Additivity of Higher Rho Invariants and Nonrigidity of Topological Manifolds

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 group of X to the analytic structure group of X . Here X is the universal cover of X , Γ = π₁X is the fundamental group of X , and is a certain C^* ‐algebra. In fact, we introduce a higher rho invariant map on the homology manifold structure group of a closed oriented connected topological manifold, and prove its additivity. This higher rho invariant map restricts to the higher rho invariant map on the topological structure group. More generally, the same techniques developed in this paper can be applied to define a higher rho invariant map on the homology manifold structure group of a closed oriented connected homology manifold. As an application, we use the additivity of the higher rho invariant map to study nonrigidity of topological manifolds. More precisely, we give a lower bound for the free rank of the algebraically reduced structure group of X by the number of torsion elements in π₁X . Here the algebraically reduced structure group of X is the quotient of modulo a certain action of self‐homotopy equivalences of X . We also introduce a notion of homological higher rho invariant, which can be used to detect many elements in the structure group of a closed oriented topological manifold, even when the fundamental group of the manifold is torsion free. In particular, we apply this homological higher rho invariant to show that the structure group is not finitely generated for a class of manifolds. © 2020 Wiley Periodicals LLC     

A Liouville Theorem for the Complex Monge-Ampère Equation on Product Manifolds

Let Y be a closed Calabi-Yau manifold. Let ω be the Kähler form of a Ricci-flat Kähler metric on . We prove that if ω is uniformly bounded above and below by constant multiples of , where is the standard flat Kähler form on and ω_Y is any Kähler form on Y, then ω is a product Kähler form up to a certain automorphism of . © 2018 Wiley Periodicals, Inc.     

Identification of the Polaron Measure I: Fixed Coupling Regime and the Central Limit Theorem for Large Times

We consider the Fröhlich model of the polaron, whose path integral formulation leads to the transformed path measure with respect to ℙ that governs the law of the increments of the three-dimensional Brownian motion on a finite interval [−T, T] , and Z_{α, T} is the partition function or the normalizing constant and α > 0 is a constant, or the coupling parameter. The polaron measure reflects a self-attractive interaction. According to a conjecture of Pekar that was proved in [9], exists and has a variational formula. In this article we show that when α > 0 is either sufficiently small or sufficiently large, the limit exists, which is also identified explicitly. As a corollary, we deduce the central limit theorem for under and obtain an expression for the limiting variance. © 2019 the Authors. Communications on Pure and Applied Mathematics is published by the Courant Institute of Mathematical Sciences and Wiley Periodicals, LLC.     

Pseudospectral Bound and Transition Threshold for the 3D Kolmogorov Flow

In this paper, we study pseudospectral bounds for the linearized operator of the Navier-Stokes equations around the 3D Kolmogorov flow. Using the pseudospectral bound and the wave operator method introduced in [22], we prove the sharp enhanced dissipation rate for the linearized Navier-Stokes equations. As an application, we prove that if the initial velocity satisfies (ν the viscosity coefficient) and k_f ∈ (0, 1), then the solution does not transition away from the Kolmogorov flow. © 2019 Wiley Periodicals, Inc.     

On Heavy-Tail Phenomena in Some Large-Deviations Problems

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 for a certain class of functionals , where is some metric space, under the n -fold probability measure , where α ∈ (0, 2] , for which the large deviations are due to translations. We retrieve, as an application, the large-deviations principles known for the Wigner matrices without Gaussian tails, in works by Bordenave and Caputo on one hand, and the author on the other hand, of the empirical spectral measure, the largest eigenvalue, and traces of polynomials. We also apply our large-deviations result to the last-passage time, which yields a large-deviations principle when the weights follow the law , with α ∈ (0, 1) . © 2020 Wiley Periodicals LLC     

Global Regularity of 2D Density Patches for Viscous Inhomogeneous Incompressible Flow with General Density: Low Regularity Case

This paper presents some progress toward an open question proposed by P.-L. Lions [26] concerning the propagation of regularities of density patches for viscous inhomogeneous incompressible flow. We first establish the global-in-time well-posedness of the two-dimensional inhomogeneous incompressible Navier-Stokes system with initial density . Here is any pair of positive constants and Ω₀ is a bounded, simply connected domain. We then prove that for any positive time t, the density , with the domain Ω(t) preserving the -boundary regularity. © 2018 Wiley Periodicals, Inc.     

Upper Tail Large Deviations of Regular Subgraph Counts in Erdős-Rényi Graphs in the Full Localized Regime

For a -regular connected graph H the problem of determining the upper tail large deviation for the number of copies of H in , an Erdős-Rényi graph on n vertices with edge probability p, has generated significant interest. For and , where is the number of vertices in H, the upper tail large deviation event is believed to occur due to the presence of localized structures. In this regime the large deviation event that the number of copies of H in exceeds its expectation by a constant factor is predicted to hold at a speed , and the rate function is conjectured to be given by the solution of a mean-field variational problem. After a series of developments in recent years, covering progressively broader ranges of p, the upper tail large deviations for cliques of fixed size were proved by Harel, Mousset, and Samotij in the entire localized regime. This paper establishes the conjecture for all connected regular graphs in the whole localized regime. © 2021 Wiley Periodicals LLC.     

Sharp Estimate of Global Coulomb Gauge

Let A be a W^{1, 2} -connection on a principal SU(2) -bundle P over a compact 4 -manifold M whose curvature F_A 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 ℒ^{4, ∞} is a new function space we introduce in this paper that satisfies L⁴(M) ⊊ ℒ^{4, ∞}(M) ⊊ L^{4 − ϵ}(M) for all ϵ > 0 . More precisely, ℒ^{4, ∞}(M) is the collection of measurable function u such that , where L^{4, ∞} is the classical Lorentz space and s_u is the L⁴ -integrability radius function associated to u defined by Briefly speaking, we achieve the estimate of by showing that σ^*A is effectively L⁴ -integrable away from controllably many points on M . © 2020 Wiley Periodicals LLC     

Robustness of Liouville Measure under a Family of Stable Diffusions

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 Δ^s 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 stationary probability measure for the leafwise diffusion of converge to the normalized Liouville measure on SM . © 2020 Wiley Periodicals LLC     

A De Giorgi–Type Conjecture for Minimal Solutions to a Nonlinear Stokes Equation

We study the one-dimensional symmetry of solutions to the nonlinear Stokes equation which are periodic in the d − 1 last variables (living on the torus 𝕋^d−1) and globally minimize the corresponding energy in Ω = ℝ × 𝕋^d−1, i.e., Namely, we find a class of nonlinear potentials W ≥ 0 such that any global minimizer u of E connecting two zeros of W as x₁ → ± ∞ is one-dimensional; i.e., u depends only on the x₁ -variable. In particular, this class includes in dimension d = 2 the nonlinearities with w being a harmonic function or a solution to the wave equation, while in dimension d ≥ 3 , this class contains a perturbation of the Ginzburg-Landau potential as well as potentials W having d + 1 wells with prescribed transition cost between the wells. For that, we develop a theory of calibrations relying on the notion of entropy (coming from scalar conservation laws). We also study the problem of the existence of global minimizers of E for general potentials W providing in particular compactness results for uniformly finite energy maps u in Ω connecting two wells of W as x₁ → ± ∞ . © 2019 Wiley Periodicals, Inc.     

Unique Continuation at the Boundary for Harmonic Functions in C1 Domains and Lipschitz Domains with Small Constant

Let be a domain, or more generally, a Lipschitz domain with small local Lipschitz constant. In this paper it is shown that if u is a function harmonic in and continuous in , which vanishes in a relatively open subset ; moreover, the normal derivative vanishes in a subset of with positive surface measure; then u is identically zero. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.     

Branching Brownian Motion with Decay of Mass and the Nonlocal Fisher-KPP Equation

In this work we study a nonlocal version of the Fisher-KPP equation, and its relation to a branching Brownian motion with decay of mass as introduced in Addario-Berry and Penington (2015) , i.e., a particle system consisting of a standard branching Brownian motion (BBM) with a competitive interaction between nearby particles. Particles in the BBM with decay of mass have a position in ℝ and a mass, branch at rate 1 into two daughter particles of the same mass and position, and move independently as Brownian motions. Particles lose mass at a rate proportional to the mass in a neighborhood around them (as measured by the function φ). We obtain two types of results. First, we study the behavior of solutions to the partial differential equation above. We show that, under suitable conditions on φ and u₀, the solutions converge to 1 behind the front and are globally bounded, improving recent results of Hamel and Ryzhik. Second, we show that the hydrodynamic limit of the BBM with decay of mass is the solution of the nonlocal Fisher-KPP equation. We then harness this to obtain several new results concerning the behavior of the particle system. © 2019 Wiley Periodicals, Inc.     

Upper Bounds on Liouville First-Passage Percolation and Watabiki's Prediction

Given a planar continuum Gaussian free field h^𝒰 in a domain 𝒰 with Dirichlet boundary condition and any δ > 0, we let be a real-valued smooth Gaussian process where is the average of h^𝒰 along a circle of radius δ with center v. For γ > 0, we study the Liouville first-passage percolation (in scale δ), i.e., the shortest path distance in 𝒰 where the weight of each path P is given by . We show that the distance between two typical points is for all sufficiently small but fixed γ > 0 and some constant c^* > 0. In addition, we obtain similar upper bounds on the Liouville first-passage percolation for discrete Gaussian free fields, as well as the Liouville graph distance, which roughly speaking is the minimal number of euclidean balls with comparable Liouville quantum gravity measure whose union contains a continuous path between two endpoints. Our results contradict some reasonable interpretations of Watabiki's prediction (1993) on the random distance of Liouville quantum gravity at high temperatures.© 2019 Wiley Periodicals, Inc.     

Inviscid Limit of Vorticity Distributions in the Yudovich Class

We prove that given initial data , forcing and any T > 0, the solutions u^ν 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 L^p vorticity topology. The main tool in these proofs is a uniformly controlled loss of regularity property of the linear transport by Yudovich solutions. Our results provide a partial foundation for the Miller-Robert statistical equilibrium theory of vortices as it applies to slightly viscous fluids. © 2020 Wiley Periodicals LLC.     

Large ∣k∣ Behavior of Complex Geometric Optics Solutions to d-bar Problems

Complex geometric optics solutions to a system of d-bar equations appearing in the context of electrical impedance tomography and the scattering theory of the integrable Davey-Stewartson II equations are studied for large values of the spectral parameter k. For potentials $q\in {⟨\cdot ⟩}^{-2}{H}^s\left(ℂ\right)$ for some $s\in \left]1,2\right]$, it is shown that the solution converges as the geometric series in $1/{\left|k\right|}^{s-1}$. For potentials q being the characteristic function of a strictly convex open set with smooth boundary, this still holds with s = 3/2, i.e., with $1/\sqrt{\mid k\mid }$ instead of $1/{\left|k\right|}^{s-1}$. The leading-order contributions are computed explicitly. Numerical simulations show the applicability of the asymptotic formulae for the example of the characteristic function of the disk. © 2022 Courant Institute of Mathematics and Wiley Periodicals LLC.     

From 1 to 6: A Finer Analysis of Perturbed Branching Brownian Motion

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.i.d. case , and when . This is due to the localization of extremal particles at the time of speed change, which depends on α and differs from the one in standard branching Brownian motion. We also establish in all cases the asymptotic law of the maximum and characterize the extremal process, which turns out to coincide essentially with that of standard branching Brownian motion. © 2020 the Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC     

Online Prediction with History-Dependent Experts: The General Case

We study the problem of prediction of binary sequences with expert advice in the online setting, which is a classic example of online machine learning. We interpret the binary sequence as the price history of a stock, and view the predictor as an investor, which converts the problem into a stock prediction problem. In this framework, an investor, who predicts the daily movements of a stock, and an adversarial market, who controls the stock, play against each other over N turns. The investor combines the predictions of $n\ge 2$ experts in order to make a decision about how much to invest at each turn, and aims to minimize their regret with respect to the best-performing expert at the end of the game. We consider the problem with history-dependent experts, in which each expert uses the previous d days of history of the market in making their predictions. We prove that the value function for this game, rescaled appropriately, converges as $N\to \mathrm{\infty }$ at a rate of $O\left(N^{-1/6}\right)$ to the viscosity solution of a nonlinear degenerate elliptic PDE, which can be understood as the Hamilton-Jacobi-Issacs equation for the two-person game. As a result, we are able to deduce asymptotically optimal strategies for the investor. Our results extend those established by the first author and R.V. Kohn [14] for $n=2$ experts and $d\le 4$ days of history. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.     

Sigma Delta Quantization for Images

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 and practical merits as their one-dimensional counterparts, and (ii) we show that both the traditional 1-dimensional and the new 2-dimensional quantization schemes can effectively quantize signals with jump discontinuities, which immediately enable the usage of adaptive quantization on images. Under mild conditions, we show that by using adaptivity, the proposed method is able to reduce the quantization error of images from the presently best $O\left(\sqrt{P}\right)$ to the much smaller $O\left(\sqrt{s}\right)$, where s is the number of jump discontinuities in the image and P ($P\gg s$) is the total number of samples. This $\sqrt{P/s}$-fold error reduction is achieved via applying a total variation norm regularized decoder, whose formulation is inspired by the mathematical super-resolution theory in the field of compressed sensing. Compared to the super-resolution setting, our error reduction is achieved without requiring adjacent spikes/discontinuities to be well-separated, which ensures its broad scope of application. We numerically demonstrate the efficacy of the new scheme on medical and natural images. We observe that for images with small pixel intensity values, the new method can significantly increase image quality over the state-of-the-art method. © 2022 Wiley Periodicals, Inc.