Publications

As context windows in large language models continue to expand, it is essential to characterize how attention behaves at extreme sequence lengths. We introduce token-sample complexity: the rate at which attention computed on $n$ tokens converges to its infinite-token limit. We estimate finite-$n$ convergence bounds at two levels: pointwise uniform convergence of the attention map, and convergence of moments for the transformed token distribution. For compactly supported (and more generally sub-Gaussian) distributions, our first result shows that the attention map converges uniformly on a ball of radius $R$ at rate $C(R)/\sqrt{n}$, where $C(R)$ grows exponentially with $R$. For large $R$, this estimate loses practical value, and our second result addresses this issue by establishing convergence rates for the moments of the transformed distribution (the token output of the attention layer). In this case, the rate is $C'(R)/n^β$ with $β<\tfrac{1}{2}$, and $C'(R)$ depends polynomially on the size of the support of the distribution. The exponent $β$ depends on the attention geometry and the spectral properties of the tokens distribution. We also examine the regime in which the attention parameter tends to infinity and the softmax approaches a hardmax, and in this setting, we establish a logarithmic rate of convergence. Experiments on synthetic Gaussian data and real BERT models on Wikipedia text confirm our predictions.

The work presented here is concerned with breaking water waves, a well-known phenomenon arising as an oceanic wave approaches the shore: its crest starts to move faster than the trough up front, which ultimately leads to the appearance of an overhanging region that quickly curls over while falling down until it collides with the water lying below. An important contemporary issue concerns the incorporation of the viscous dissipation associated with the breaking into the many models that have been introduced to describe the ocean. This is mostly done empirically. In the present work, we follow a different path: we aim at modelling wave breaking up to the free surface self-intersection (the splash singularity), relying thus on a more geometrical approach to the subject. The first part of this thesis will be devoted to the motivation of a set of equations that describes overhanging waves in the inviscid irrotational regime, with either a one-dimensional or a two-dimensional free surface. This is done by setting aside the commonly used Eulerian framework and working in (pseudo)Lagrangian coordinates instead. This should be seen as an extension of the Zakharov-Craig-Sulem formulation of the Water Waves problem. The non-canonical Hamiltonian structure of these partial differential equations is investigated and it is shown that in the absence of breaking, they can be reduced to the usual set of equations. Emphasis is put on the various physical assumptions that are made along the way. In a second moment, we come back to these very hypotheses and put them to the test. This is done numerically using a Navier-Stokes based computational framework based on the Finite-Element Method (FEM). The major novelty compared to other studies lies in the use of the Arbitrary Lagrangian-Eulerian method (ALE), which diminishes the interpolation error greatly. The viscosity can therefore be decreased to values that allow the comparison with the inviscid solution (computed using another method, based on potential theory in the complex plane) to be carried out. Over a flat topography, it is found that both the free-surface and bed boundary layers are sufficiently well-behaved as to not perturb the bulk irrotational flow. Water being characterised by a relatively small viscosity, the consequence is that, in this regime the inviscid models accurately describe the oceanic flow. We do not prove this assertion rigorously, however. Difficulties seem to arise, however, when a non-flat topography is considered. Indeed, the typical velocities associated with the wave are high enough to eventually trigger boundary layer separation near curved-enough portions of the bed, resulting in vorticity being shed in the initially irrotational flow, far from the topography. The convergence to the inviscid solution is therefore compromised.

We classify the unimodular Euclidean integral lattices of rank 29 by developing an elementary, yet very efficient, inductive method. As an application, we determine the isometry classes of even lattices of rank at most 28 and prime (half-)determinant at most 7. We also provide new isometry invariants allowing for independent verification of the completeness of our lists, and we give conceptual explanations of some unique orbit phenomena discovered during our computations. Some of the genera classified here are orders of magnitude larger than any genus previously classified. In a forthcoming companion paper, we use these computations to study the cohomology of GL_n(Z).

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 first revisit these results in the context of Sobolev spaces modelled on L^2 and then explore the endpoint Besov case B_{p,1}^{d/p}. We also exemplify our method on the SKT system, showing the existence of local, non-negative, strong solutions.

We prove a bumpy metric theorem in the sense of Ma\~{n}e for non-convex Hamiltonians that are satisfying a certain geometric property.

We construct self-similar solutions to the 2D Navier--Stokes equations evolving from arbitrarily large $-1$--homogeneous initial data and present numerical evidence for their non-uniqueness.

A fundamental challenge in the theory of deep learning is to understand whether gradient-based training in high-dimensional parameter spaces can be captured by simpler, lower-dimensional structures, leading to so-called implicit bias. As a stepping stone, we study when a gradient flow on a high-dimensional variable $θ$ implies an intrinsic gradient flow on a lower-dimensional variable $z = ϕ(θ)$, for an architecture-related function $ϕ$. We express a so-called intrinsic dynamic property and show how it is related to the study of conservation laws associated with the factorization $ϕ$. This leads to a simple criterion based on the inclusion of kernels of linear maps which yields a necessary condition for this property to hold. We then apply our theory to general ReLU networks of arbitrary depth and show that, for any initialization, it is possible to rewrite the flow as an intrinsic dynamic in a lower dimension that depends only on $z$ and the initialization, when $ϕ$ is the so-called path-lifting. In the case of linear networks with $ϕ$ the product of weight matrices, so-called balanced initializations are also known to enable such a dimensionality reduction; we generalize this result to a broader class of {\em relaxed balanced} initializations, showing that, in certain configurations, these are the \emph{only} initializations that ensure the intrinsic dynamic property. Finally, for the linear neural ODE associated with the limit of infinitely deep linear networks, with relaxed balanced initialization, we explicitly express the corresponding intrinsic dynamics.

189 pages, comments welcome. Figures added. Numerous typos corrected. Contents of Sections 4.4, 6.2, 9.1 and 9.2 of Version 1 corrected and completed

This paper is devoted to the asymptotic analysis of strongly rotating and stratified fluids, under a $\beta$-plane approximation, and within a three-dimensional spatial domain with strong topography. Our purpose is to propose a linear idealized model, which is able to capture one of the key features of western boundary currents, in spite of its simplicity: the separation of the currents from the coast. Our simplified framework allows us to perform explicit computations, and to highlight the intricate links between rotation, stratification and bathymetry. In fact, we are able to construct approximate solutions at any order for our system, and to justify their validity. Each term in the asymptotic expansion is the sum of an interior part and of two boundary layer parts: a ``Munk'' type boundary layer, which is quasi-geostrophic, and an ``Ekman part'', which is not. Even though the Munk part of the approximation bears some similarity with previously studied 2D models, the analysis of the Ekman part is completely new, and several of its properties differ strongly from the ones of classical Ekman layers. Our theoretical analysis is supplemented with numerical illustrations, which exhibit the desired separation behavior.

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.

Relative entropy, as a divergence metric between two distributions, can be used for offline change-point detection and extends classical methods that mainly rely on moment-based discrepancies. To build a statistical test suitable for this context, we study the distribution of empirical relative entropy and derive several types of approximations: concentration inequalities for finite samples, asymptotic distributions, and Berry-Esseen bounds in a pre-asymptotic regime. For the latter, we introduce a new approach to obtain Berry-Esseen inequalities for nonlinear functions of sum statistics under some convexity assumptions. Our theoretical contributions cover both one-and two-sample empirical relative entropies. We then detail a change-point detection procedure built on relative entropy and compare it, through extensive simulations, with classical methods based on moments or on information criteria. Finally, we illustrate its practical relevance on two real datasets involving temperature series and volatility of stock indices.

In this thesis, we study interpretable groups and fields in various theories of enriched fields, using tools from geometric model theory. The work is divided into the following three parts. The group configuration theorem for generically stable types. Following the proof of the group configuration theorem in the usual stable setting, we generalize it to the case of generically stable group configurations in arbitrary theories. More explicitly, we show how one can construct a type-definable group (action) from a generically stable sextuple of points satisfying the usual algebraicity and independence properties of a group configuration. On groups and fields interpretable in NTP2 fields. We show that, in NTP2 theories of enriched fields, under mild model-theoretic and algebraic assumptions, any definably amenable interpretable group admits a definable morphism to an algebraic group with purely imaginary kernel, i.e. that does not admit definable maps to the field sort with infinite image. We deduce a structure theorem for interpretable fields, which we instantiate for henselian valued fields of characteristic 0. We also extend these results to NIP (possibly enriched) differential fields, and prove a full classification of interpretable fields for differentially closed valued fields. In passing, we prove that in arbitrary theories, if K and F are definable fields such that the group of affine transformations F+ ⋊ F × can be definably embedded into an algebraic group over K, then F admits a definable field embedding into a finite extension of the field K. On groups and fields definable in D-henselian fields. Finally, we focus on the theory of D-henselian valued fields with differentially closed residue field and divisible value group, studied by Scanlon and Rideau-Kikuchi. Adapting the proof of Hrushovski’s p-configuration theorem, we prove that groups definable in the valued field sort with generically stable generics orthogonal to all differentially algebraic types, admit definable group homomorphisms to alge- braic groups, with kernels of finite rank. We then show that any definable field, in the valued field sort, with a generating subring admitting such a generic, is definably isomorphic to the valued field itself, assuming its Kolchin closure is of infinite rank.

In this thesis, we study interpretable groups and fields in various theories of enriched fields, using tools from geometric model theory. The work is divided into the following three parts. The group configuration theorem for generically stable types. Following the proof of the group configuration theorem in the usual stable setting, we generalize it to the case of generically stable group configurations in arbitrary theories. More explicitly, we show how one can construct a type-definable group (action) from a generically stable sextuple of points satisfying the usual algebraicity and independence properties of a group configuration. On groups and fields interpretable in NTP2 fields. We show that, in NTP2 theories of enriched fields, under mild model-theoretic and algebraic assumptions, any definably amenable interpretable group admits a definable morphism to an algebraic group with purely imaginary kernel, i.e. that does not admit definable maps to the field sort with infinite image. We deduce a structure theorem for interpretable fields, which we instantiate for henselian valued fields of characteristic 0. We also extend these results to NIP (possibly enriched) differential fields, and prove a full classification of interpretable fields for differentially closed valued fields. In passing, we prove that in arbitrary theories, if K and F are definable fields such that the group of affine transformations F+ ⋊ F × can be definably embedded into an algebraic group over K, then F admits a definable field embedding into a finite extension of the field K. On groups and fields definable in D-henselian fields. Finally, we focus on the theory of D-henselian valued fields with differentially closed residue field and divisible value group, studied by Scanlon and Rideau-Kikuchi. Adapting the proof of Hrushovski’s p-configuration theorem, we prove that groups definable in the valued field sort with generically stable generics orthogonal to all differentially algebraic types, admit definable group homomorphisms to alge- braic groups, with kernels of finite rank. We then show that any definable field, in the valued field sort, with a generating subring admitting such a generic, is definably isomorphic to the valued field itself, assuming its Kolchin closure is of infinite rank.

We prove convergence of solutions of Dirichlet problems and Green's functions on Tutte's harmonic embeddings to those of the linearized Monge-Ampère equation $\mathcal{L}_φh=0$. The potential $φ$ appears as the limit of piecewise linear potentials associated with the embeddings and the only assumption that we use is the uniform convexity of $φ$. Even if $φ$ is quadratic, this setup significantly generalizes known results for discrete harmonic functions on orthodiagonal tilings. Motivated by potential applications to the analysis of 2d lattice models on irregular graphs, we also study the situation in which the limits are harmonic in a different complex structure.

We prove convergence of solutions of Dirichlet problems and Green's functions on Tutte's harmonic embeddings to those of the linearized Monge-Ampère equation $\mathcal{L}_φh=0$. The potential $φ$ appears as the limit of piecewise linear potentials associated with the embeddings and the only assumption that we use is the uniform convexity of $φ$. Even if $φ$ is quadratic, this setup significantly generalizes known results for discrete harmonic functions on orthodiagonal tilings. Motivated by potential applications to the analysis of 2d lattice models on irregular graphs, we also study the situation in which the limits are harmonic in a different complex structure.

Numerical experiments of dynamo action designed to understand the generation of Earth's magnetic field produce different regime branches identified within bifurcation diagrams. Notable are distinct branches where the resultant magnetic field is either weak or strong. Weak‐field solutions are identified by the prominent role of viscosity (and/or inertia) on the motion, whereas the magnetic field has a leading‐order effect on the flow in strong‐field solutions. We demonstrate the persistence of the strong‐field branch, preserving the expected force balance of Earth's core, and provide scaling laws governing its onset as parameters move toward values appropriate for the Geodynamo. We introduce a new output parameter, based on dynamically important parts of rotational and magnetic forces, that captures expected values of strong‐field solutions throughout input parameter space. This new measure of the field strength and our bounds on scaling laws can guide future studies in locating strong‐field dynamos in parameter space.

Recent numerical experiments of dynamo action relevant to the generation of the geomagnetic field have produced different regime branches identified within bifurcation diagrams. Notable are separate branches in which the resultant magnetic field is either weak or strong. Weak-field solutions can be identified by the prominent role of viscosity on the motion whereas the magnetic field has a leading order effect on the flow in strong-field solutions. For a given Ekman number, E (measuring the ratio of viscosity to rotational effects), the existence of these branches and bistability between them is reliant on a small enough magnetic Ekman number, E m (measuring the ratio of magnetic diffusion to rotational effects, so E / E m = P m , the magnetic Prandtl number). Both branches are known to produce large scale dipolar magnetic fields but do not exhibit an expected scale separation between the flow and magnetic field. In this work, by reducing E m , we identify a variety of dynamo states on the weak-field branch beyond the known dipolar solutions. Specifically, hemispherical and nondipolar dynamos were found, in addition to the usual dipolar solutions. Some solutions exhibit clear scale separation between small-scale flow and large-scale magnetic field, despite the large ratio of viscosity to magnetic diffusion. Numerical solutions in this regime have not been observed before and they offer a first connection with earlier theoretical work based on mean-field theory.

Convection is the main heat transport mechanism in the Earth's liquid core and is thought to power the dynamo that generates the geomagnetic field. Core convection is strongly constrained by rotation while being turbulent. Given the difficulty in modeling these conditions, some key properties of core convection are still debated, including the dominant energy‐carrying lengthscale. Different regimes of rapidly rotating, unmagnetized, turbulent convection exist depending on the importance of viscous and inertial forces in the dynamics, and hence different theoretical predictions for the dominant flow lengthscale have been proposed. Here we study the transition from viscously dominated to inertia‐dominated regimes using numerical simulations in spherical and planar geometries. We find that the cross‐over occurs when the inertial lengthscale approximately equals the viscous lengthscale. This suggests that core convection in the absence of magnetic fields is dominated by the inertial scale, which is hundred times larger than the viscous scale.

Nous proposons une exploration de la notion de distance, sous les regards croisés d'un mathématicien et d'un géographe.

Asymptotic solutions are investigated for the travelling wave consisting of infectives I ( x − c t ) propagating at speed c into a region of uninfected susceptibles S = S + , on the basis that S + is large. In the moving frame, three domains are identified. In the narrow leading frontal region, the infectives terminate relatively abruptly. Conditions ahead (increasing x ) of the front control the speed c of the front advance. In the trailing region (decreasing x ), the number of infectives decay relatively slowly. Our asymptotic development focuses on the dependence of I on S in the central region. Then, the apparently simple problem is complicated by the presence of both algebraic and logarithmic dependencies. Still, we can construct an asymptotic expansion to a high order of accuracy that embeds the trailing region solution. A proper solution in the frontal region is numerical, but here the central region solution works well too. We also investigated numerically the evolution from an initial state to a travelling wave. Following the decay of transients, the speed adopted by the wave is fast, though the slowest of those admissible. The asymptotic solutions are compared with the numerical solutions and display excellent agreement.

A classical approach to the Calderón problem is to estimate the unknown conductivity by solving a nonlinear least-squares problem. It leads to a nonconvex optimization problem which is generally believed to be riddled with bad local minimums. We revisit this issue in the case of piecewise constant radial conductivities and prove that, contrary to previous claims, there are no spurious critical points in the case of two scalar unknowns with no measurement noise. We also provide a partial proof of this result in the general setting which holds under a numerically verifiable assumption. Finally, we investigate whether a recently proposed approach based on convexification yields better reconstructions. For the first time, we propose a way to implement it in practice and show that it is consistently outperformed by some least squares solvers, which are also faster and require less measurements.

We prove existence, uniqueness and regularity of weak solutions of Kolmogorov--Fokker--Planck equations with either local or non-local diffusion in the velocity variable and rough diffusion coefficients or kernels. Our results cover the Cauchy problem and allow a broad class of source terms under minimal assumptions. The core of the analysis is a set of sharp kinetic embeddings \`a la Lions and transfer-of-regularity results \`a la Bouchut--H\"ormander. We formulate these tools in a homogeneous, scale-invariant form, available for a large range of regularity parameters.

In this paper, we are interested in the $\beta$-ensembles (or 1D log-gas) with Freud weights, namely with a potential of the form $|x|^{p}$ with $p \geq 2$. Since this potential is not of class $\mathcal{C}^{3}$ when $p \in (2,3]$, most of the literature does not apply. In this singular setting, we prove the central limit theorem for linear statistics with general test-functions and compute the subleading correction to the free energy. Our strategy relies on establishing an optimal local law in the spirit of [Bourgade, Mody, Pain 22']. Our results allow us to give a large $N$ expansion up to $o(N)$ of the log-volume of the unit balls of $N\times N$ self-adjoint matrices for the $p$-Schatten norms and to give a consistency check of the KLS conjecture. For the latter, we consider the functions $f(X)=\mathrm{Tr}(X^r)^q$ and the uniform distributions on these same Schatten balls for $N$ large enough. While the case $p>3$, $q=1, r=2$, was proven in [Dadoun, Fradelizi, Guédon, Zitt 23'], we address in the present paper the case $p\geq2$, $q\geq1$ and $r\geq2$ an even integer. The proofs are based on a link between the moments of norms of uniform laws on $p$-Schatten balls and the $\beta$-ensembles with Freud weights.

We propose a model theoretic interpretation of the theorems about the equivalence between mixed characteristic perfectoid spaces and their tilts.

We study groups definable in existentially closed geometric fields with commuting derivations. Our main result is that such a group can be definably embedded in a group interpretable in the underlying geometric field. Compared to earlier work of the first two authors toguether with K. Peterzil, the novelty is that we also deal with infinite dimensional groups.

In this thesis, we investigate combinatorial, geometric, and probabilistic properties of wreath products and other group extensions. The work is divided into the following two parts. [1] Non-extendable geodesics in Cayley graphs. We study the property of having unbounded depth in Cayley graphs of wreath products. That is, whether there exist elements at arbitrarily large distance from other elements of larger word length. We prove that for any finite group A and any finitely generated group B, the wreath product A ≀ B admits a standard generating set with unbounded depth. If B is abelian, then the above is true for every standard generating set. This generalizes the case B = ℤ, due to Cleary and Taback. When B = H ∗ K for two finite groups H and K, we characterize which standard generators of A ≀ B have unbounded depth in terms of a geometrical constant related to the Cayley graphs of H and K. [2] Random walks and Poisson boundaries of groups. First, we study random walks on the lampshuffler group FSym(H) ⋊ H, where H is a finitely generated group and FSym(H) is the group of finitary permutations of H. We show that for any step distribution µ with a finite first moment that induces a transient random walk on H, the permutation coordinate of the random walk almost surely stabilizes pointwise to a limit function. Our main result states that for H = ℤ, the Poisson boundary of the random walk (FSym(ℤ)⋊ℤ, μ) is equal to the space of limit functions endowed with the hitting measure. Our result provides new examples of completely described non-trivial Poisson boundaries of elementary amenable groups. Next, in collaboration with Joshua Frisch, we completely describe the Poisson boundary of the wreath product A ≀ B of countable groups A and B, for all probability measures µ with finite entropy and such that the lamp configurations stabilize almost surely along sample paths. If in addition the projection of µ to B is Liouville, we prove that the Poisson boundary of (A ≀ B, µ) coincides with the space of limit lamp configurations, endowed with the corresponding hitting measure. This improves earlier results by Lyons-Peres and, in particular, we answer an open question asked by Kaimanovich and Lyons-Peres for B = ℤᵈ, d ≥ 3, and measures µ with a finite first moment.

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.

Most existing learning-based methods for solving imaging inverse problems can be roughly divided into two classes: iterative algorithms, such as plug-and-play and diffusion methods leveraging pretrained denoisers, and unrolled architectures that are trained end-to-end for specific imaging problems. Iterative methods in the first class are computationally costly and often yield suboptimal reconstruction performance, whereas unrolled architectures are generally problem-specific and require expensive training. In this work, we propose a novel non-iterative, lightweight architecture that incorporates knowledge about the forward operator (acquisition physics and noise parameters) without relying on unrolling. Our model is trained to solve a wide range of inverse problems, such as deblurring, magnetic resonance imaging, computed tomography, inpainting, and super-resolution, and handles arbitrary image sizes and channels, such as grayscale, complex, and color data. The proposed model can be easily adapted to unseen inverse problems or datasets with a few fine-tuning steps (up to a few images) in a self-supervised way, without ground-truth references. Throughout a series of experiments, we demonstrate state-of-the-art performance from medical imaging to low-photon imaging and microscopy. Our code is available at https://github.com/matthieutrs/ram.

DeepInverse is an open-source PyTorch-based library for imaging inverse problems. DeepInverse implements all steps for image reconstruction, including efficient forward operators, defining and solving variational problems and designing and training advanced neural networks, for a wide set of domains (medical imaging, astronomical imaging, remote sensing, computational photography, compressed sensing and more).

Theoretical results on water waves almost always start by assuming irrotationality of the flow in order to simplify the formulation. In this work, we investigate the well-foundedness of this hypothesis via numerical simulations of the free-surface Navier-Stokes equations. We show that, in the presence of a non-flat bathymetry, either angular or smooth, a gravity wave of finite amplitude can shed vortex pairs from the bottom boundary layer into the bulk of the flow. As these eddies approach the free surface they modify the shape of the wave. It is found that this perturbation does not vanish as the Reynolds number is increased. The vanishing viscosity limit of water waves is therefore singular when no-slip boundary conditions are enforced on the bottom.

We prove that fields of meromorphic functions on Stein surfaces have cohomological dimension 2, and solve the period-index problem and Serre's conjecture II for these fields. We obtain analogous results for fields of real meromorphic functions on Stein surfaces equipped with an antiholomorphic involution. We deduce an optimal quantitative solution to Hilbert's 17th problem on analytic surfaces.

We show that any sum of squares in a field of transcendence degree 1 over Q is a sum of squares, answering a question of Pop and Pfister. We deduce this result from representation theorem, in k(C), for quadratic forms of rank at least 5 with coefficients in k, where C is a curve over a number field k.

We prove that holomorphic maps from an open subset of a complex smooth projective curve to a complex smooth projective rationally simply connected variety can be approximated by algebraic maps for the compact-open topology. This theorem can be applied in particular when the target is a smooth hypersurface of degree d in P^n with n greater than or equal to d^2-1. We deduce it from a more general result: the tight approximation property holds for rationally simply connected varieties over function fields of complex curves.

We study the rationality of some geometrically rational three-dimensional conic and quadric surface bundles, defined over the reals and more general real closed fields, for which the real locus is connected and the intermediate Jacobian obstructions to rationality vanish. We obtain both negative and positive results, using unramified cohomology and birational rigidity techniques, as well as concrete rationality constructions.

The problem of the geodynamo is simple to formulate (Why does the Earth possess a magnetic field?), yet it proves surprisingly hard to address. As with most geophysical flows, the fluid flow of molten iron in the Earth's core is strongly influenced by the Coriolis effect. Because the liquid is electrically conducting, it is also strongly influenced by the Lorentz force. The balance is unusual in that, whereas each of these effects considered separately tends to impede the flow, the magnetic field in the Earth's core relaxes the effect of the rapid rotation and allows the development of a large-scale flow in the core that in turn regenerates the field. This review covers some recent developments regarding the interplay between rotation and magnetic fields and how it affects the flow in the Earth's core.

Transformers are deep architectures that define "in-context mappings" which enable predicting new tokens based on a given set of tokens (such as a prompt in NLP applications or a set of patches for a vision transformer). In this work, we study in particular the ability of these architectures to handle an arbitrarily large number of context tokens. To mathematically, uniformly address their expressivity, we consider the case that the mappings are conditioned on a context represented by a probability distribution of tokens which becomes discrete for a finite number of these. The relevant notion of smoothness then corresponds to continuity in terms of the Wasserstein distance between these contexts. We demonstrate that deep transformers are universal and can approximate continuous in-context mappings to arbitrary precision, uniformly over compact token domains. A key aspect of our results, compared to existing findings, is that for a fixed precision, a single transformer can operate on an arbitrary (even infinite) number of tokens. Additionally, it operates with a fixed embedding dimension of tokens (this dimension does not increase with precision) and a fixed number of heads (proportional to the dimension). The use of MLPs between multi-head attention layers is also explicitly controlled. We consider both unmasked attentions (as used for the vision transformer) and masked causal attentions (as used for NLP and time series applications). We tackle the causal setting leveraging a space-time lifting to analyze causal attention as a mapping over probability distributions of tokens.

Causal Transformers are trained to predict the next token for a given context. While it is widely accepted that self-attention is crucial for encoding the causal structure of sequences, the precise underlying mechanism behind this in-context autoregressive learning ability remains unclear. In this paper, we take a step towards understanding this phenomenon by studying the approximation ability of Transformers for next-token prediction. Specifically, we explore the capacity of causal Transformers to predict the next token xt+1 given an autoregressive sequence (x1,…,xt) as a prompt, where xt+1=f(xt), and f is a context-dependent function that varies with each sequence. On the theoretical side, we focus on specific instances, namely when f is linear or when (xt)t≥1 is periodic. We explicitly construct a Transformer (with linear, exponential, or softmax attention) that learns the mapping f in-context through a causal kernel descent method. The causal kernel descent method we propose provably estimates xt+1 based solely on past and current observations (x1,…,xt), with connections to the Kaczmarz algorithm in Hilbert spaces. We present experimental results that validate our theoretical findings and suggest their applicability to more general mappings f.

Super-resolution of pointwise sources is of utmost importance in various areas of imaging sciences. Specific instances of this problem arise in single molecule fluorescence, spike sorting in neuroscience, astrophysical imaging, radar imaging, and nuclear resonance imaging. In all these applications, the Lasso method (also known as Basis Pursuit or \( \ell^1 \)-regularization) is the de facto baseline method for recovering sparse vectors from low-resolution measurements. This approach requires discretization of the domain, which leads to quantization artifacts and consequently, an overestimation of the number of sources. While grid-less methods, such as Prony-type methods or non-convex optimization over the source position, can mitigate this, the Lasso remains a strong baseline due to its versatility and simplicity. In this work, we introduce a simple extension of the Lasso, termed ``super-resolved Lasso" (SR-Lasso). Inspired by the Continuous Basis Pursuit (C-BP) method, our approach introduces an extra parameter to account for the shift of the sources between grid locations. Our method is more comprehensive than C-BP, accommodating both arbitrary real-valued or complex-valued sources. Furthermore, it can be solved similarly to the Lasso as it boils down to solving a group-Lasso problem. A notable advantage of SR-Lasso is its theoretical properties, akin to grid-less methods. Given a separation condition on the sources and a restriction on the shift magnitude outside the grid, SR-Lasso precisely estimates the correct number of sources.

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We have reached a point where many bio foundation models exist across 4 different scales, from molecules to molecular chains, cells, and tissues. However, while related in many ways, these models do not yet bridge these scales. We present a framework and architecture called Xpressor that enables cross-scale learning by (1) using a novel cross-attention mechanism to compress high-dimensional gene representations into lower-dimensional cell-state vectors, and (2) implementing a multi-scale fine-tuning approach that allows cell models to leverage and adapt protein-level representations. Using a cell Foundation Model as an example, we demonstrate that our architecture improves model performance across multiple tasks, including cell-type prediction (+12%) and embedding quality (+8%). Together, these advances represent first steps toward models that can understad and bridge different scales of biological organization.

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.

Uniform attachment with freezing is an extension of the classical model of random recursive trees, in which trees are recursively built by attaching new vertices to old ones. In the model of uniform attachment with freezing, vertices are allowed to freeze, in the sense that new vertices cannot be attached to already frozen ones. We study the impact of removing attachment and/or freezing steps on the height of the trees. We show in particular that removing an attachment step can increase the expected height, and that freezing cannot substantially decrease the height of random recursive trees. Our methods are based on coupling arguments.

We are interested in the geometry of the ``infection tree'' in a stochastic SIR (Susceptible-Infectious-Recovered) model, starting with a single infectious individual. This tree is constructed by drawing an edge between two individuals when one infects the other. We focus on the regime where the infectious period before recovery follows an exponential distribution with rate 1, and infections occur at a rate λn∼λ/n where n is the initial number of healthy individuals with λ>1. We show that provided that the infection does not quickly die out, the height of the infection tree is asymptotically κ(λ)logn as n→∞, where κ(λ) is a continuous function in λ that undergoes a second-order phase transition at λc≃1.8038. Our main tools include a connection with the model of uniform attachment trees with freezing and the application of martingale techniques to control profiles of random trees.

We are interested in the geometry of the ``infection tree'' in a stochastic SIR (Susceptible-Infectious-Recovered) model, starting with a single infectious individual. This tree is constructed by drawing an edge between two individuals when one infects the other. We focus on the regime where the infectious period before recovery follows an exponential distribution with rate 1, and infections occur at a rate λn∼λ/n where n is the initial number of healthy individuals with λ>1. We show that provided that the infection does not quickly die out, the height of the infection tree is asymptotically κ(λ)logn as n→∞, where κ(λ) is a continuous function in λ that undergoes a second-order phase transition at λc≃1.8038. Our main tools include a connection with the model of uniform attachment trees with freezing and the application of martingale techniques to control profiles of random trees.

This short paper presents a general approach for computing robust Wasserstein barycenters [2], [78], [79] of persistence diagrams. The classical method consists in computing assignment arithmetic means after finding the optimal transport plans between the barycenter and the persistence diagrams. However, this procedure only works for the transportation cost related to the q-Wasserstein distance W q when q = 2. We adapt an alternative fixed-point method [74] to compute a barycenter diagram for generic transportation costs (q > 1), in particular those robust to outliers, q ∈ (1, 2). We show the utility of our work in two applications: (i) the clustering of persistence diagrams on their metric space and (ii) the dictionary encoding of persistence diagrams [71]. In both scenarios, we demonstrate the added robustness to outliers provided by our generalized framework. Our Python implementation is available at this address: https://github.com/Keanu-Sisouk/RobustBarycenter.