force chains, while the ultrasound-induced decrease of interparticle friction occurs in the weak contact forces perpendicular to the strong-force chains. This interparticle-friction reduction initiates local rearrangements at the grain scale that eventually lead to a continuous flow through a percolation process at the macroscopic scale—with a delay depending the proximity to the failure. Consitent with the experiment, we show that ultrasound-induced flow appears more uniform in space than pure gravity-driven flow, indicating the role of an effective temperature by ultrasonic vibration.

Understanding the mechanisms behind the remote triggering of landslides by seismic waves at micro-strain amplitude is essential for quantifying seismic hazards. Granular materials provide a relevant model system to investigate landslides within the unjamming transition framework, from solid to liquid states. Furthermore, recent laboratory experiments have revealed that ultrasound-induced granular avalanches can be related to a reduction in the interparticle friction through shear acoustic lubrication of contacts. However, investigating slip at the scale of grain contacts within an optically opaque granular medium remains a challenging issue. Here, we propose an original coupling model and numerically investigate 2D dense granular flows triggered by basal acoustic waves. We model the triggering dynamics at two separated time-scales—one for grain motion (milliseconds) and the other for ultrasound (10 microseconds)—relying the computation of vibrational modes with a discrete element method through the reduction of the local friction. We show that ultrasound predominantly propagates through the strong-force chains, while the ultrasound-induced decrease of interparticle friction occurs in the weak contact forces perpendicular to the strong-force chains. This interparticle-friction reduction initiates local rearrangements at the grain scale that eventually lead to a continuous flow through a percolation process at the macroscopic scale—with a delay depending the proximity to the failure. Consitent with the experiment, we show that ultrasound-induced flow appears more uniform in space than pure gravity-driven flow, indicating the role of an effective temperature by ultrasonic vibration.

We present a rigorous proof of the Dorn, Otto, Zamolodchikov, Zamolodchikov formula (the DOZZ formula) for the 3 point structure constants of Liouville Conformal Field Theory (LCFT) starting from a rigorous probabilistic construction of the functional integral defining LCFT given earlier by the authors and David. A crucial ingredient in our argument is a probabilistic derivation of the reflection relation in LCFT based on a refined tail analysis of Gaussian multiplicative chaos measures.

We consider a Schrödinger equation with a nonlinearity which is a general perturbation of a power" nonlinearity. We construct a profile decomposition adapted to this nonlinearity.We also prove global existence and scattering in a general defocusing setting, assuming thatthe critical Sobolev norm is bounded in the energy-supercritical case. This generalizes severalprevious works on double-power nonlinearities.

Caffarelli's contraction theorem states that probability measures with uniformly logconcave densities on R d can be realized as the image of a standard Gaussian measure by a globally Lipschitz transport map. We discuss some counterexamples and obstructions that prevent a similar result from holding on the half-sphere endowed with a uniform measure, answering a question of Beck and Jerison.

Comparing probability distributions is a fundamental problem in data sciences. Simple norms and divergences such as the total variation and the relative entropy only compare densities in a point-wise manner and fail to capture the geometric nature of the problem. In sharp contrast, Maximum Mean Discrepancies (MMD) and Optimal Transport distances (OT) are two classes of distances between measures that take into account the geometry of the underlying space and metrize the convergence in law. This paper studies the Sinkhorn divergences, a family of geometric divergences that interpolates between MMD and OT. Relying on a new notion of geometric entropy, we provide theoretical guarantees for these divergences: positivity, convexity and metrization of the convergence in law. On the practical side, we detail a numerical scheme that enables the large scale application of these divergences for machine learning: on the GPU, gradients of the Sinkhorn loss can be computed for batches of a million samples.

In this paper, we investigate the impact of stochasticity and large stepsizes on the implicit regularisation of gradient descent (GD) and stochastic gradient descent (SGD) over diagonal linear networks. We prove the convergence of GD and SGD with macroscopic stepsizes in an overparametrised regression setting and characterise their solutions through an implicit regularisation problem. Our crisp characterisation leads to qualitative insights about the impact of stochasticity and stepsizes on the recovered solution. Specifically, we show that large stepsizes consistently benefit SGD for sparse regression problems, while they can hinder the recovery of sparse solutions for GD. These effects are magnified for stepsizes in a tight window just below the divergence threshold, in the "edge of stability" regime. Our findings are supported by experimental results.

We construct solutions to the constraint equations in general relativity using the limit equation criterion introduced by Dahl, Humbert and the first author. We focus on solutions over compact 3-manifolds admitting a $\bS^1$-symmetry group. When the quotient manifold has genus greater than 2, we obtain strong far from CMC results.

We study the parabolic defocusing stochastic quantization equation with both mutliplicative spatial white noise and an independant space-time white noise forcing, on compact surfaces, with polynomial nonlinearity. After renormalizing the nonlinearity, we construct the random Gibbs measure as an absolutely continuous measure with respect to the law of the Anderson Gaussian Free Field for fixed realization of the spatial white noise. Then, when the initial data is distributed according to the Gibbs measure, we prove almost sure global well-posedness for the dynamics and invariance of the Gibbs measure.

We prove a local version of Gowers' Ramsey-type theorem [25], as well as local versions both of the Banach space first dichotomy (the "unconditional/HI" dichotomy) of Gowers [25] and of the third dichotomy (the "minimal/tight" dichotomy) due to Ferenczi-Rosendal [22]. This means that we obtain versions of these dichotomies restricted to certain families of subspaces called D-families, of which several concrete examples are given. As a main example, non-Hilbertian spaces form D-families; therefore versions of the above properties for non-Hilbertian spaces appear in new Banach space dichotomies. As a consequence we obtain new information on the number of subspaces of non-Hilbertian Banach spaces, making some progress towards the "ergodic" conjecture of Ferenczi-Rosendal and towards a question of Johnson.

We give new and simple proofs of some classical properties of hereditarily indecomposable Banach spaces, including the result by W. T. Gowers and B. Maurey that a hereditarily indecomposable Banach space cannot be isomorphic to a proper subspace of itself. These proofs do not make use of spectral theory and therefore, they work in real spaces as well as in complex spaces. We use our method to prove some new results. For example, we give a quantitative version of the latter result by Gowers and Maurey and deduce that Banach spaces that are isometric to all of their subspaces should have an unconditional basis with unconditional constant arbitrarily close to $1$. We also study the homotopy relation between into isomorphisms from hereditarily indecomposable spaces.

We develop a general framework for infinite-dimensional Ramsey theory with and without pigeonhole principle, inspired by Gowers’ Ramsey-type theorem for block sequences in Banach spaces and by its exact version proved by Rosendal. In this framework, we prove the adversarial Ramsey principle for Borel sets, a result conjectured by Rosendal that generalizes at the same time his version of Gowers’ theorem and Borel determinacy of games on integers.

In this paper, we show how to extend the twin blow-up method recently developped by the authors (Comptes Rendus. Math., 2024), in order to obtain a new comparison principle for an evolution coercive Hamilton-Jacobi equation posed in a domain of an Euclidian space of any dimension and supplemented with a boundary condition. The method allows dealing with the case where tangential variables and the variable corresponding to the normal gradient of the solution are strongly coupled at the boundary. We elaborate on a method introduced by P.-L. Lions and P. Souganidis (Atti Accad. Naz. Lincei, 2017). Their argument relies on a single blow-up procedure after rescaling the semi-solutions to be compared while two simultaneous blow-ups are performed in this work, one for each variable of the classical doubling variable technique. A one-sided Lipschitz estimate satisfied by a combination of the two blow-up limits plays a key role.

We consider Markov processes with generator of the form $γ\mathcal{L}_{1} + \mathcal{L}_{0}$, in which $\mathcal{L}_{1}$ generates a so-called dominant process that converges at large times towards a random point in a fixed subset called the effective state space. Using the usual characterization through martingales problems, we give general conditions under which homogenization holds true: the original process converges, when $γ$ is large and for the Meyer-Zheng pseudo-path topology and for finite-dimensional time marginals, towards an identified effective Markov process on the effective space. Few simple model examples for diffusions are studied.

Considering an integer d > 0, we show the existence of convex-cocompact representations of surface groups into SO(4, 1) admitting an embedded minimal map with curvatures in (−1, 1) and whose associated hyperbolic 4-manifolds are disk bundles of degree d over the surface, provided the genus g of the surface is large enough. We also show that we can realize these representations as complex variation of Hodge structures. This gives examples of quasicircles in S³ bounding superminimal disks in H⁴ of arbitrarily small second fundamental form. Those are examples of generalized almost-Fuchsian representations which are not deformations of Fuchsian representations.

We prove universality of spin correlations in the scaling limit of the planar Ising model on isoradial graphs with uniformly bounded angles and Z-invariant weights. Specifically, we show that in the massive scaling limit, i.e., as the mesh size $\delta$ tends to zero at the same rate as the inverse temperature goes to the critical one, the two-point spin correlations in the full plane behave as \[ \delta^{-\frac{1}{4}}\mathbb{E}\left[\sigma_{u_{1}}\sigma_{u_{2}}\right]\ \to\ C_{\sigma}^{2}\cdot\Xi\left(|u_{1}-u_{2}|,m\right)\quad\text{as}\quad\delta\to0, \] where the universal constant $C_{\sigma}$ and the function $\Xi(|u_{1}-u_{2}|,m)$ are independent of the lattice. The mass $m$ is defined by the relation $k'-1\sim 4m\delta$, where $k'$ is the Baxter elliptic parameter. This includes $m$ of both signs as well as the critical case when $\Xi(r,0)=r^{-1/4}.$ These results, together with techniques developed to obtain them, are sufficient to extend to isoradial graphs the convergence of multi-point spin correlations in finite planar domains on the square grid, which was established in a joint work of the first two authors and C. Hongler at criticality, and by S.C. Park in the sub-critical massive regime. We also give a simple proof of the fact that the infinite-volume magnetization in the Z-invariant model is independent of the site and of the lattice. As compared to techniques already existing in the literature, we streamline the analysis of discrete (massive) holomorphic spinors near their ramification points, relying only upon discrete analogues of the kernel $z^{-1/2}$ for $m=0$ and of $z^{-1/2}e^{\pm 2m|z|}$ for $m\ne 0$. Enabling the generalization to isoradial graphs and providing a solid ground for further generalizations, our approach also considerably simplifies the proofs in the square lattice setup.

Given a sequence of random variables $\left\{ X_k : k \geq 1\right\}$ uniformly distributed in $(0,1)$ and independent, we consider the following random sets of directions $$\Omega_{\text{rand},\text{lin}} := \left\{ \frac{\pi X_k}{k}: k \geq 1\right\}$$ and $$\Omega_{\text{rand},\text{lac}} := \left\{ \frac{ \pi X_k}{2^k} : k\geq 1 \right\}.$$ We prove that almost surely the directional maximal operators associated to those sets of directions are not bounded on $L^p(\mathbb{R}^2)$ for any $1 < p < \infty$.

We prove that the set of singular times for weak solutions of the homogeneous Boltzmann equation with very soft potentials constructed as in Villani (1998) has Hausdorff dimension at most $\frac{|\gamma+2s|}{2s}$ with $\gamma \in [-4s,-2s)$ and $s \in (0,1)$.

We present a rigorous mathematical analysis of the modeling of inviscid water waves. The free-surface is described as a parametrized curve. We introduce a numerically stable algorithm which accounts for its evolution with time. The method is shown to converge using approximate solutions, such as Stokes waves and Green-Naghdi solitary waves. It is finally tested on a wave breaking problem, for which an odd-even coupling suffices to achieve numerical convergence up to the splash without the need for additional filtering.