Publications

We prove that the category of Stein spaces and holomorphic maps is anti-equivalent to the category of Stein algebras and $\mathbb{C}$-algebra morphisms. This removes a finite dimensionality hypothesis from a theorem of Forster.

Artin vanishing theorems for Stein spaces refer to the vanishing of some of their (co)homology groups in degrees higher than the dimension. We obtain new positive and negative results concerning Artin vanishing for the cohomology of a Stein space relative to a Runge open subset. We also prove an Artin vanishing theorem for the Gal(C/R)-equivariant cohomology of a Gal(C/R)-equivariant Stein space relative to the fixed locus.

We consider the problem of smoothing algebraic cycles with rational coefficients on smooth projective complex varieties up to homological equivalence. We show that a solution to this problem would be incompatible with the validity of the Hartshorne conjecture on complete intersections in projective space. We also solve unconditionally a symplectic variant of this problem.

We construct and study relations between Chern classes and Galois cohomology classes in the Gal(C/R)-equivariant cohomology of real algebraic varieties with no real points. We give applications to the topology of their sets of complex points, and to sums of squares problems. In particular, we show that -1 is a sum of 2 squares in the function field of any smooth projective real algebraic surface with no real points and with vanishing geometric genus, as well as higher-dimensional generalizations of this result.

The abundance of unpaired multimodal single-cell data has motivated a growing body of research into the development of diagonal integration methods. However, the state-of-the-art suffers from the loss of biological information due to feature conversion and struggles with modality-specific populations. To overcome these crucial limitations, we here introduce scConfluence, a method for single-cell diagonal integration. scConfluence combines uncoupled autoencoders on the complete set of features with regularized Inverse Optimal Transport on weakly connected features. We extensively benchmark scConfluence in several single-cell integration scenarios proving that it outperforms the state-of-the-art. We then demonstrate the biological relevance of scConfluence in three applications. We predict spatial patterns for Scgn, Synpr and Olah in scRNA-smFISH integration. We improve the classification of B cells and Monocytes in highly heterogeneous scRNA-scATAC-CyTOF integration. Finally, we reveal the joint contribution of Fezf2 and apical dendrite morphology in Intra Telencephalic neurons, based on morphological images and scRNA.

We investigate the mixing time of the asymmetric Zero Range process on the segment with a non-decreasing rate. We show that the cutoff holds in the totally asymmetric case with a convex flux, and also with a concave flux if the asymmetry is strong enough. We show that the mixing occurs when the macroscopic system reaches equilibrium. A key ingredient of the proof, of independent interest, is the hydrodynamic limit for irregular initial data.

For a quantum system in a macroscopically large volume V, prepared in a pure state and subject to maximally noisy or ergodic unitary dynamics, the reduced density matrix of any sub-system v≪V is almost surely totally mixed. We show that the fluctuations around this limiting value, evaluated according to the invariant measure of these unitary flows, are captured by the Gaussian unitary ensemble (GUE) of random matrix theory. An extension of this statement, applicable when the unitary transformations conserve the energy but are maximally noisy or ergodic on any energy shell, allows to decipher the fluctuations around canonical typicality. According to typicality, if the large system is prepared in a generic pure state in a given energy shell, the reduced density matrix of the sub-system is almost surely the canonical Gibbs state of that sub-system. We show that the fluctuations around the Gibbs state are encoded in a deformation of the GUE whose covariance is specified by the Gibbs state. Contact with the eigenstate thermalization hypothesis is discussed.

We present a new description of the known large deviation function of the classical symmetric simple exclusion process by exploiting its connection with the quantum symmetric simple exclusion processes and using tools from free probability. This may seem paradoxal as free probability usually deals with non commutative probability while the simple exclusion process belongs to the realm of classical probability. On the way, we give a new formula for the free energy -- alias the logarithm of the Laplace transform of the probability distribution -- of correlated Bernoulli variables in terms of the set of their cumulants with non-coinciding indices. This latter result is obtained either by developing a combinatorial approach for cumulants of products of random variables or by borrowing techniques from Feynman graphs.

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

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 cannot analyze spatially resolved transcriptomic data.

Here, we propose STORIES, a method that employs 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 as Nptx1 in neuron regeneration and Aldh1l1 in gliogenesis and additional putative drivers.

The purpose of this paper is to investigate the well-posedness of several linear and nonlinear equations with a parabolic forward-backward structure, and to highlight the similarities and differences between them. The epitomal linear example will be the stationary Kolmogorov equation $y u_x - u_{yy}=f$, which we investigate in a rectangle $(x_0,x_1)\times(-1,1)$, supplemented with boundary conditions on the ``parabolic boundary'' of the domain: the top and lower boundaries $\{y=\pm 1\}$, and the lateral boundaries $\{x_0\}\times (0,1)$ and $\{x_1\}\times (-1,0)$. We first prove that this equation admits a finite number of singular solutions associated with regular data. These singular solutions, of which we provide an explicit construction, are localized in the vicinity of the points $(x_0,0)$ and $(x_1,0)$. Hence, the solutions to the Kolmogorov equation associated with a smooth source term $f$ are regular if and only if $f$ satisfies a finite number of orthogonality conditions. This is similar to well-known phenomena in elliptic problems in polygonal domains. We then extend this theory to the Vlasov--Poisson--Fokker--Planck system $y u_x + E[u] u_y - u_{yy}=f$, and to two quasilinear equations: the Burgers type equation $u u_x - u_{yy} = f$ in the vicinity of the linear shear flow, and the Prandtl system in the vicinity of a recirculating solution, close to the curve where the horizontal velocity changes sign. We therefore revisit part of a recent work by Iyer and Masmoudi. For the two latter quasilinear equations, we introduce a geometric change of variables which simplifies the analysis. In these new variables, the linear differential operator is very close to the Kolmogorov operator $y \partial_x - \partial_{yy}$. Stepping on the linear theory, we prove existence and uniqueness of regular solutions for data within a manifold of finite codimension, corresponding to some nonlinear orthogonality conditions. Treating these three nonlinear problems in a unified way also allows us to compare their structures. In particular, we show that the vorticity formulation of the Prandtl system, in an adequate set of variables, is very similar to the Burgers one. As a consequence, solutions of the Prandtl system are actually smoother than the ones of Burgers, which allows us to have a theory of weak solutions of the Prandtl system close to the recirculation zone.

We provide a new description of the scaling limit of dimer fluctuations in homogeneous Aztec diamonds via the intrinsic conformal structure of a space-like maximal surface in the three-dimensional Minkowski space R^{2,1}. This surface naturally appears as the limit of the graphs of origami maps associated to symmetric t-embeddings of Aztec diamonds, fitting the framework recently developed in [7,8].

Transformers have achieved state-of-the-art performance in language modeling tasks. However, the reasons behind their tremendous success are still unclear. In this paper, towards a better understanding, we train a Transformer model on a simple next token prediction task, where sequences are generated as a first-order autoregressive process $s_{t+1} = W s_t$. We show how a trained Transformer predicts the next token by first learning $W$ in-context, then applying a prediction mapping. We call the resulting procedure in-context autoregressive learning. More precisely, focusing on commuting orthogonal matrices $W$, we first show that a trained one-layer linear Transformer implements one step of gradient descent for the minimization of an inner objective function, when considering augmented tokens. When the tokens are not augmented, we characterize the global minima of a one-layer diagonal linear multi-head Transformer. Importantly, we exhibit orthogonality between heads and show that positional encoding captures trigonometric relations in the data. On the experimental side, we consider the general case of non-commuting orthogonal matrices and generalize our theoretical findings.

Bilevel optimization aims to optimize an outer objective function that depends on the solution to an inner optimization problem. It is routinely used in Machine Learning, notably for hyperparameter tuning. The conventional method to compute the so-called hypergradient of the outer problem is to use the Implicit Function Theorem (IFT). As a function of the error of the inner problem resolution, we study the error of the IFT method. We analyze two strategies to reduce this error: preconditioning the IFT formula and reparameterizing the inner problem. We give a detailed account of the impact of these two modifications on the error, highlighting the role played by higher-order derivatives of the functionals at stake. Our theoretical findings explain when super efficiency, namely reaching an error on the hypergradient that depends quadratically on the error on the inner problem, is achievable and compare the two approaches when this is impossible. Numerical evaluations on hyperparameter tuning for regression problems substantiate our theoretical findings.

The profiling of multiple molecular layers from the same set of cells has recently become possible. There is thus a growing need for multi-view learning methods able to jointly analyze these data. We here present Multi-Omics Wasserstein inteGrative anaLysIs (Mowgli), a novel method for the integration of paired multi-omics data with any type and number of omics. Of note, Mowgli combines integrative Nonnegative Matrix Factorization and Optimal Transport, enhancing at the same time the clustering performance and interpretability of integrative Nonnegative Matrix Factorization. We apply Mowgli to multiple paired single-cell multi-omics data profiled with 10X Multiome, CITE-seq, and TEA-seq. Our in-depth benchmark demonstrates that Mowgli’s performance is competitive with the state-of-the-art in cell clustering and superior to the state-of-the-art once considering biological interpretability. Mowgli is implemented as a Python package seamlessly integrated within the scverse ecosystem and it is available at http://github.com/cantinilab/mowgli.

In this article, we establish a "Gehring lemma" for a real function satisfying a reverse H\"older inequality on all "kinetic cylinders" contained in a large one: it asserts that the integrability degree of the function improves under such an assumption. The kinetic cylinders are derived from the non-commutative group of invariances of the Kolmogorov equation. Our contributions here are (1) the extension of Gehring's Lemma to this kinetic (hypoelliptic) scaling used to generate the cylinders, (2) the localisation of the lemma in this hypoelliptic context (using ideas from the elliptic theory), (3) the streamlining of a short and quantitative proof. We then use this lemma to establish that the velocity gradient of weak solutions to linear kinetic equations of Fokker-Planck type with rough coefficients have Lebesgue integrability strictly greater than two, while the natural energy estimate merely ensures that it is square integrable. Our argument here is new but relies on Poincaré-type inequalities established in previous works.

Our goal is to highlight some of the deep links 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 a non-reversible dynamic, so that one is interested in schemes only involving 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~$u$ which is a finite sum of Dirac masses. The general goal is then to find a control such that the flow of $f_0 + u(t) f_1$ 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 concerning numerical splitting methods, and we prove a handful of new ones, with an emphasis on splittings with additional positivity conditions on the coefficients. First, we show that there exist numerical schemes of any arbitrary order involving only forward flows of $f_0$ if one allows complex coefficients for the flows of $f_1$. Equivalently, for complex-valued controls, we prove that the Lie algebra rank condition is equivalent to the small-time local controllability of a system. Second, for real-valued coefficients, we show that the well-known order restrictions are linked with so-called "bad" Lie brackets from control theory, which are known to yield obstructions to small-time local controllability. We use our recent basis of the free Lie algebra to precisely identify the conditions under which high-order methods exist.

Recently, Vadim Kaimanovich presented a particular example of a measure on a product of two standard lamplighter groups such that the Poisson boundary of the induced random walk is non-trivial, but the boundary on the marginals is trivial. This was surprising since such behavior is not possible for measures of finite entropy. As we show in this paper, this secret-sharing phenomenon is possible precisely for pairs of amenable groups with non-trivial ICC-factors.

It is a classical result of Kaimanovich and Vershik and independently of Rosenblatt that a non-amenable group admits a non-degenerate symmetric measure such that the Poisson boundary is trivial. Most if not all examples to date of non-free actions of countable groups on their Poisson boundaries had the stabilizers sitting inside the amenable radical. We show that every countable non-C*-simple group admits a symmetric measure of full support with non-trivial stabilizers. For a class of non-C*-simple groups with trivial amenable radical, which is non-empty as was shown by le Boudec, this gives a wealth of examples with non-normal stabilizers.

A recent study (Lee et al., 2021) has shown that contrails are the main contributor to aviation-related 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.

A recent study (Lee et al., 2021) has shown that contrails are the main contributor to aviation-related 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.

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

The aim of this paper is to provide a fast and efficient procedure for (real-time) target identification in imaging based on matching on a dictionary of precomputed generalized polarization tensors (GPTs). The approach is based on some important properties of the GPTs and new invariants. A new shape representation is given and numerically tested in the presence of measurement noise. The stability and resolution of the proposed identification algorithm is numerically quantified. We compare the proposed GPT-based shape representation with a moment-based one.

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.

Les travaux présentés dans ce mémoire concernent différentes équations de la théorie cinétique des gaz (Boltzmann, Landau, Fokker-Planck) et s'articulent autour de deux grands axes :- développer une théorie de Cauchy dans un cadre perturbatif de solutions fortes et étudier le comportement en temps grand des solutions,- étudier des problèmes de limites hydrodynamiques.Qu'elles s'inscrivent dans l'un ou l'autre de ces deux axes, les études menées reposent sur l'analyse de problèmes linéaires au moyen de différentes théories : théorie des semi- groupes, théorie spectrale, hypocoercivité, élargissement d'espace, argument de perturbation, hypoellipticité etc... Ainsi, nous appliquons et/ou étendons ces techniques à de nouvelles situations : EDP de nouvelle nature, équations discrètes, nouveau cadre fonctionnel notamment.

We investigate the asymptotic behaviour of fast rotating incompressible fluids with vanishing viscosity, in a {three dimensional} domain with topography including the case of land area. Assuming the initial data is well-prepared, we prove a convergence theorem of the velocity fields to a two-dimensional vector field solving a linear, damped ordinary differential equation. The proof is based on a weak-strong uniqueness argument, combined with an abstract result implying that the weak convergence of a family of weak solutions to the Navier-Stokes-Coriolis system can be translated into a form of uniform-in-time convergence. This argument yields strong convergence of the velocity fields, without a precise rate though.

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.

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.

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.