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.

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.

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 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

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 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

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

Based on the a priori estimates in our previous work, we continue to investigate the water‐waves problem in a bounded two‐dimensional corner domain in this paper. We prove the local well‐posedness of the solution to the water‐waves system when the contact angles are less than π /16. © 2020 Wiley Periodicals LLC

We study noncollapsed Gromov‐Hausdorff limits of Kähler manifolds with Ricci curvature bounded below. Our main result is that each tangent cone is homeomorphic to a normal affine variety. This extends a result of Donaldson‐Sun, who considered noncollapsed limits of polarized Kähler manifolds with two‐sided Ricci curvature bounds. © 2019 Wiley Periodicals LLC

Let P N be a uniform random N  × N permutation matrix and let χ N (z ) = det(zI N  − P N ) denote its characteristic polynomial. We prove a law of large numbers for the maximum modulus of χ N on the unit circle, specifically, with probability tending to 1 as N  → ∞, for a numerical constant x 0 ≈ 0.652. The main idea of the proof is to uncover a logarithmic correlation structure for the distribution

We consider nonlinear, uniformly elliptic equations with random, highly oscillating coefficients satisfying a finite range of dependence. We prove that homogenization and linearization commute in the sense that the linearized equation (linearized around an arbitrary solution) homogenizes to the linearization of the homogenized equation (linearized around the corresponding solution of the homogenized

Analytic functions in the Hardy class H2 over the upper half‐plane ℍ+ are uniquely determined by their values on any curve Γ lying in the interior or on the boundary of ℍ+. The goal of this paper is to provide a sharp quantitative version of this statement. We answer the following question: Given f of a unit H2‐norm that is small on Γ (say, its L2‐norm is of order ϵ), how large can f be at a point

We consider a 2D vorticity configuration where vorticity is highly concentrated around a curve and exponentially decaying away from it: the intensity of the vorticity is O(1/ε) on the curve while it decays on an O(ε) distance from the curve itself. We prove that, if the initial datum is of vortex‐layer type, Euler solutions preserve this structure for a time that does not depend on ε. Moreover, the

In this paper we prove that the Benjamin‐Ono equation, when considered on the torus, is an integrable (pseudo)differential equation in the strongest possible sense: this equation admits global Birkhoff coordinates on the space of real‐valued, 2π‐periodic, L2‐integrable functions of mean 0. These are coordinates that allow us to integrate it by quadrature and hence are also referred to as nonlinear

The triple‐deck model is a classical high‐order boundary layer model that has been proposed to describe flow regimes where the Prandtl theory is expected to fail. At first sight the model appears to lose two derivatives through the pressure‐displacement relation that links pressure to the tangential slip. In order to overcome this, we split the triple‐deck system into two coupled equations: a Prandtl‐type

We provide a detailed analysis of the boundary layers for mixed hyperbolic‐parabolic systems in one space dimension and small amplitude regimes. As an application of our results, we describe the solution of the so‐called boundary Riemann problem recovered as the zero viscosity limit of the physical viscous approximation. In particular, we tackle the so‐called doubly characteristic case, which is considerably

We exploit the so‐called atomic condition, recently defined by De Philippis, De Rosa, and Ghiraldin and proved to be necessary and sufficient for the validity of the anisotropic counterpart of the Allard rectifiability theorem. In particular, we address an open question of this seminal work, showing that the atomic condition implies the strict Almgren geometric ellipticity condition. © 2020 Wiley Periodicals

We investigate in this work a recently proposed diagrammatic quantum Monte Carlo method—the inchworm Monte Carlo method—for open quantum systems. We establish its validity rigorously based on resummation of Dyson series. Moreover, we introduce an integro‐differential equation formulation for open quantum systems, which illuminates the mathematical structure of the inchworm algorithm. This new formulation

The logarithmic correction for the order of the maximum for two‐speed branching Brownian motion changes discontinuously when approaching slopes , which corresponds to standard branching Brownian motion. In this article we study this transition more closely by choosing and . We show that the logarithmic correction for the order of the maximum now smoothly interpolates between the correction in the i

This paper investigates the stability of traveling wave solutions to the free boundary Euler equations with a submerged point vortex. We prove that sufficiently small‐amplitude waves with small enough vortex strength are conditionally orbitally stable. In the process of obtaining this result, we develop a quite general stability/instability theory for bound state solutions of a large class of infinite‐dimensional

We consider nonuniformly elliptic variational problems and give optimal conditions guaranteeing the local Lipschitz regularity of solutions in terms of the regularity of the given data. The analysis catches the main model cases in the literature. Integrals with fast, exponential‐type growth conditions as well as integrals with unbalanced polynomial growth conditions are covered. Our criteria involve

We consider a subclass of those quasilinear wave equations in 3 + 1 space‐time dimensions that satisfy the “weak null condition” as defined by Lindblad and Rodnianski , and study the large‐time behavior of solutions to the Cauchy problem. The prototype for the class of equations considered is . Global solutions for such equations have been constructed by Lindblad and Alinhac. Our main results are the

The space of Kähler metrics, on the one hand, can be approximated by subspaces of algebraic metrics, while, on the other hand, it can also be enlarged to finite‐energy spaces arising in pluripotential theory. The latter spaces are realized as metric completions of Finsler structures on the space of Kähler metrics. The former spaces are the finite‐dimensional spaces of Fubini‐Study metrics of Kähler

We investigate the homogeneous Dirichlet problem for the fast diffusion equation ut = Δum, posed in a smooth bounded domain Ω ⊂ ℝN, in the exponent range ms = (N − 2)+/(N + 2) < m < 1. It is known that bounded positive solutions extinguish in a finite time T > 0, and also that they approach a separate variable solution u(t, x) ∼ (T − t)1/(1 − m)S(x) as t → T−, where S belongs to the set of solutions

We consider the Ising model at its critical temperature with external magnetic field ha15/8 on the square lattice with lattice spacing a. We show that the truncated two‐point function in this model decays exponentially with a rate independent of a as a ↓ 0. As a consequence, we show exponential decay in the near‐critical scaling limit Euclidean magnetization field. For the lattice model with a = 1

We study inverse problems for the Einstein equations with source fields in a general form. Under a microlocal linearization stability condition, we show that by generating small gravitational perturbations and measuring the responses near a freely falling observer, one can uniquely determine the background Lorentzian metric up to isometries in a region where the gravitational perturbations can travel

We study the solvability of second boundary value problems of fourth‐order equations of Abreu type arising from approximation of convex functionals whose Lagrangians depend on the gradient variable, subject to a convexity constraint. These functionals arise in different scientific disciplines such as Newton's problem of minimal resistance in physics and the monopolist's problem in economics. The right‐hand

We study the following control problem. A fish with bounded aquatic locomotion speed swims in fast waters. Can this fish, under reasonable assumptions, get to a desired destination? It can, even if the flow is time dependent. Moreover, given a prescribed sufficiently large time t, it can be there at exactly the time t. The major difference from our previous work is the time dependence of the flow.

We consider random walk in a uniformly elliptic i.i.d. random environment in ℤd for d ≥ 2. It is believed that whenever the random walk is transient in a given direction it is necessarily ballistic. In order to quantify the gap which would be needed to prove this equivalence, several ballisticity conditions have been introduced. In particular, Sznitman defined the so‐called conditions (T) and (T′)

We study local regularity properties for solutions of linear, nonuniformly elliptic equations. Assuming certain integrability conditions on the coefficient field, we prove local boundedness and Harnack inequality. The assumed integrability assumptions are essentially sharp and improve upon classical results by Trudinger. We then apply the deterministic regularity results to the corrector equation in

We study the Gibbs measure of mixed spherical p ‐spin glass models at low temperature, in (part of) the 1‐RSB regime, including, in particular, models close to pure in an appropriate sense. We show that the Gibbs measure concentrates on spherical bands around deep critical points of the (extended) Hamiltonian restricted to the sphere of radius , where is the rightmost point in the support of the overlap

Slender body theory facilitates computational simulations of thin fibers immersed in a viscous fluid by approximating each fiber using only the geometry of the fiber centerline curve and the line force density along it. However, it has been unclear how well slender body theory actually approximates Stokes flow about a thin but truly three‐dimensional fiber, in part due to the fact that simply prescribing

Matrices Φ  ∈ ℝ n  × p satisfying the restricted isometry property (RIP) are an important ingredient of the compressive sensing methods. While it is known that random matrices satisfy the RIP with high probability even for n  = logO (1)p , the explicit deteministic construction of such matrices defied the repeated efforts, and most of the known approaches hit the so‐called sparsity bottleneck. The

We review some widely studied models and firing dynamics for neuronal systems, both at the single cell and network level, and dynamical systems techniques to study them. In particular, we focus on two topics in mathematical neuroscience that have attracted the attention of mathematicians for decades: single-cell excitability and bursting. We review the mathematical framework for three types of excitability

We introduce vector diffusion maps (VDM), a new mathematical framework for organizing and analyzing massive high-dimensional data sets, images, and shapes. VDM is a mathematical and algorithmic generalization of diffusion maps and other nonlinear dimensionality reduction methods, such as LLE, ISOMAP, and Laplacian eigenmaps. While existing methods are either directly or indirectly related to the heat

We study the noneuclidean (incompatible) elastic energy functionals in the description of prestressed thin films at their singular limits (Γ ‐limits) as h  → 0 in the film's thickness h . First, we extend the prior results to arbitrary incompatibility metrics that depend on both the midplate and the transversal variables (the “nonoscillatory” case). Second, we analyze a more general class of incompatibilities

We rigorously construct continuous curves of rotating vortex patch solutions to the two‐dimensional Euler equations. The curves are large in that, as the parameter tends to infinity, the minimum along the interface of the angular fluid velocity in the rotating frame becomes arbitrarily small. This is consistent with the conjectured existence [30, 38] of singular limiting patches with 90 corners at

We prove a regularity result for Lagrangian flows of Sobolev vector fields over RCD(K, N) metric measure spaces; regularity is understood with respect to a newly defined quasi‐metric built from the Green function of the Laplacian. Its main application is that RCD(K, N) spaces have constant dimension. In this way we generalize to such an abstract framework a result proved by Colding‐Naber for Ricci

We study diffusion and mixing in different linear fluid dynamics models, mainly related to incompressible flows. In this setting, mixing is a purely advective effect that causes a transfer of energy to high frequencies. When diffusion is present, mixing enhances the dissipative forces. This phenomenon is referred to as enhanced dissipation, namely the identification of a timescale faster than the purely