Publications

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.

In this paper, we investigate the link between kinetic equations (including Boltzmann with or without cutoff assumption and Landau equations) and the incompressible Navier-Stokes equation. We work with strong solutions and we treat all the cases in a unified framework. The main purpose of this work is to be as accurate as possible in terms of functional spaces. More precisely, it is well-known that the Navier-Stokes equation can be solved in a lower regularity setting (in the space variable) than kinetic equations. Our main result allows to get a rigorous link between solutions to the Navier-Stokes equation with such low regularity data and kinetic equations.

In this paper we present a unifying framework of residual domination for henselian valued fields of equicharacteristic zero, encompassing some valued fields with operators. We introduce a general definition for residual domination and show that it is well behaved. For instance, we prove a change of base theorem for residual domination over algebraically closed sets. We show that the class of residually dominated types coincides with the types that are orthogonal to the value group, and with the class of types whose reduct to $\ACVF$ (the theory of algebraically closed valued fields with a non-trivial valuation) are generically stable. If the residue field is stable, we show that an $\A$-invariant type concentrated on the field sort is orthogonal to the value group if and only if it is generically stable. If the residue field is simple, the theory of the valued field is $\NTP_{2}$ and algebraically closed sets of imaginary parameters are extension basis, we show that an $\A$-invariant type concentrated on the field sort is orthogonal to the value group if and only if it is generically simple. Examples of the simple case, among others, include the limit theory $\VFA_{0}$ of the Frobenius automorphism acting on an algebraically closed valued field of characteristic $\p$ (where $\p$ tends to infinity), as well as non-principal ultraproducts of the $\p$-adics.

In this paper we present a unifying framework of residual domination for henselian valued fields of equicharacteristic zero, encompassing some valued fields with operators. We introduce a general definition for residual domination and show that it is well behaved. For instance, we prove a change of base theorem for residual domination over algebraically closed sets. We show that the class of residually dominated types coincides with the types that are orthogonal to the value group, and with the class of types whose reduct to $\ACVF$ (the theory of algebraically closed valued fields with a non-trivial valuation) are generically stable. If the residue field is stable, we show that an $\A$-invariant type concentrated on the field sort is orthogonal to the value group if and only if it is generically stable. If the residue field is simple, the theory of the valued field is $\NTP_{2}$ and algebraically closed sets of imaginary parameters are extension basis, we show that an $\A$-invariant type concentrated on the field sort is orthogonal to the value group if and only if it is generically simple. Examples of the simple case, among others, include the limit theory $\VFA_{0}$ of the Frobenius automorphism acting on an algebraically closed valued field of characteristic $\p$ (where $\p$ tends to infinity), as well as non-principal ultraproducts of the $\p$-adics.

The low-Reynolds-number Stokes flow driven by rotation of two parallel cylinders of equal unit radius is investigated by both analytical and numerical techniques. In Part I, the case of counterrotating cylinders is considered. A numerical (finite-element) solution is obtained by enclosing the system in an outer cylinder of radius R₀ >>1, on which the no-slip condition is imposed. A model problem with the same symmetries is first solved exactly, and the limit of validity of the Stokes approximation is determined; this model has some relevance for ciliary propulsion. For the twocylinder problem, attention is focused on the small-gap situation ε ≪ 1. An exact analytic solution is obtained in the contact limit ε = 0, and a net force F_c acting on the pair of cylinders in this contact limit is identified; this contributes to the torque that each cylinder experiences about its axis. The far-field torque doublet ('torquelet') is also identified. Part II treats the case of co-rotating cylinders, for which again a finite-element numerical solution is obtained for R₀ >>1. The theory of Watson (1995, Mathematika, 42, 105-126) is elucidated and shown to agree well with the numerical solution. In contrast to the counter-rotating case, inertia effects are negligible throughout the fluid domain, however large, provided R_e << 1. In the concluding section, the main results for both cases are summarised, and the situation when the fluid is unbounded (R₀ = ∞) is discussed. If the cylinders are free to move (while rotating about their axes), in the counter-rotating case they will then translate relative to the fluid at infinity with constant velocity, the drag force exactly compensating the self-induced force due to the counterrotation. In the co-rotating case, if the cylinders are free to move, they will rotate as a pair relative to the fluid at infinity and the net torque on the cylinder pair is zero; the flow relative to the fluid at infinity is identified as a 'radial quadrupole'. If, on the other hand, the cylinder axes are held fixed, the Stokes flow in the counter-rotating case extends only for a distance r ∼ Re^-1 log [Re^-1] from the cylinders, and it is argued that the cylinders then experience a (dimensionless) force Fy ∼ 1/ log [Re^-1 log [Re^-1]]; in the co-rotating case, the cylinder pair experiences a (dimensionless) torque T , which tends to 17.2587 as ε ↓ 0; this torque is associated with a vortex-type flow ∼ r^-1 that is established in the far field. Situations that can be described by the condition ε < 0 are treated for both counter-and corotating cases in the supplementary material.

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 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 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 extend the results about Hensel minimality to include also the mixed characteristic case. This completes our axiomatic framework for tame non-archimedean geometry over Henselian valued fields of characteristic zero. We prove a Jacobian property, a strong form of Taylor approximations of definable functions, resplendency results and cell decomposition, all under 1-h-minimality. We obtain a diophantine application of counting rational points of bounded height on Hensel minimal curves.

We present a framework for tame geometry on Henselian valued fields which we call Hensel minimality. In the spirit of o-minimality, which is key to real geometry and several diophantine applications, we develop geometric results and applications for Hensel minimal structures that were previously known only under stronger, less axiomatic assumptions. We show existence of t-stratifications in Hensel minimal structures and Taylor approximation results which are key to non-archimedean versions of Pila-Wilkie point counting, Yomdin's parameterization results and to motivic integration. In this first paper we work in equi-characteristic zero; in the sequel paper, we develop the mixed characteristic case and a diophantine application.

We present a framework for tame geometry on Henselian valued fields which we call Hensel minimality. In the spirit of o-minimality, which is key to real geometry and several diophantine applications, we develop geometric results and applications for Hensel minimal structures that were previously known only under stronger, less axiomatic assumptions. We show existence of t-stratifications in Hensel minimal structures and Taylor approximation results which are key to non-archimedean versions of Pila-Wilkie point counting, Yomdin's parameterization results and to motivic integration. In this first paper we work in equi-characteristic zero; in the sequel paper, we develop the mixed characteristic case and a diophantine application.

Abstract The existence of bound states for the magnetic Laplacian in unbounded domains can be quite challenging in the case of a homogeneous magnetic field. We provide an affirmative answer for almost flat corners and slightly curved half‐planes when the total curvature of the boundary is positive.

We introduce a weak division-like property for noncommutative rings: a nontrivial ring is fadelian if for all nonzero a, x there exist b, c such that x=ab+ca. We prove properties of fadelian rings, and construct examples of such rings which are not division rings, as well as non-Noetherian and non-Ore examples. Some of these results are formalized using the Lean proof assistant.

Most theoretical and numerical results regarding water waves rely on the irrotationality of the flow in order to simplify significantly the mathematical formulation. This assumption is rarely discussed in details. In this work we investigate the well-foundedness of this important hypothesis using numerical simulations of the free-surface Navier-Stokes equation, using a scheme introduced in Riquier and Dormy (2024). We show that, in the presence of an irregular bottom, a gravity wave of non-negligible height can effectively destabilise the bottom boundary layer, stripping off vortices into the main flow. As a vortex approaches the surface the solution of the Navier-Stokes flow in the limit of vanishing viscosity is shown to differ from the inviscid Euler solution.

We prove a formula which gives the number of occurrences of certain labels and local configurations inside two-step puzzles introduced by Buch, Kresch, Purbhoo and Tamvakis from the work of Knutson. Puzzles are tilings of the triangular lattice by edge labeled tiles and are known to compute the Schubert structure constants of the cohomology of two-step flag varieties. The formula that we obtain depends only on the boundary conditions of the puzzle. The proof is based on the study of color maps which are tilings of the triangular lattice by edge labeled tiles obtained from puzzles.

We prove a formula which gives the number of occurrences of certain labels and local configurations inside two-step puzzles introduced by Buch, Kresch, Purbhoo and Tamvakis from the work of Knutson. Puzzles are tilings of the triangular lattice by edge labeled tiles and are known to compute the Schubert structure constants of the cohomology of two-step flag varieties. The formula that we obtain depends only on the boundary conditions of the puzzle. The proof is based on the study of color maps which are tilings of the triangular lattice by edge labeled tiles obtained from puzzles.

Pursuing ideas in a recent work of the second author, we determine the isometry classes of unimodular lattices of rank 28, as well as the isometry classes of unimodular lattices of rank 29 without nonzero vectors of norm <=2.

We develop a method initiated by Bacher and Venkov, and based on a study of the Kneser neighbors of the standard lattice Z^n , which allows to classify the integral unimodular Euclidean lattices of rank n. As an application, of computational flavour, we determine the isometry classes of unimodular lattices of rank 26 and 27.

In this paper, we provide the first rigorous derivation of hydrodynamic equations from the Boltzmann equation for inelastic hard spheres with small inelasticity. The hydrodynamic system that we obtain is an incompressible Navier-Stokes-Fourier system with self-consistent forcing terms and, to our knowledge, it is thus the first hydrodynamic system that properly describes rapid granular flows consistent with the kinetic formulation. To this end, we write our Boltzmann equation in a non dimensional form using the dimensionless Knudsen number which is intended to be sent to $0$. There are several difficulties in such derivation, the first one coming from the fact that the original Boltzmann equation is free-cooling and, thus, requires a self-similar change of variables to introduce an homogeneous steady state. Such a homogeneous state is not explicit and is heavy-tailed, which is a major obstacle to adapting energy estimates and spectral analysis. Additionally, a central challenge is to understand the relation between the restitution coefficient, which quantifies the energy loss at the microscopic level, and the Knudsen number. This is achieved by identifying the correct nearly elastic regime to capture nontrivial hydrodynamic behavior. We are, then, able to prove exponential stability uniformly with respect to the Knudsen number for solutions of the rescaled Boltzmann equation in a close to equilibrium regime. Finally, we prove that solutions to the Boltzmann equation converge in a specific weak sense towards a hydrodynamic limit which depends on time and space variables only through macroscopic quantities. Such macroscopic quantities are solutions to a suitable modification of the incompressible Navier-Stokes-Fourier system which appears to be new in this context.

In this paper, we provide the first rigorous derivation of hydrodynamic equations from the Boltzmann equation for inelastic hard spheres with small inelasticity. The hydrodynamic system that we obtain is an incompressible Navier-Stokes-Fourier system with self-consistent forcing terms and, to our knowledge, it is thus the first hydrodynamic system that properly describes rapid granular flows consistent with the kinetic formulation. To this end, we write our Boltzmann equation in a non dimensional form using the dimensionless Knudsen number which is intended to be sent to $0$. There are several difficulties in such derivation, the first one coming from the fact that the original Boltzmann equation is free-cooling and, thus, requires a self-similar change of variables to introduce an homogeneous steady state. Such a homogeneous state is not explicit and is heavy-tailed, which is a major obstacle to adapting energy estimates and spectral analysis. Additionally, a central challenge is to understand the relation between the restitution coefficient, which quantifies the energy loss at the microscopic level, and the Knudsen number. This is achieved by identifying the correct nearly elastic regime to capture nontrivial hydrodynamic behavior. We are, then, able to prove exponential stability uniformly with respect to the Knudsen number for solutions of the rescaled Boltzmann equation in a close to equilibrium regime. Finally, we prove that solutions to the Boltzmann equation converge in a specific weak sense towards a hydrodynamic limit which depends on time and space variables only through macroscopic quantities. Such macroscopic quantities are solutions to a suitable modification of the incompressible Navier-Stokes-Fourier system which appears to be new in this context.

We consider a system of diffusion processes interacting through their empirical distribution. Assuming that the empirical average of a given observable can be observed at any time, we derive regularity and quantitative stability results for the optimal solutions in the associated version of the Gibbs conditioning principle. The proofs rely on the analysis of a McKean-Vlasov control problem with distributional constraints. Some new estimates are derived for Hamilton-Jacobi-Bellman equations and the Hessian of the log-density of diffusion processes, which are of independent interest.

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.

These notes focus on the minimization of convex functionals using first-order optimization methods, which are fundamental in many areas of applied mathematics and engineering. The primary goal of this document is to introduce and analyze the most classical first-order optimization algorithms. We aim to provide readers with both a practical and theoretical understanding in how and why these algorithms converge to minimizers of convex functions. The main algorithms covered in these notes include gradient descent, Forward-Backward splitting, Douglas-Rachford splitting, the Alternating Direction Method of Multipliers (ADMM), and Primal-Dual algorithms. All these algorithms fall into the class of first order methods, as they only involve gradients and subdifferentials, that are first order derivatives of the functions to optimize. For each method, we provide convergence theorems, with precise assumptions and conditions under which the convergence holds, accompanied by complete proofs. Beyond convex optimization, the final part of this manuscript extends the analysis to nonconvex problems, where we discuss the convergence behavior of these same first-order methods under broader assumptions. To contextualize the theory, we also include a selection of practical examples illustrating how these algorithms are applied in different image processing problems.

We extend the Gibbs conditioning principle to an abstract setting combining infinitely many linear equality constraints and non-linear inequality constraints, which need not be convex. A conditional large large deviation principle (LDP) is proved in a Wassersteintype topology, and optimality conditions are written in this abstract setting. This setting encompasses versions of the Schrödinger bridge problem with marginal non-linear inequality constraints at every time. In the case of convex constraints, stability results for perturbations both in the constraints and the reference measure are proved. We then specify our results when the reference measure is the path-law of a continuous diffusion process, whose law is constrained at each time. We obtain a complete description of the constrained process through an atypical mean-field PDE system involving a Lagrange multiplier.

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.

It was proved by Bahouri et al. [9] that the Schrödinger equation on the Heisenberg group Hd, involving the sublaplacian, is an example of a totally non-dispersive evolution equation: for this reason global dispersive estimates cannot hold. This paper aims at establishing local dispersive estimates on Hd for the linear Schrödinger equation, by a refined study of the Schrödinger kernel St on Hd. The sharpness of these estimates is discussed through several examples. Our approach, based on the explicit formula of the heat kernel on Hd derived by Gaveau [19], is achieved by combining complex analysis and Fourier-Heisenberg tools. As a by-product of our results we establish local Strichartz estimates and prove that the kernel St concentrates on quantized horizontal hyperplanes of Hd.

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.