Orthogonal systems in L2(ℝ), once implemented in spectral methods, enjoy a number of important advantages if their differentiation matrix is skew‐symmetric and highly structured. Such systems, where the differentiation matrix is skew‐symmetric, tridiagonal, and irreducible, have been recently fully characterised. In this paper we go a step further, imposing the extra requirement of fast computation:

We prove that, at every positive temperature, the infinite‐volume free energy of the one‐dimensional log‐gas, or beta‐ensemble, has a unique minimizer, which is the Sine‐beta process arising from random matrix theory. We rely on a quantitative displacement convexity argument at the level of point processes, and on the screening procedure introduced by Sandier‐Serfaty. © 2020 Wiley Periodicals, Inc

We prove asymptotic stability of point vortex solutions to the full Euler equation in two dimensions. More precisely, we show that a small, Gevrey smooth, and compactly supported perturbation of a point vortex leads to a global solution of the Euler equation in 2D, which converges weakly as t → ∞ to a radial profile with respect to the vortex. The position of the point vortex, which is time dependent

In this paper we introduce a control variate estimator for a quantity of interest that can be expressed as the expectation of a function of a random process, that is itself the solution of a differential equation driven by fast mean‐reverting ergodic random forces. The control variate is built with the same function and with the limit diffusion process that approximates the original random process

Erwin Schrödinger posed—and to a large extent solved—in 1931/32 the problem of finding the most likely random evolution between two continuous probability distributions. This article considers this problem in the case when only samples of the two distributions are available. A novel iterative procedure is proposed, inspired by Fortet‐IPF‐Sinkhorn type algorithms. Since only samples of the marginals

A framework is proposed that addresses both conditional density estimation and latent variable discovery. The objective function maximizes explanation of variability in the data, achieved through the optimal transport barycenter generalized to a collection of conditional distributions indexed by a covariate—either given or latent—in any suitable space. Theoretical results establish the existence of

Consider an infinite cloud of hard spheres sedimenting in a Stokes flow in the whole space ℝd. Despite many contributions in fluid mechanics and applied mathematics, there is so far no rigorous definition of the associated effective sedimentation velocity. Calculations by Caflisch and Luke in dimension d = 3 suggest that the effective velocity is well‐defined for hard spheres distributed according

We extend the mathematical theory of rigidity of frameworks (graphs embedded in d‐dimensional space) to consider nonlocal rigidity and flexibility properties. We provide conditions on a framework under which (I) as the framework flexes continuously it must remain inside a small ball, a property we call “almost‐rigidity”; (II) any other framework with the same edge lengths must lie outside a much larger

Inspired by the numerical immersed boundary method, we introduce regularized Stokes immersed boundary problems in two dimensions to describe regularized motion of a 1‐D closed elastic string in a 2‐D Stokes flow, in which a regularized δ‐function is used to mollify the flow field and singular forcing. We establish global well‐posedness of the regularized problems and prove that as the regularization

We introduce new obstructions to rationality for geometrically rational threefolds arising from the geometry of curves and their cycle maps. © 2020 Wiley Periodicals LLC

We obtain the best known quantitative estimates for the Lp‐Poincaré and log‐Sobolev inequalities on domains in various sub‐Riemannian manifolds, including ideal Carnot groups and in particular ideal generalized H‐type Carnot groups and the Heisenberg groups, corank 1 Carnot groups, the Grushin plane, and various H‐type foliations, Sasakian and 3‐Sasakian manifolds. Moreover, this constitutes the first

This paper concerns the existence of transonic shock solutions to the 2‐D steady compressible Euler system in an almost flat finite nozzle (in the sense that it is a generic small perturbation of a flat one), under physical boundary conditions proposed by Courant‐Friedrichs in [14], in which the receiver pressure is prescribed at the exit of the nozzle. In the resulting free boundary problem, the location

We establish a theory of Q‐valued functions minimizing a suitable generalization of the Dirichlet integral. In a second paper the theory will be used to approximate efficiently area minimizing currents mod(p) when p = 2Q, and to establish a first general partial regularity theorem for every p in any dimension and codimension . © 2020 The Authors. Communications on Pure and Applied Mathematics published

We present a general approach to study a class of random growth models in n‐dimensional Euclidean space. These models are designed to capture basic growth features that are expected to manifest at the mesoscopic level for several classical self‐interacting processes originally defined at the microscopic scale. It includes once‐reinforced random walk with strong reinforcement, origin‐excited random

Let X be a closed oriented connected topological manifold of dimension n ≥ 5. The structure group is the abelian group of equivalence classes of all pairs (f, M) such that M is a closed oriented manifold and f : M → X is an orientation‐preserving homotopy equivalence. The main purpose of this article is to prove that a higher rho invariant map defines a group homomorphism from the topological structure

We show the existence of smoothly embedded closed minimal surfaces in infinite volume hyperbolic 3‐manifolds except for some special cases. © 2020Wiley Periodicals LLC

We investigate Sineβ, the universal point process arising as the thermodynamic limit of the microscopic scale behavior in the bulk of one‐dimensional log‐gases, or β‐ensembles, at inverse temperature β > 0. We adopt a statistical physics perspective, and give a description of Sineβ using the Dobrushin‐Lanford‐Ruelle (DLR) formalism by proving that it satisfies the DLR equations: the restriction of

Dobrushin in 1972 showed that the interface of a 3D Ising model with minus boundary conditions above the xy‐plane and plus below is rigid (has O(1) fluctuations) at every sufficiently low temperature. Since then, basic features of this interface—such as the asymptotics of its maximum—were only identified in more tractable random surface models that approximate the Ising interface at low temperatures

Low‐rank matrix recovery from structured measurements has been a topic of intense study in the last decade and many important problems like matrix completion and blind deconvolution have been formulated in this framework. An important benchmark method to solve these problems is to minimize the nuclear norm, a convex proxy for the rank. A common approach to establish recovery guarantees for this convex

We consider the 2D isentropic compressible Euler equations, with pressure law p(ρ) = (1γ)ργ, with γ > 1. We provide an elementary constructive proof of shock formation from smooth initial data of finite energy, with no vacuum regions, and with nontrivial vorticity. We prove that for initial data which has minimum slope −1ε, for ε > 0 taken sufficiently small relative to the amplitude, there exist smooth

We prove that a solution of an elliptic operator with periodic coefficients behaves on large scales like an analytic function in the sense of approximation by polynomials with periodic corrections. Equivalently, the constants in the large‐scale Ck, 1 estimate scale exponentially in k, just as for the classical estimate for harmonic functions, and the minimal scale grows at most linearly in k. As a

We introduce a numerical method for the solution of the time‐dependent Schrödinger equation with a smooth potential, based on its reformulation as a Volterra integral equation. We present versions of the method both for periodic boundary conditions, and for free space problems with compactly supported initial data and potential. A spatially uniform electric field may be included, making the solver

We consider compact ancient solutions to the three‐dimensional Ricci flow that are κ‐noncollapsed. We prove that such a solution either is a family of shrinking round spheres or has a unique asymptotic behavior as t → − ∞, which we describe. This analysis applies in particular to the ancient solution constructed by Perelman. © 2020 Wiley Periodicals LLC

Let A be a W1, 2‐connection on a principal SU(2)‐bundle P over a compact 4‐manifold M whose curvature FA satisfies . Our main result is the existence of a global section σ : M → P with finite singularities on M such that the connection form σ*A satisfies the Coulomb equation d*(σ*A) = 0 and admits a sharp estimate . Here ℒ4, ∞ is a new function space we introduce in this paper that satisfies L4(M) ⊊ ℒ4

Consider a C∞ closed connected Riemannian manifold (M, g) with negative sectional curvature. The unit tangent bundle SM is foliated by the (weak) stable foliation of the geodesic flow. Let Δs be the leafwise Laplacian for and let X be the geodesic spray, i.e., the vector field that generates the geodesic flow. For each ρ, the operator generates a diffusion for . We show that, as ρ → − ∞, the unique

Soap films at equilibrium are modeled, rather than as surfaces, as regions of small total volume through the introduction of a capillarity problem with a homotopic spanning condition. This point of view introduces a length scale in the classical Plateau's problem, which is in turn recovered in the vanishing volume limit. This approximation of area minimizing hypersurfaces leads to an energy based selection

A classical result of Aubin states that the constant in the Moser‐Trudinger‐Onofri inequality on can be improved for functions with zero first‐order moments of the area element. We generalize it to the higher‐order moments case. These new inequalities bear similarity to a sequence of Lebedev‐Milin‐type inequalities on coming from the work of Grenander‐Szego on Toeplitz determinants (as pointed out

This work is concerned with an asymptotic analysis, in the sense of Γ‐convergence, of a sequence of variational models of brittle damage in the context of linearized elasticity. The study is performed as the damaged zone concentrates into a set of zero volume and, at the same time and to the same order ε, the stiffness of the damaged material becomes small. Three main features make the analysis highly

We deal with a parabolic‐elliptic chemotaxis system. It is known that finite‐time blowup occurs for a large class of initial data. However, there have been no results on exact blowup rate or detailed blowup behavior except a special radial solution given just formally in [13] and rigorously in [19, 10, 9]. Our aim is to show that for all radial blowup solutions, their blowup rate, and blowup profile

We consider a nonlocal aggregation equation with degenerate diffusion, which describes the mean‐field limit of interacting particles driven by nonlocal interactions and localized repulsion. When the interaction potential is attractive, it is previously known that all steady states must be radially decreasing up to a translation, but uniqueness (for a given mass) within the radial class was open, except

This paper is devoted to the asymptotic analysis of a thin film equation that describes the evolution of a thin liquid droplet on a solid support driven by capillary forces. We propose an analytic framework to rigorously investigate the connection between this model and Tanner's law, which claims: the edge velocity of a spreading thin film on a prewetted solid is approximately proportional to the cube

We prove a sharp logarithmic Sobolev inequality that holds for submanifolds in Euclidean space of arbitrary dimension and codimension. Like the Michael‐Simon Sobolev inequality, this inequality includes a term involving the mean curvature. © 2020 Wiley Periodicals LLC

Using toric geometry we give an explicit construction of the compact steady solitons for pluriclosed flow first constructed in by the first author in 2019. This construction also reveals that these solitons are generalized Kähler in two distinct ways, with vanishing and nonvanishing Poisson structure. This gives the first examples of generalized Kähler structures with nonvanishing Poisson structure

In this paper, we study the transition threshold of the 3D Couette flow in Sobolev space at high Reynolds number Re. It was proved that if the initial velocity v0 satisfies for some c0 > 0 independent of Re, then the solution of the 3D Navier‐Stokes equations is global in time and does not transition away from the Couette flow. This result confirms the transition threshold conjecture proposed by Trefethen

We consider the Fröhlich polaron model in the strong coupling limit. It is well‐known that to leading order the ground state energy is given by the (classical) Pekar energy. In this work, we establish the subleading correction, describing quantum fluctuation about the classical limit. Our proof applies to a model of a confined polaron, where both the electron and the polarization field are restricted

For M a simple surface, the nonlinear statistical inverse problem of recovering a matrix field from discrete, noisy measurements of the SO(n)‐valued scattering data CΦ of a solution of a matrix ODE is considered (n ≥ 2). Injectivity of the map Φ ↦ CΦ was established by Paternain, Salo, and Uhlmann in 2012. A statistical algorithm for the solution of this inverse problem based on Gaussian process priors

We provide the first nontrivial upper bound for the chemical distance exponent in two‐dimensional critical percolation. Specifically, we prove that the expected length of the shortest horizontal crossing path of a box of side length n in critical percolation on ℤ2 is bounded by Cn2 − δπ3(n) for some δ > 0, where π3(n) is the “three‐arm probability to distance n.” This implies that the ratio of this

We prove that given initial data , forcing and any T > 0, the solutions uν of Navier‐Stokes converge strongly in for any p ∈ [1, ∞) to the unique Yudovich weak solution u of the Euler equations. A consequence is that vorticity distribution functions converge to their inviscid counterparts. As a by‐product of the proof, we establish continuity of the Euler solution map for Yudovich solutions in the

We compute the topological simple structure set of closed manifolds that occur as total spaces of flat bundles over lens spaces Sl/(ℤ/p) with fiber Tn for an odd prime p and l ≥ 3 provided that the induced ℤ/p‐action on π1(Tn) = ℤn is free outside the origin. To the best of our knowledge this is the first computation of the structure set of a topological manifold whose fundamental group is not obtained

We study asymptotic behaviors of positive solutions to the Yamabe equation and the σk‐Yamabe equation near isolated singular points and establish expansions up to arbitrary orders. Such results generalize an earlier pioneering work by Caffarelli, Gidas, and Spruck and a work by Korevaar, Mazzeo, Pacard, and Schoen on the Yamabe equation and a work by Han, Li, and Teixeira on the σk‐Yamabe equation

The detailed behavior of solutions to Stokes equations on regions with corners has been historically difficult to characterize. The solutions to Stokes equations on regions with corners are known to develop singularities in the vicinity of corners; in particular, the solutions are known to have infinite oscillations along almost every ray that meet at the corner. While the nature of singularities for

Despite great progress in the study of critical percolation on ℤd for d large, properties of critical clusters in high‐dimensional fractional spaces and boxes remain poorly understood, unlike the situation in two dimensions. Closely related models such as critical branching random walk give natural conjectures for the value of the relevant high‐dimensional critical exponents; see in particular the

We prove the Yau‐Tian‐Donaldson conjecture for any ℚ‐Fano variety that has a log smooth resolution of singularities such that a negative linear combination of exceptional divisors is relatively ample and the discrepancies of all exceptional divisors are nonpositive. In other words, if such a Fano variety is K‐polystable, then it admits a Kähler‐Einstein metric. This extends the previous result for

We consider a driven tagged particle in a symmetric exclusion process on ℤ with a removal rule. In this process, untagged particles are removed once they jump to the left of the tagged particle. We investigate the behavior of the displacement of the tagged particle and prove limit theorems of it: an (annealed) law of large numbers, a central limit theorem, and a large deviation principle. We also characterize

By constructing an infinite‐dimensional KAM theorem of the normal frequencies being dense at a finite point, we show that some shallow water equations such as the Benjamin‐Bona‐Mahony equation and the generalized d ‐dimensional Pochhammer‐Chree equation subject to some boundary conditions possess many (a family of initial values of positive Lebesgue measure of finite dimension) smooth solutions that

The main result of this paper gives a new construction of extremal Kähler metrics on the total space of certain holomorphic submersions, giving a vast generalisation and unification of results of Hong, Fine and others. The principal new ingredient is a novel geometric partial differential equation on such fibrations, which we call the optimal symplectic connection equation.

It is well‐known that when the geometry and/or coefficients allow stable trapped rays, the outgoing solution operator of the Helmholtz equation grows exponentially through a sequence of real frequencies tending to infinity.

We prove an a priori bound for solutions of the dynamic equation. This bound provides a control on solutions on a compact space‐time set only in terms of the realisation of the noise on an enlargement of this set, and it does not depend on any choice of space‐time boundary conditions.

We prove a sharp transition for the infinitesimal distribution of a periodically banded GUE matrix. For bandwidths , we further prove that our model is infinitesimally free from the matrix units and the normalized all‐1’s matrix. Our results allow us to extend previous work of Shlyakhtenko on finite‐rank perturbations of Wigner matrices in the infinitesimal framework. For finite‐rank perturbations

Exterior problems for the maximal surface equation are studied. We obtain the precise asymptotic behavior of the exterior solution at infinity. We also prove that the exterior Dirichlet problem is uniquely solvable for admissible boundary data and prescribed asymptotic behavior at infinity. © 2020 Wiley Periodicals LLC.

We establish an optimal regularity result for parametrized two‐dimensional stationary varifolds. Namely, we show that the parametrization map is a smooth minimal branched immersion and that the multiplicity function is constant.

We derive a multiscale generalisation of the Bakry‐Émery criterion for a measure to satisfy a log‐Sobolev inequality. Our criterion relies on the control of an associated PDE well‐known in renormalisation theory: the Polchinski equation. It implies the usual Bakry‐Émery criterion, but we show that it remains effective for measures that are far from log‐concave. Indeed, using our criterion, we prove

Let K be a convex polyhedron and ℱ its Wulff energy, and let denote the set of convex polyhedra close to K whose faces are parallel to those of K . We show that, for sufficiently small ε , all ε ‐minimizers belong to

Following van der Poorten, we consider a family of nonlinear maps that are generated from the continued fraction expansion of a function on a hyperelliptic curve of genus g. Using the connection with the classical theory of J ‐fractions and orthogonal polynomials, we show that in the simplest case g = 1 this provides a straightforward derivation of Hankel determinant formulae for the terms of a general

Many real‐world processes and phenomena are modeled using systems of ordinary differential equations with parameters. Given such a system, we say that a parameter is globally identifiable if it can be uniquely recovered from input and output data. The main contribution of this paper is to provide theory, an algorithm, and software for deciding global identifiability. First, we rigorously derive an

We focus on spherical spin glasses whose Parisi distribution has support of the form [0, q ]. For such models we construct paths from the origin to the sphere that consistently remain close to the ground‐state energy on the sphere of corresponding radius. The construction uses a greedy strategy, which always follows a direction corresponding to the most negative eigenvalues of the Hessian of the Hamiltonian

This paper proves the existence of small‐amplitude global‐in‐time unique mild solutions to both the Landau equation including the Coulomb potential and the Boltzmann equation without angular cutoff. Since the well‐known works [45] and [3, 43] on the construction of classical solutions in smooth Sobolev spaces which in particular are regular in the spatial variables, it still remains an open problem

We revisit the proof of the large‐deviations principle of Wiener chaoses partially given by Borell and then by Ledoux in its full form. We show that some heavy‐tail phenomena observed in large deviations can be explained by the same mechanism as for the Wiener chaoses, meaning that the deviations are created, in a sense, by translations. More precisely, we prove a general large‐deviations principle

We show in full generality the stability of optimal transport paths in branched transport: namely, we prove that any limit of optimal transport paths is optimal as well. This solves an open problem in the field (cf. Open problem 1 in the book Optimal transportation networks by Bernot, Caselles, and Morel), which has been addressed up to now only under restrictive assumptions. © 2020 Wiley Periodicals

We present a new proof of the sphere covering inequality in the spirit of comparison geometry, and as a by‐product we find another sphere covering inequality that can be viewed as the dual of the original one. We also prove sphere covering inequalities on surfaces satisfying general isoperimetric inequalities, and discuss their applications to elliptic equations with exponential nonlinearities in dimension