Publications

Many tasks in Machine Learning, or other fields such as physics or economics, amount to the comparison between a data distribution and a model, called density fitting. This problem is formulated as the minimization of a distance between these two measures, and optimal transport distances (OT) appear as a powerful tool for this task, since they leverage an underlying geometric information in the data. However, their practical application is challenging, because they are computationally intensive to solve, sensitive to noisy data and can only compare probabilities defined on the same space. To cope with each one of these issues, variants of OT called entropically regularized, unbalanced and Gromov-Wasserstein have been proposed in the literature. The scope of this thesis is to develop combinations of those variants so as to add priors of robustness to noise and/or an ability to handle change of spaces when comparing measures, while remaining computationally tractable. Thus the contributions of this thesis focus on providing algorithms and implementations with guarantees that the computational burden of OT variants is alleviated, and proving metric properties to ensure the comparison of distributions with respect to those variants remains insightful.

We show the convergence of the characteristic polynomial for random permutation matrices sampled from the generalized Ewens distribution. Under this distribution, the measure of a given permutation depends only on its cycle structure, according to certain weights assigned to each cycle length. The proof is based on uniform control of the characteristic polynomial using results from the singularity analysis of generating functions, together with the convergence of traces to explicit random variables expressed via a Poisson family. The limit function is the exponential of a Poisson series which has already appeared in the case of uniform permutation matrices. It is the Poisson analog of the Gaussian Holomorphic Chaos, related to the limit of characteristic polynomials for other matrix models such as Circular Ensembles, i.i.d. matrices, and Gaussian elliptic matrices.

We study the discretization, convergence, and numerical implementation of recent reformulations of the quadratic porous medium equation (multidimensional and anisotropic) and Burgers' equation (one-dimensional, with optional viscosity), as forward in time variants of the Benamou-Brenier formulation of optimal transport. This approach turns those evolution problems into global optimization problems in time and space, of which we introduce a discretization, one of whose originalities lies in the harmonic interpolation of the densities involved. We prove that the resulting schemes are unconditionally stable w.r.t. the space and time steps, and we establish a quadratic convergence rate for the dual PDE solution, under suitable assumptions. We also show that the schemes can be efficiently solved numerically using a proximal splitting method and a global space-time fast Fourier transform, and we illustrate our results with numerical experiments.

We study the second Huber Theorem in dimensions 2 and 4. In dimension 2, we prove the optimal regularity for the conformal factor using Coulomb frames. In dimension 4, we introduce another Coulomb-type condition which is similar to the case of Yang--Mills connections. We obtain a generalization of the two-dimensional case that can be applied to study the singularities of Bach-flat metrics.

We study the martingale property and moment explosions of a signature volatility model, where the volatility process of the log-price is given by a linear form of the signature of a time-extended Brownian motion. Excluding trivial cases, we demonstrate that the price process is a true martingale if and only if the order of the linear form is odd and a correlation parameter is negative. The proof involves a fine analysis of the explosion time of a signature stochastic differential equation. This result is of key practical relevance, as it highlights that, when used for approximation purposes, the linear combination of signature elements must be taken of odd order to preserve the martingale property. Once martingality is established, we also characterize the existence of higher moments of the price process in terms of a condition on a correlation parameter.

This thesis studies the persistent homology of real-valued continuous functions f on compact topological spaces X. The introduction of homological indices and homological dimensions allows us to link persistence theory to metric quantities of the compact space X, such as its upper-box dimension. These quantities give a precise framework to the Wasserstein p-stability results known in the literature, but also extend them to Hölder functions on more general spaces (including all compact Riemannian manifolds) with explicit constants and whose regime for p is optimal. In degree zero of homology, a more in-depth study can be made using trees associated to f, which generalize the merge trees which are definable when f is Morse. It is possible to link the dimension of these trees to the persistence index of f and to its barcode. We apply these deterministic results to the stochastic setting to draw consequences about the barcodes of random functions of prescribed regularity. These consequences also allow us to develop distributional discrimination tests for the processes, of which we present a particular example. Finally, we define the zeta-functions associated with a stochastic process and compute these functions and other related quantities for several processes in dimension one, including the Brownian motion and the alpha-stable Lévy processes.

This thesis studies the persistent homology of real-valued continuous functions f on compact topological spaces X. The introduction of homological indices and homological dimensions allows us to link persistence theory to metric quantities of the compact space X, such as its upper-box dimension. These quantities give a precise framework to the Wasserstein p-stability results known in the literature, but also extend them to Hölder functions on more general spaces (including all compact Riemannian manifolds) with explicit constants and whose regime for p is optimal. In degree zero of homology, a more in-depth study can be made using trees associated to f, which generalize the merge trees which are definable when f is Morse. It is possible to link the dimension of these trees to the persistence index of f and to its barcode. We apply these deterministic results to the stochastic setting to draw consequences about the barcodes of random functions of prescribed regularity. These consequences also allow us to develop distributional discrimination tests for the processes, of which we present a particular example. Finally, we define the zeta-functions associated with a stochastic process and compute these functions and other related quantities for several processes in dimension one, including the Brownian motion and the alpha-stable Lévy processes.

The cutoff phenomenon, conceptualized at the origin for finite Markov chains, states that for a parametric family of evolution equations, started from a point, the distance towards a long time equilibrium may become more and more abrupt for certain choices of initial conditions, when the parameter tends to infinity. This threshold phenomenon can be seen as a critical competition between trend to equilibrium and worst initial condition. In this note, we investigate this phenomenon beyond stochastic processes, in the context of the analysis of nonlinear partial differential equations, by proving cutoff for the fast diffusion and porous medium Fokker-Planck equations on the Euclidean space, when the dimension tends to infinity. We formulate the phenomenon using quadratic Wasserstein distance, as well as using specific relative entropy and Fisher information. Our high dimensional asymptotic analysis uses the exact solvability of the model involving Barenblatt profiles. It includes the Ornstein-Uhlenbeck dynamics as a special linear case.

Ce recueil regroupe les traductions en français des textes officiels de présentation des sept problèmes mathématiques du millénaire. Il donne donc un petit aperçu des mathématiques contemporaines.

The initial aim of CroMoSim was to enable reproducibility of the results present in the book “Crowds in equations: an introduction to the microscopic modeling of crowds” published in 2018 by Bertrand Maury and Sylvain Faure, and to provide a starting point for people wishing to model crowd movements. The aim was to propose several mathematical models, as well as common tools enabling the use of complex building plans. The software is currently available as a package written in Python, which can be installed via PyPI (pip). The complete code is available on Github (\href{https://github.com/sylvain-faure/cromosim}{cromosim}), and several examples of use are available. Cromosim enables calculations to be made on "real" building plans, by defining a color code characterizing the destinations of individuals, walls or furniture... Several microscopic mathematical models of crowd movements can thus be used: social force or granular-inspired models, cellular automata, compartment models. The initial target audience was mainly students and researchers, as all the "ingredients" of the models are accessible and modifiable.

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 the interest of reproducible research, this is exactly the version of the code used in the paper "Abide by the Law and Follow the Flow: Conservation Laws for Gradient Flows" by the same authors, available at https://inria.hal.science/hal-04150576 with its detailed bibliographical notice. Any updates to this code will be available at https://github.com/sibyllema/Conservation_laws.

In this paper, we investigate the properties of the Sliced Wasserstein Distance (SW) when employed as an objective functional. The SW metric has gained significant interest in the optimal transport and machine learning literature, due to its ability to capture intricate geometric properties of probability distributions while remaining computationally tractable, making it a valuable tool for various applications, including generative modeling and domain adaptation. Our study aims to provide a rigorous analysis of the critical points arising from the optimization of the SW objective. By computing explicit perturbations, we establish that stable critical points of SW cannot concentrate on segments. This stability analysis is crucial for understanding the behaviour of optimization algorithms for models trained using the SW objective. Furthermore, we investigate the properties of the SW objective, shedding light on the existence and convergence behavior of critical points. We illustrate our theoretical results through numerical experiments.

Turbulent cascades characterize the transfer of energy injected by a random force at large scales towards the small scales. We construct a linear equation that mimics the phenomenology of energy cascades when the external force is a statistically homogeneous and stationary stochastic process. In the Fourier variable, this equation can be seen as a wave equation, which corresponds to a wave operator of degree 0 in physical space. Our results give a complete characterization of the solution: it is smooth at any finite time, and, up to smaller order corrections, it converges to a fractional Gaussian field at infinite time. We apply a finite volume method in the Fourier variables formulation in order to reach the invariant measure of the equation.

Dans cette note, nous expliquons la construction de variétés de Hecke p-adiques associées aux groupes unitaires définis d'un corps de nombres CM et décrivons des propriétés de la famille de représentations galoisiennes portée par ces variétés. Nous donnons une application à la construction de certaines représentations galoisiennes qui joue un rôle important dans cette série de livres. Les variétés de Hecke ci-dessus sont aussi utilisées dans [BCh2]. La construction des variétés de Hecke que nous donnons est une combinaison essentiellement triviale des méthodes de [Ch] (cas d'un corps quadratique imaginaire, revisité dans [BCh, §7]) et de [Bu] (cas d'une algèbre de quaternions sur un corps totalement réel, voir aussi [Y]). Elle pourrait aussi se déduire comme cas particulier des travaux d'Emerton [E], et elle est aussi contenue dans les travaux récents de Loeffler [Loe]. En ce qui concerne les propriétés galoisiennes, nous utilisons et étendons certains résultats de [BCh, §7].

We develop a functional analytic approach to the study of the Kramers and kinetic Fokker-Planck equations which parallels the classical $H^1$ theory of uniformly elliptic equations. In particular, we identify a function space analogous to $H^1$ and develop a well-posedness theory for weak solutions in this space. In the case of a conservative force, we identify the weak solution as the minimizer of a uniformly convex functional. We prove new functional inequalities of Poincaré and Hörmander type and combine them with basic energy estimates (analogous to the Caccioppoli inequality) in an iteration procedure to obtain the~$C^\infty$ regularity of weak solutions. We also use the Poincaré-type inequality to give an elementary proof of the exponential convergence to equilibrium for solutions of the kinetic Fokker-Planck equation which mirrors the classic dissipative estimate for the heat equation. Finally, we prove enhanced dissipation in a weakly collisional limit.

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

To a bicomplex one can associate two natural filtrations, the column and row filtrations, and then two associated spectral sequences. This can be generalized to $N$-multicomplexes. We present a family of model category structures on the category of $N$-multicomplexes where the weak equivalences are the morphisms inducing a quasi-isomorphism at a fixed page $r$ of the first spectral sequence and at a fixed page $s$ of the second spectral sequence. Such weak equivalences arise naturally in complex geometry. In particular, the model structures presented here establish a basis for studying homotopy types of almost and generalized complex manifolds.

We study homotopy theory of the category of spectral sequences with respect to the class of weak equivalences given by maps which are quasi-isomorphisms on a fixed page. We introduce the category of extended spectral sequences and show that this is bicomplete by analysis of a certain linear presheaf category modelled on discs. We endow the category of extended spectral sequences with various model category structures, restricting to give the almost Brown category structures on spectral sequences of our earlier work. One of these has the property that spectral sequences is a homotopically full subcategory. By results of Meier, this exhibits the category of spectral sequences as a fibrant object in the Barwick-Kan model structure on relative categories, that is, it gives a model for an infinity category of spectral sequences. We also use the presheaf approach to define two décalage functors on spectral sequences, left and right adjoint to a shift functor, thereby clarifying prior use of the term décalage in connection with spectral sequences.

We show that the momentum, the density, and the electromagnetic field associated with the massive Klein-Gordon-Maxwell equations converge in the semi-classical limit towards their respective equivalents associated with the relativistic Euler-Maxwell equations. The proof relies on a modulated stress-energy method and a compactness argument. We also give a proof of the well-posedness of the relativistic Euler-Maxwell equations and show how this system, and so the semi-classical limit of Klein-Gordon-Maxwell, is related to the relativistic massive Vlasov-Maxwell equations.

We show that the momentum, the density, and the electromagnetic field associated with the massive Klein-Gordon-Maxwell equations converge in the semi-classical limit towards their respective equivalents associated with the relativistic Euler-Maxwell equations. The proof relies on a modulated stress-energy method and a compactness argument. We also give a proof of the well-posedness of the relativistic Euler-Maxwell equations and show how this system, and so the semi-classical limit of Klein-Gordon-Maxwell, is related to the relativistic massive Vlasov-Maxwell equations.

This article studies the problem of estimating the state variable of non-smooth subdifferential dynamics constrained in a bounded convex domain given some real-time observation. On the one hand, we show that the value function of the estimation problem is a viscosity solution of a Hamilton Jacobi Bellman equation whose sub and super solutions have different Neumann type boundary conditions. This intricacy arises from the non-reversibility in time of the non-smooth dynamics, and hinders the derivation of a comparison principle and the uniqueness of the solution in general. Nonetheless, we identify conditions on the drift (including zero drift) coefficient in the non-smooth dynamics that make such a derivation possible. On the other hand, we show in a general situation that the value function appears in the small noise limit of the corresponding stochastic filtering problem by establishing a large deviation result. We also give quantitative approximation results when replacing the non-smooth dynamics with a smooth penalised one.

Benchmarks are important tools for tracking the rapid advancements in large language model (LLM) capabilities. However, benchmarks are not keeping pace in difficulty: LLMs now achieve over 90% accuracy on popular benchmarks like MMLU, limiting informed measurement of state-of-the-art LLM capabilities. In response, we introduce Humanity's Last Exam (HLE), a multi-modal benchmark at the frontier of human knowledge, designed to be the final closed-ended academic benchmark of its kind with broad subject coverage. HLE consists of 3,000 questions across dozens of subjects, including mathematics, humanities, and the natural sciences. HLE is developed globally by subject-matter experts and consists of multiple-choice and short-answer questions suitable for automated grading. Each question has a known solution that is unambiguous and easily verifiable, but cannot be quickly answered via internet retrieval. State-of-the-art LLMs demonstrate low accuracy and calibration on HLE, highlighting a significant gap between current LLM capabilities and the expert human frontier on closed-ended academic questions. To inform research and policymaking upon a clear understanding of model capabilities, we publicly release HLE at https://lastexam.ai.

Python Programming for Mathematics focuses on the practical use of the Python language in a range of different areas of mathematics. Through fifty-five exercises of increasing difficulty, the book provides an expansive overview of the power of using programming to solve complex mathematical problems. This book is intended for undergraduate and graduate students who already have learned the basics of Python programming and would like to learn how to apply that programming skill in mathematics.

Python est un langage de programmation phare dans le monde scientifique. Il est parfaitement adapté pour programmer des problèmes mathématiques. Cet ouvrage est consacré à l’utilisation pratique du langage Python dans différents domaines des mathématiques : les suites, l’algèbre linéaire, l’intégration, la théorie des graphes, le calcul symbolique, la recherche de zéros de fonctions, les probabilités, les statistiques, les équations différentielles, la science des données et la théorie des nombres.Cette nouvelle édition augmentée de chapitres sur la science des données et l’intelligence artificielle dresse un panorama des applications de la programmation en Python dans les mathématiques. À travers 55 exercices de difficulté croissante et corrigés en détail, vous pourrez acquérir les compétences nécessaires pour résoudre des problèmes complexes. Les codes sources de l’ouvrage sont disponibles en ligne.

In this thesis, we study the problem of extending a Morse function on a closed manifold to a Lefschetz fibration on the cotangent bundle of said manifold, as well as certain Floer-theoretic questions related to this fibration. In the first chapter, we present the extension theorem due to E. Giroux, we extend it to the real-analytic setting and prove a theorem in the local model on the result of parallel transport along the unit semi-circle. In chapters 2 and 3, we re-examine a conjecture of P. Seidel in this context, which establishes a relation between the flow category of the Morse function and the directed Donaldson-Fukaya category of the Lefschetz fibration, particularly in dimension 2, resp. 3. In the last chapter, we give a negative answer to a natural question of E. Giroux, on a relation between the Heegaard-Floer homology of a 3-manifold and the Lefschetz fibration which extends a Morse function which induces a Heegaard diagram.

We study the Maximum Zero-Sum Partition problem (or MZSP), defined as follows: given a multiset S={a1,a2,…,an} of integers ai∈Z⁎ (where Z⁎ denotes the set of non-zero integers) such that ∑i=1nai=0, find a maximum cardinality partition {S1,S2,…,Sk} of S such that, for every 1≤i≤k, ∑aj∈Siaj=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 consider Riesz energy problems with radial external fields. We study the question of whether or not the equilibrium is the uniform distribution on a sphere. We develop general necessary as well as general sufficient conditions on the external field that apply to powers of the Euclidean norm as well as certain Lennard--Jones type fields. Additionally, in the former case, we completely characterize the values of the power for which dimension reduction occurs in the sense that the support of the equilibrium measure becomes a sphere. We also briefly discuss the relation between these problems and certain constrained optimization problems. Our approach involves the Frostman characterization, the Funk--Hecke formula, and the calculus of hypergeometric functions.

It is known that a cutoff phenomenon occurs in high dimension for certain positively curved overdamped Langevin diffusions in the Euclidean space, including the Ornstein-Uhlenbeck process and its Dyson version. In this note, we provide a structural explanation of this phenomenon, and we extend the result to a wide class of non-Gaussian and non-product models with a convex interaction. The key observation is a relation to a spectral rigidity result of Cheng and Zhou, linked to the presence of a Gaussian factor. We formulate the phenomenon using a Wasserstein coupling distance, and we deduce from it the formulation for total variation distance and relative entropy divergence. Furthermore, we discuss a natural extension to Riemannian manifolds, and ask about a possible extension or stability by perturbation.

This work is concerned with the appearance of decay bounds in the velocity variable for solutions of the space-inhomogeneous Boltzmann equation without cutoff posed in a domain in the case of hard and moderately soft potentials. Such bounds are derived for general non-negative suitable weak subsolutions. These estimates hold true as long as mass, energy and entropy density functions are under control. The following boundary conditions are treated: in-flow, bounce-back, specular reflection, diffuse reflection and Maxwell reflection. The proof relies on a family of Truncated Convex Inequalities that is inspired by the one recently introduced by F. Golse, L. Silvestre and the first author (2023). To the best of our knowledge, the generation of arbitrary polynomial decay in the velocity variable for the Boltzmann equation without cutoff is new in the case of soft potentials, even for classical solutions.

We prove an identity in integral geometry, showing that if $P_x$ and $Q_x$ are two polynomials, $\int dx \, \delta(P_x) \otimes \delta(Q_x)$ is proportional to $\delta(R)$ where $R$ is the resultant of $P_x$ and $Q_x$.

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

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 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 study and classify the emergence of protected edge modes at the junction of one-dimensional materials. Using symmetries of Lagrangian planes in boundary symplectic spaces, we present a novel proof of the periodic table of topological insulators in one dimension. We show that edge modes necessarily arise at the junction of two materials having different topological indices. Our approach provides a systematic framework for understanding symmetry-protected modes in one-dimension. It does not rely on periodic nor ergodicity and covers a wide range of operators which includes both continuous and discrete models.

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.

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 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 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.

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.