We are grateful to Peter Thompson for pointing out an error in [1, Lemma 3.5, p. 1848]. The original proof worked only under the assumption that θ ̂ $\hat{\theta }$ is a vector of constants. However, some of the components of θ ̂ $\hat{\bm{\theta }}$ could be the states of the dynamic under consideration, and the lemma was used in such a setup (i.e., with θ ̂ $\hat{\bm{\theta }}$ involving states)

Consider the tight binding model of graphene, sharply terminated along an edge l parallel to a direction of translational symmetry of the underlying period lattice. We classify such edges l into those of “zigzag type” and those of “armchair type”, generalizing the classical zigzag and armchair edges. We prove that zero energy / flat band edge states arise for edges of zigzag type, but never for those

We show that for an area minimizing m-dimensional integral current T of codimension at least two inside a sufficiently regular Riemannian manifold, the upper Minkowski dimension of the interior singular set is at most m − 2 $m-2$ . This provides a strengthening of the existing ( m − 2 ) $(m-2)$ -dimensional Hausdorff dimension bound due to Almgren and De Lellis & Spadaro. As a by-product of the proof

Using compactifications in the logarithmic cotangent bundle, we obtain a formula for the Chern classes of the pushforward of Lagrangian cycles under an open embedding with normal crossing complement. This generalizes earlier results of Aluffi and Wu-Zhou. The first application of our formula is a geometric description of Chern-Mather classes of an arbitrary very affine variety, generalizing earlier

This paper characterizes the annealed, topological complexity (both of total critical points and of local minima) of the elastic manifold. This classical model of a disordered elastic system captures point configurations with self-interactions in a random medium. We establish the simple versus glassy phase diagram in the model parameters, with these phases separated by a physical boundary known as

We use the Riemann–Hilbert approach, together with string and Toda equations, to study the topological expansion in the quartic random matrix model. The coefficients of the topological expansion are generating functions for the numbers Nj⁡(g)$\mathcal {N}_j(g)$ of 4-valent connected graphs with j vertices on a compact Riemann surface of genus g. We explicitly evaluate these numbers for Riemann surfaces

We study a large family of Riesz-type singular interaction potentials with anisotropy in two dimensions. Their associated global energy minimizers are given by explicit formulas whose supports are determined by ellipses under certain assumptions. More precisely, by parameterizing the strength of the anisotropic part we characterize the sharp range in which these explicit ellipse-supported configurations

Motivated by the physics literature on “photonic doping” of scatterers made from “epsilon-near-zero” (ENZ) materials, we consider how the scattering of time-harmonic TM electromagnetic waves by a cylindrical ENZ region Ω × R $\Omega \times \mathbb {R}$ is affected by the presence of a “dopant” D ⊂ Ω $D \subset \Omega$ in which the dielectric permittivity is not near zero. Mathematically, this reduces

We study the propagation of wavepackets along curved interfaces between topological, magnetic materials. Our Hamiltonian is a massive Dirac operator with a magnetic potential. We construct semiclassical wavepackets propagating along the curved interface as adiabatic modulations of straight edge states under constant magnetic fields. While in the magnetic-free case, the wavepackets propagate coherently

We complete the verification of the 1952 Yang and Lee proposal that thermodynamic singularities are exactly the limits in R ${\mathbb {R}}$ of finite-volume singularities in C ${\mathbb {C}}$ . For the Ising model defined on a finite Λ ⊂ Z d $\Lambda \subset \mathbb {Z}^d$ at inverse temperature β ≥ 0 $\beta \ge 0$ and external field h, let α 1 ( Λ , β ) $\alpha _1(\Lambda ,\beta )$ be the modulus

Advances in compressive sensing (CS) provided reconstruction algorithms of sparse signals from linear measurements with optimal sample complexity, but natural extensions of this methodology to nonlinear inverse problems have been met with potentially fundamental sample complexity bottlenecks. In particular, tractable algorithms for compressive phase retrieval with sparsity priors have not been able

We consider the (discrete) parabolic Anderson model ∂ u ( t , x ) / ∂ t = Δ u ( t , x ) + ξ t ( x ) u ( t , x ) $\partial u(t,x)/\partial t=\Delta u(t,x) +\xi _t(x) u(t,x)$ , t ≥ 0 $t\ge 0$ , x ∈ Z d $x\in \mathbb {Z}^d$ , where the ξ-field is R $\mathbb {R}$ -valued and plays the role of a dynamic random environment, and Δ is the discrete Laplacian. We focus on the case in which ξ is given by a properly

Let Γ = q 1 Z ⊕ q 2 Z ⊕ … ⊕ q d Z $\Gamma =q_1\mathbb {Z}\oplus q_2 \mathbb {Z}\oplus \ldots \oplus q_d\mathbb {Z}$ , where q l ∈ Z + $q_l\in \mathbb {Z}_+$ , l = 1 , 2 , … , d $l=1,2,\ldots ,d$ , are pairwise coprime. Let Δ + V $\Delta +V$ be the discrete Schrödinger operator, where Δ is the discrete Laplacian on Z d $\mathbb {Z}^d$ and the potential V : Z d → C $V:\mathbb {Z}^d\rightarrow \mathbb

We study the problem where a one-dimensional elastic string is immersed in a two-dimensional steady Stokes fluid. This is known as the Stokes immersed boundary problem and also as the Peskin problem. We consider the case with equal viscosities and with a fully non-linear tension law; this model has been called the fully nonlinear Peskin problem. In this case we prove local in time wellposedness for

We prove the existence of small amplitude time quasi-periodic solutions of the pure gravity water waves equations with constant vorticity, for a bidimensional fluid over a flat bottom delimited by a space periodic free interface. Using a Nash-Moser implicit function iterative scheme we construct traveling nonlinear waves which pass through each other slightly deforming and retaining forever a quasiperiodic

Given a hermitian line bundle L → M $L\rightarrow M$ on a closed Riemannian manifold ( M n , g ) $(M^n,g)$ , the self-dual Yang–Mills–Higgs energies are a natural family of functionals

The regularity of the free boundary in optimal transportation is equivalent to that of the potential function along the free boundary. By establishing new geometric estimates of the free boundary and studying the second boundary value problem of the Monge-Ampère equation, we obtain the C 2 , α $C^{2,\alpha }$ regularity of the potential function as well as that of the free boundary, thereby resolve

We prove that various spaces of constrained positive scalar curvature metrics on compact three-manifolds with boundary, when not empty, are contractible. The constraints we mostly focus on are given in terms of local conditions on the mean curvature of the boundary, and our treatment includes both the mean-convex and the minimal case. We then discuss the implications of these results on the topology

The problem of the prescribed curvature measure is one of the important problems in differential geometry and nonlinear partial differential equations. In this paper, we consider the prescribed curvature measure problem in the hyperbolic space. We obtain the existence of star-shaped k-convex bodies with prescribed (n-k)-th curvature measures ( k < n ) $(k

For the thin obstacle problem in R n $\mathbb {R}^n$ , n ≥ 2 $n\ge 2$ , we prove that at all free boundary points, with the exception of a ( n − 3 ) $(n-3)$ -dimensional set, the solution differs from its blow-up by higher order corrections. This expansion entails a C1, 1-type free boundary regularity result, up to a codimension 3 set.

We establish the local existence and uniqueness of multi-dimensional contact discontinuities for the ideal compressible magnetohydrodynamics (MHD) in Sobolev spaces, which are most typical interfacial waves for astrophysical plasmas and prototypical fundamental waves for hyperbolic systems of conservation laws. Such waves are characteristic discontinuities for which there is no flow across the discontinuity

We consider ancient noncollapsed mean curvature flows in R 4 $\mathbb {R}^4$ whose tangent flow at − ∞ $-\infty$ is a bubble-sheet. We carry out a fine spectral analysis for the bubble-sheet function u that measures the deviation of the renormalized flow from the round cylinder R 2 × S 1 ( 2 ) $\mathbb {R}^2 \times S^1(\sqrt {2})$ and prove that for τ → − ∞ $\tau \rightarrow -\infty$ we have the fine

We study the (characteristic) Cauchy problem for the Maxwell-Bloch equations of light-matter interaction via asymptotics, under assumptions that prevent the generation of solitons. Our analysis clarifies some features of the sense in which physically-motivated initial-boundary conditions are satisfied. In particular, we present a proper Riemann-Hilbert problem that generates the unique causal solution

The Theorem of Bonnet–Myers implies that manifolds with topology M n − 1 × S 1 $M^{n-1} \times \mathbb {S}^1$ do not admit a metric of positive Ricci curvature, while the resolution of Geroch's conjecture implies that the torus T n $\mathbb {T}^n$ does not admit a metric of positive scalar curvature. In this work we introduce a new notion of curvature interpolating between Ricci and scalar curvature

Gromov proved a quadratic decay inequality of scalar curvature for a class of complete manifolds. In this paper, we prove that for any uniformly contractible manifold with finite asymptotic dimension, its scalar curvature decays to zero at a rate depending only on the contractibility radius of the manifold and the diameter control of the asymptotic dimension. We construct examples of uniformly contractible

Let X be a compact, Kähler, Calabi-Yau threefold and suppose X ↦ X ̲ ⇝ X t $X\mapsto \underline{X}\leadsto X_t$ , for t ∈ Δ $t\in \Delta$ , is a conifold transition obtained by contracting finitely many disjoint ( − 1 , − 1 ) $(-1,-1)$ curves in X and then smoothing the resulting ordinary double point singularities. We show that, for | t | ≪ 1 $|t|\ll 1$ sufficiently small, the tangent bundle T 1

Koopman operators are infinite-dimensional operators that globally linearize nonlinear dynamical systems, making their spectral information valuable for understanding dynamics. However, Koopman operators can have continuous spectra and infinite-dimensional invariant subspaces, making computing their spectral information a considerable challenge. This paper describes data-driven algorithms with rigorous

We study the question of the integrability of Einstein deformations and relate it to the question of the desingularization of Einstein metrics. Our main application is a negative answer to the long-standing question of whether or not every Einstein 4-orbifold (which is an Einstein metric space in a synthetic sense) is limit of smooth Einstein 4-manifolds. We more precisely show that spherical and hyperbolic

Our goal in this work is to better understand the relationship between replica symmetry breaking, shattering, and metastability. To this end, we study the static and dynamic behaviour of spherical pure p-spin glasses above the replica symmetry breaking temperature T s $T_{s}$ . In this regime, we find that there are at least two distinct temperatures related to non-trivial behaviour. First we prove

We study stability of a spherical vortex introduced by M. Hill in 1894, which is an explicit solution of the three-dimensional incompressible Euler equations. The flow is axi-symmetric with no swirl, the vortex core is simply a ball sliding on the axis of symmetry with a constant speed, and the vorticity in the core is proportional to the distance from the symmetry axis. We use the variational setting

We present a new and simple proof for the classic results of Imbrie (1985) and Bricmont–Kupiainen (1988) that for the random field Ising model in dimension three and above there is long range order at low temperatures with presence of weak disorder. With the same method, we obtain a couple of new results: (1) we prove that long range order exists for the random field Potts model at low temperatures

We consider finite Morse index solutions to semilinear elliptic questions, and we investigate their smoothness. It is well-known that:

For the thin obstacle problem, we develop a unified approach that leads to rates of convergence to blow-up profiles at contact points with integer frequencies. For these points, we also obtain a stratification result.

In this paper, we present a probabilistic study of rare phenomena of the cubic nonlinear Schrödinger equation on the torus in a weakly nonlinear setting. This equation has been used as a model to numerically study the formation of rogue waves in deep sea. Our results are twofold: first, we introduce a notion of criticality and prove a Large Deviations Principle (LDP) for the subcritical and critical

We establish interior regularity for convex viscosity solutions of the special Lagrangian equation. Our result states that all such solutions are real analytic in the interior of the domain.

In this paper we examine necessary conditions for an inhomogeneity to be non-scattering, or equivalently, by negation, sufficient conditions for it to be scattering. These conditions are formulated in terms of the regularity of the boundary of the inhomogeneity. We examine broad classes of incident waves in both two and three dimensions. Our analysis is greatly influenced by the analysis carried out

This paper contains a new proof of the short-time existence for the flow by curvature of a network of curves in the plane. Appearing initially in metallurgy and as a model for the evolution of grain boundaries, this flow was later treated by Brakke using varifold methods. There is good reason to treat this problem by a direct PDE approach, but doing so requires one to deal with the singular nature

The free boundary problem for a two-dimensional fluid permeating a porous medium is studied. This is known as the one-phase Muskat problem and is mathematically equivalent to the vertical Hele-Shaw problem driven by gravity force. We prove that if the initial free boundary is the graph of a periodic Lipschitz function, then there exists a global-in-time Lipschitz solution in the strong L t ∞ L x 2

It has been known since Lanford that the dynamics of a hard-sphere gas is described in the low density limit by the Boltzmann equation, at least for short times. The classical strategy of proof fails for longer times, even close to equilibrium. In this paper, we introduce a weak convergence method coupled with a sampling argument to prove that the covariance of the fluctuation field around equilibrium

Early research in aerodynamics and biological propulsion was dramatically advanced by the analytical solutions of Theodorsen, von Kármán, Wu and others. While these classical solutions apply only to isolated swimmers, the flow interactions between multiple swimmers are relevant to many practical applications, including the schooling and flocking of animal collectives. In this work, we derive a class

Beginning with the work of Landau, Pollak and Slepian in the 1960s on time-band limiting, commuting pairs of integral and differential operators have played a key role in signal processing, random matrix theory, and integrable systems. Previously, such pairs were constructed by ad hoc methods, which essentially worked because a commuting operator of low order could be found by a direct calculation

Finite-time coherent sets (FTCSs) are distinguished regions of phase space that resist mixing with the surrounding space for some finite period of time; physical manifestations include eddies and vortices in the ocean and atmosphere, respectively. The boundaries of FTCSs are examples of Lagrangian coherent structures (LCSs). The selection of the time duration over which FTCS and LCS computations are

We investigate the long-time properties of the two-dimensional inviscid Boussinesq equations near a stably stratified Couette flow, for an initial Gevrey perturbation of size ε. Under the classical Miles-Howard stability condition on the Richardson number, we prove that the system experiences a shear-buoyancy instability: the density variation and velocity undergo an O ( t − 1 / 2 ) $O(t^{-1/2})$ inviscid

We consider the orthogonal polynomial pn(z) with respect to the planar measure supported on the whole complex plane

While it is well known that constant rotation induces linear dispersive effects in various fluid models, we study here its effect on long time nonlinear dynamics in the inviscid setting. More precisely, we investigate stability in the 3d rotating Euler equations in R 3 $\mathbb {R}^3$ with a fixed speed of rotation. We show that for any M > 0 $\mathcal {M}> 0$ , axisymmetric initial data of sufficiently

The goal of this paper is to characterize function distributions that general neural networks trained by descent algorithms (GD/SGD), can or cannot learn in polytime. The results are: (1) The paradigm of general neural networks trained by SGD is poly-time universal: any function distribution that can be learned from samples in polytime can also be learned by a poly-size neural net trained by SGD with

We consider the zero-average Gaussian free field on a certain class of finite d-regular graphs for fixed d ≥ 3 $d\ge 3$ . This class includes d-regular expanders of large girth and typical realisations of random d-regular graphs. We show that the level set of the zero-average Gaussian free field above level h has a giant component in the whole supercritical phase, that is for all h < h ★ $h

We prove the existence of branched immersed constant mean curvature (CMC) 2-spheres in an arbitrary Riemannian 3-sphere for almost every prescribed mean curvature, and moreover for all prescribed mean curvatures when the 3-sphere is positively curved. To achieve this, we develop a min-max scheme for a weighted Dirichlet energy functional. There are three main ingredients in our approach: a bi-harmonic

We are concerned with the Cauchy problem for the two-dimensional compressible Navier-Stokes equations supplemented with general H1 initial velocity and bounded initial density not necessarily strictly positive: it may be the characteristic function of any set, for instance. In the perfect gas case, we establish global-in-time existence and uniqueness, provided the volume (bulk) viscosity coefficient

The Bessel process models the local eigenvalue statistics near 0 of certain large positive definite matrices. In this work, we consider the probability

We analyse the case of a dense modified Korteweg–de Vries (mKdV) soliton gas and its large time behaviour in the presence of a single trial soliton. We show that the solution can be expressed in terms of Fredholm determinants as well as in terms of a Riemann–Hilbert problem. We then show that the solution can be decomposed as the sum of the background gas solution (a modulated elliptic wave), plus

Let the viscosity ε → 0 $\varepsilon \rightarrow 0$ for the 2D steady Navier-Stokes equations in the region 0 ≤ x ≤ L $0\le x\le L$ and 0 ≤ y < ∞ $0\le y<\infty$ with no slip boundary conditions at y = 0 $y=0$ . For L < < 1 $L<<1$ , we justify the validity of the steady Prandtl layer expansion for scaled Prandtl layers, including the celebrated Blasius boundary layer. Our uniform estimates in ε are

We study the two-dimensional Anisotropic KPZ equation (AKPZ) formally given by

We study the one-dimensional KPZ equation on a large torus, started at equilibrium. The main results are optimal variance bounds in the super-relaxation regime and part of the relaxation regime.

The magnetization ripple is a microstructure formed in thin ferromagnetic films. It can be described by minimizers of a nonconvex energy functional leading to a nonlocal and nonlinear elliptic SPDE in two dimensions driven by white noise, which is singular. We address the universal character of the magnetization ripple using variational methods based on Γ-convergence. Due to the infinite energy of

We prove the instability of the Couette flow if the disturbances is less smooth than the Gevrey space of class 2. This shows that this is the critical regularity for this problem since it was proved in [5] that stability and inviscid damping hold for disturbances which are smoother than the Gevrey space of class 2. A big novelty is that this critical space is due to an instability mechanism which is

We consider solutions of the Navier-Stokes equations in 3d with vortex filament initial data of arbitrary circulation, that is, initial vorticity given by a divergence-free vector-valued measure of arbitrary mass supported on a smooth curve. First, we prove global well-posedness for perturbations of the Oseen vortex column in scaling-critical spaces. Second, we prove local well-posedness (in a sense

Comparison results of Talenti type for elliptic problems with Dirichlet boundary conditions have been widely investigated in recent decades. In this paper, we deal with Robin boundary conditions. Surprisingly, contrary to the Dirichlet case, Robin boundary conditions make the comparison sensitive to the dimension, and while the planar case seems to be completely settled, in higher dimensions some open

In signal quantization, it is well-known that introducing adaptivity to quantization schemes can improve their stability and accuracy in quantizing bandlimited signals. However, adaptive quantization has only been designed for one-dimensional signals. The contribution of this paper is two-fold: (i) we propose the first family of two-dimensional adaptive quantization schemes that maintain the same mathematical

Let Γ be a nonelementary hyperbolic group with a word metric d and ∂Γ its hyperbolic boundary equipped with a visual metric d a for some parameter a > 1 . Fix a superexponential symmetric probability μ on Γ whose support generates Γ as a semigroup, and denote by ρ the spectral radius of the random walk Y on Γ with step distribution μ. Let ν be a probability on 1,2,3 , … with mean λ = ∑ k = 1 ∞ kν k