Showing papers in "Duke Mathematical Journal in 2018"

Conserved energies for the cubic nonlinear Schrödinger equation in one dimension

Abstract: Author(s): Koch, H; Tataru, D | Abstract: We consider the cubic nonlinear Schrodinger (NLS) equation as well as the modified Korteweg-de Vries (mKdV) equation in one space dimension. We prove that for each s g -1/2 there exists a conserved energy which is equivalent to the Hs-norm of the solution. For the Korteweg-de Vries (KdV) equation, there is a similar conserved energy for every s ≥-1.

Stochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion process

Abstract: We consider the asymmetric simple exclusion process (ASEP) on the positive integers with an open boundary condition. We show that, when starting devoid of particles and for a certain boundary condition, the height function at the origin fluctuates asymptotically (in large time τ) according to the Tracy–Widom Gaussian orthogonal ensemble distribution on the τ1/3-scale. This is the first example of Kardar–Parisi–Zhang asymptotics for a half-space system outside the class of free-fermionic/determinantal/Pfaffian models. Our main tool in this analysis is a new class of probability measures on Young diagrams that we call half-space Macdonald processes, as well as two surprising relations. The first relates a special (Hall–Littlewood) case of these measures to the half-space stochastic six-vertex model (which further limits to the ASEP) using a Yang–Baxter graphical argument. The second relates certain averages under these measures to their half-space (or Pfaffian) Schur process analogues via a refined Littlewood identity.

Monodromy dependence and connection formulae for isomonodromic tau functions

Abstract: We discuss an extension of the Jimbo–Miwa–Ueno differential 1-form to a form closed on the full space of extended monodromy data of systems of linear ordinary differential equations with rational coefficients. This extension is based on the results of M. Bertola, generalizing a previous construction by B. Malgrange. We show how this 1-form can be used to solve a long-standing problem of evaluation of the connection formulae for the isomonodromic tau functions which would include an explicit computation of the relevant constant factors. We explain how this scheme works for Fuchsian systems and, in particular, calculate the connection constant for the generic Painleve VI tau function. The result proves the conjectural formula for this constant proposed by Iorgov, Lisovyy, and Tykhyy. We also apply the method to non-Fuchsian systems and evaluate constant factors in the asymptotics of the Painleve II tau function.

Universal dynamics for the defocusing logarithmic Schrödinger equation

Abstract: We consider the nonlinear Schrodinger equation with a logarithmic nonlinearity in a dispersive regime. We show that the presence of the nonlinearity affects the large time behavior of the solution: the dispersion is faster than usual by a logarithmic factor in time and the positive Sobolev norms of the solution grow logarithmically in time. Moreover, after rescaling in space by the dispersion rate, the modulus of the solution converges to a universal Gaussian profile. These properties are suggested by explicit computations in the case of Gaussian initial data, and remain when an extra power-like nonlinearity is present in the equation. One of the key steps of the proof consists in using the Madelung transform to reduce the equation to a variant of the isothermal compressible Euler equation, whose large time behavior turns out to be governed by a parabolic equation involving a Fokker-Planck operator.

Current fluctuations of the stationary ASEP and six-vertex model

Abstract: Our results in this article are twofold. First, we consider current fluctuations of the stationary asymmetric simple exclusion process (ASEP), run for some long time T, and show that they are of order T1/3 along a characteristic line. Upon scaling by T1/3, we establish that these fluctuations converge to the long-time height fluctuations of the stationary Kardar–Parisi–Zhang (KPZ) equation, that is, to the Baik–Rains distribution. This result has long been predicted under the context of KPZ universality and in particular extends upon a number of results in the field, including the work of Ferrari and Spohn from 2005 (when they established the same result for the TASEP) and the work of Balazs and Seppalainen from 2010 (when they established the T1/3-scaling for the general ASEP). Second, we introduce a class of translation-invariant Gibbs measures that characterizes a one-parameter family of slopes for an arbitrary ferroelectric, symmetric six-vertex model. This family of slopes corresponds to what is known as the conical singularity (or tricritical point) in the free-energy profile for the ferroelectric six-vertex model. We consider fluctuations of the height function of this model on a large grid of size T and show that they too are of order T1/3 along a certain characteristic line; this confirms a prediction of Bukman and Shore from 1995, stating that the ferroelectric six-vertex model should exhibit KPZ growth at the conical singularity. Upon scaling the height fluctuations by T1/3, we again recover the Baik–Rains distribution in the large T limit. Recasting this statement in terms of the (asymmetric) stochastic six-vertex model confirms a prediction of Gwa and Spohn from 1992.

Gevrey stability of Prandtl expansions for $2$-dimensional Navier–Stokes flows

Abstract: We investigate the stability of boundary layer solutions of the 2-dimensional incompressible Navier–Stokes equations. We consider shear flow solutions of Prandtl type: uν(t,x,y)=(UE(t,y)+UBL(t,yν),0), 0<ν≪1. We show that if UBL is monotonic and concave in Y=y/ν, then uν is stable over some time interval (0,T), T independent of ν, under perturbations with Gevrey regularity in x and Sobolev regularity in y. We improve in this way the classical stability results of Sammartino and Caflisch in analytic class (both in x and y). Moreover, in the case where UBL is steady and strictly concave, our Gevrey exponent for stability is optimal. The proof relies on new and sharp resolvent estimates for the linearized Orr–Sommerfeld operator.

Compactification of strata of Abelian differentials

Abstract: We describe the closure of the strata of Abelian differentials with prescribed type of zeros and poles, in the projectivized Hodge bundle over the Deligne–Mumford moduli space of stable curves with marked points. We provide an explicit characterization of pointed stable differentials in the boundary of the closure, both a complex analytic proof and a flat geometric proof for smoothing the boundary differentials, and numerous examples. The main new ingredient in our description is a global residue condition arising from a full order on the dual graph of a stable curve.

On the maximum of the C$\beta$E field

Abstract: In this article, we investigate the extremal values of (the logarithm of) the characteristic polynomial of a random unitary matrix whose spectrum is distributed according to the circular beta ensemble (CβE). More precisely, assuming that Xn is this characteristic polynomial and U is the unit circle, we prove that sup z∈UℜlogXn(z)=2β(logn−34loglogn+O(1)), as well as an analogous statement for the imaginary part. The notation O(1) means that the corresponding family of random variables, indexed by n, is tight. This answers a conjecture of Fyodorov, Hiary, and Keating, originally formulated for the β=2 case, which corresponds to the circular unitary ensemble (CUE) field.

Quadratic Chabauty and rational points, I: $p$-adic heights

Abstract: We give the first explicit examples beyond the Chabauty-Coleman method where Kim's nonabelian Chabauty program determines the set of rational points of a curve defined over $\mathbb{Q}$ or a quadratic number field. We accomplish this by studying the role of $p$-adic heights in explicit nonabelian Chabauty.

Integer homology 3-spheres admit irreducible representations in SL(2,C)

Abstract: We prove that the fundamental group of any integer homology 3-sphere different from the 3-sphere admits irreducible representations of its fundamental group in SL.(2,C). For hyperbolic integer homology spheres, this comes with the definition; for Seifert fibered integer homology spheres, this is well known. We prove that the splicing of any two nontrivial knots in S-3 admits an irreducible SU.(2)-representation. Using a result of Kuperberg, we get the corollary that the problem of 3-sphere recognition is in the complexity class coNP, provided the generalized Riemann hypothesis holds. To prove our result, we establish a topological fact about the image of the SU.(2)representation variety of a nontrivial knot complement into the representation variety of its boundary torus, a pillowcase, using holonomy perturbations of the Chern-Simons function in an exhaustive way-showing that any area-preserving self-map of the pillowcase fixing the four singular points, and which is isotopic to the identity, can be C degrees-approximated by maps realized geometrically through holonomy perturbations of the flatness equation in a thickened torus. We conclude with a stretching argument in instanton gauge theory and a nonvanishing result of Kronheimer and Mrowka for Donaldson's invariants of a 4-manifold which contains the 0-surgery of a knot as a splitting hypersurface.

A $p$-adic Waldspurger formula

Abstract: In this article, we study p-adic torus periods for certain p-adic-valued functions on Shimura curves of classical origin. We prove a p-adic Waldspurger formula for these periods as a generalization of recent work of Bertolini, Darmon, and Prasanna. In pursuing such a formula, we construct a new anti-cyclotomic p-adic L-function of Rankin–Selberg type. At a character of positive weight, the p-adic L-function interpolates the central critical value of the complex Rankin–Selberg L-function. Its value at a finite-order character, which is outside the range of interpolation, essentially computes the corresponding p-adic torus period.

The colored HOMFLYPT function is $q$-holonomic

Abstract: We prove that the HOMFLYPT polynomial of a link colored by partitions with a fixed number of rows is a q-holonomic function. By specializing to the case of knots colored by a partition with a single row, it proves the existence of an (a,q) superpolynomial of knots in 3-space, as was conjectured by string theorists. Our proof uses skew-Howe duality that reduces the evaluation of web diagrams and their ladders to a Poincare–Birkhoff–Witt computation of an auxiliary quantum group of rank the number of strings of the ladder diagram. The result is a concrete and algorithmic web evaluation algorithm that is manifestly q-holonomic.

The Breuil–Mézard conjecture when l≠p

Abstract: Let l and p be primes, let F=Qp be a finite extension with absolute Galois group GF , let F be a finite field of characteristic l, and let W GF ! GLn.F/ be a continuous representation. Let R./ be the universal framed deformation ring for . If l D p, then the Breuil–Mezard conjecture (as recently formulated by Emerton and Gee) relates the mod l reduction of certain cycles in R./ to the mod l reduction of certain representations of GLn.OF /. We state an analogue of the Breuil–Mezard conjecture when l ¤ p, and we prove it whenever l>2 using automorphy lifting theorems. We give a local proof when l is “quasibanal” for F and is tamely ramified. We also analyze the reduction modulo l of the types . / defined by Schneider and Zink.

Spectral asymptotics for sub-Riemannian Laplacians. I: quantum ergodicity and quantum limits in the 3D contact case.

Abstract: This is the first paper of a series in which we plan to study spectral asymptotics for sub-Riemannian Laplacians and to extend results that are classical in the Riemannian case concerning Weyl measures, quantum limits, quantum ergodicity, quasi-modes, trace formulae. Even if hypoelliptic operators have been well studied from the point of view of PDE's, global geometrical and dynamical aspects have not been the subject of much attention. As we will see, already in the simplest case, the statements of the results in the sub-Riemannian setting are quite different from those in the Riemannian one. Let us consider a sub-Riemannian (sR) metric on a closed three-dimensional manifold with an oriented contact distribution. There exists a privileged choice of the contact form, with an associated Reeb vector field and a canonical volume form that coincides with the Popp measure. We establish a Quantum Ergodicity (QE) theorem for the eigenfunctions of any associated sR Laplacian under the assumption that the Reeb flow is ergodic. The limit measure is given by the normalized Popp measure. This is the first time that such a result is established for a hypoelliptic operator, whereas the usual Shnirelman theorem yields QE for the Laplace-Beltrami operator on a closed Riemannian manifold with ergodic geodesic flow. To prove our theorem, we first establish a microlocal Weyl law, which allows us to identify the limit measure and to prove the microlocal concentration of the eigenfunctions on the characteristic manifold of the sR Laplacian. Then, we derive a Birkhoff normal form along this characteristic manifold, thus showing that, in some sense, all 3D contact structures are microlocally equivalent. The quantum version of this normal form provides a useful microlocal factorization of the sR Laplacian. Using the normal form, the factorization and the ergodicity assumption, we finally establish a variance estimate, from which QE follows. We also obtain a second result, which is valid without any ergodicity assumption: every Quantum Limit (QL) can be interpreted as the sum of two mutually singular measures: the first measure is supported on the unit cotangent bundle and is invariant under the sR geodesic flow, and the second measure is supported on the characteristic manifold of the sR Laplacian and is invariant under the lift of the Reeb flow. Moreover, we prove that the first measure is zero for most QL's.

Uniform rectifiability from Carleson measure estimates and ε-approximability of bounded harmonic functions

Abstract: Let Ω⊂Rn+1, n≥1, be a corkscrew domain with Ahlfors–David regular boundary. In this article we prove that ∂Ω is uniformly n-rectifiable if every bounded harmonic function on Ω is e-approximable or if every bounded harmonic function on Ω satisfies a suitable square-function Carleson measure estimate. In particular, this applies to the case when Ω=Rn+1∖E and E is Ahlfors–David regular. Our results establish a conjecture posed by Hofmann, Martell, and Mayboroda, in which they proved the converse statements. Here we also obtain two additional criteria for uniform rectifiability, one in terms of the so-called S

Hypersymplectic 4-Manifolds, The G2-laplacian flow, and extension assuming bounded scalar curvature

Abstract: A hypersymplectic structure on a 4-manifold X is a triple ω of symplectic forms which at every point span a maximal positive definite subspace of Λ2 for the wedge product. This article is motivated by a conjecture by Donaldson: when X is compact, ω can be deformed through cohomologous hypersymplectic structures to a hyper-Kahler triple. We approach this via a link with G2-geometry. A hypersymplectic structure ω on a compact manifold X defines a natural G2-structure ϕ on X×T3 which has vanishing torsion precisely when ω is a hyper-Kahler triple. We study the G2-Laplacian flow starting from ϕ, which we interpret as a flow of hypersymplectic structures. Our main result is that the flow extends as long as the scalar curvature of the corresponding G2-structure remains bounded. An application of our result is a lower bound for the maximal existence time of the flow in terms of weak bounds on the initial data (and with no assumption that scalar curvature is bounded along the flow).

A geometric characterization of toric varieties

Abstract: We prove a conjecture of Shokurov which characterises toric varieties using log pairs.

High-frequency backreaction for the Einstein equations under polarized U(1)-symmetry

Abstract: Known examples in plane symmetry or Gowdy symmetry show that, given a 1-parameter family of solutions to the vacuum Einstein equations, it may have a weak limit which does not satisfy the vacuum equations, but instead has a nontrivial stress-energy-momentum tensor. We consider this phenomenon under polarized U(1)-symmetry—a much weaker symmetry than most of the known examples—such that the stress-energy-momentum tensor can be identified with that of multiple families of null dust propagating in distinct directions. We prove that any generic local-in-time small-data polarized U(1)-symmetric solution to the Einstein–multiple null dust system can be achieved as a weak limit of vacuum solutions. Our construction allows the number of families to be arbitrarily large and appears to be the first construction of such examples with more than two families.

On the conservativity of the functor assigning to a motivic spectrum its motive

Abstract: Given a 0-connective motivic spectrum E is an element of SH(k) over a perfect field k, we determine (h) under bar (0) of the associated motive ME is an element of DM(k) in terms of (pi) under bar (0). (E). Using this, we show that if k has finite 2-etale cohomological dimension, then the functor M W SH(k) -> DM(k) is conservative when restricted to the subcategory of compact spectra and induces an injection on Picard groups. We extend the conservativity result to fields of finite virtual 2-etale cohomological dimension by considering what we call real motives.

A minimization problem with free boundary related to a cooperative system

Abstract: We study the minimum problem for the functional ∫Ω(|∇u|2+Q2χ{|u|>0})dx with the constraint ui≥0 for i=1,…,m, where Ω⊂Rn is a bounded domain and u=(u1,…,um)∈H1(Ω;Rm). First we derive the Euler equation satisfied by each component. Then we show that the noncoincidence set {|u|>0} is (locally) nontangentially accessible. Having this, we are able to establish sufficient regularity of the force term appearing in the Euler equations and derive the regularity of the free boundary Ω∩∂{|u|>0}.

Bohr sets and multiplicative Diophantine approximation

Abstract: In two dimensions, Gallagher’s theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. We prove an inhomogeneous fiber version of Gallagher’s theorem, sharpening and making unconditional a result recently obtained conditionally by Beresnevich, Haynes, and Velani. The idea is to find large generalized arithmetic progressions within inhomogeneous Bohr sets, extending a construction given by Tao. This precise structure enables us to verify the hypotheses of the Duffin–Schaeffer theorem for the problem at hand, via the geometry of numbers.

Rigidity of critical circle maps

Abstract: We prove that any two C4 critical circle maps with the same irrational rotation number and the same odd criticality are conjugate to each other by a C1 circle diffeomorphism. The conjugacy is C1+α for a full Lebesgue measure set of rotation numbers.

Espaces de Banach–Colmez et faisceaux cohérents sur la courbe de Fargues–Fontaine

Abstract: We give a new definition, simpler but equivalent, of the abelian category of Banach–Colmez spaces introduced by Colmez, and we explain the precise relationship with the category of coherent sheaves on the Fargues–Fontaine curve. One goes from one category to the other by changing the t-structure on the derived category. Along the way we obtain a description of the proetale cohomology of the open disk and the affine space, which is of independent interest.

The Sard conjecture on Martinet surfaces

Abstract: Given a totally nonholonomic distribution of rank two on a three-dimensional manifold we investigate the size of the set of points that can be reached by singular horizontal paths starting from a same point. In this setting, the Sard conjecture states that that set should be a subset of the so-called Martinet surface of 2-dimensional Hausdorff measure zero. We prove that the conjecture holds in the case where the Martinet surface is smooth. Moreover, we address the case of singular real-analytic Martinet surfaces and show that the result holds true under an assumption of non-transversality of the distribution on the singular set of the Martinet surface. Our methods rely on the control of the divergence of vector fields generating the trace of the distribution on the Martinet surface and some techniques of resolution of singularities.

Counting points of schemes over finite rings and counting representations of arithmetic lattices

Abstract: We relate the singularities of a scheme X to the asymptotics of the number of points of X over finite rings. This gives a partial answer to a question of Mustata. We use this result to count representations of arithmetic lattices. More precisely, if Γ is an arithmetic lattice whose Q-rank is greater than 1, then let rn(Γ) be the number of irreducible n-dimensional representations of Γ up to isomorphism. We prove that there is a constant C (in fact, any C>40 suffices) such that rn(Γ)=O(nC) for every such Γ. This answers a question of Larsen and Lubotzky.

Exceptional isogenies between reductions of pairs of elliptic curves

Abstract: Let E and E' be two elliptic curves over a number field. We prove that the reductions of E and E' at a finite place p are geometrically isogenous for infinitely many p, and we draw consequences for the existence of supersingular primes. This result is an analogue for distributions of Frobenius traces of known results on the density of Noether–Lefschetz loci in Hodge theory. The proof relies on dynamical properties of the Hecke correspondences on the modular curve.

Analytic torsion and R-torsion of Witt representations on manifolds with cusps

Abstract: We establish a Cheeger–Muller theorem for unimodular representations satisfying a Witt condition on a noncompact manifold with cusps. This class of spaces includes all noncompact hyperbolic spaces of finite volume, but we do not assume that the metric has constant curvature nor that the link of the cusp is a torus. We use renormalized traces in the sense of Melrose to define the analytic torsion, and we relate it to the intersection R-torsion of Dar of the natural compactification to a stratified space. Our proof relies on our recent work on the behavior of the Hodge Laplacian spectrum on a closed manifold undergoing degeneration to a manifold with fibered cusps.

On finiteness properties of the Johnson filtrations

Abstract: Let Γ be either the automorphism group of the free group of rank n≥4 or the mapping class group of an orientable surface of genus n≥12 with at most 1 boundary component, and let G be either the subgroup of IA-automorphisms or the Torelli subgroup of Γ. For N∈N denote by γNG the Nth term of the lower central series of G. We prove that (i) any subgroup of G containing γ2G=[G,G] (in particular, the Johnson kernel in the mapping class group case) is finitely generated; (ii) if N=2 or n≥8N−4 and K is any subgroup of G containing γNG (for instance, K can be the Nth term of the Johnson filtration of G), then G/[K,K] is nilpotent and hence the Abelianization of K is finitely generated; (iii) if H is any finite-index subgroup of Γ containing γNG, with N as in (ii), then H has finite Abelianization.

On the marked length spectrum of generic strictly convex billiard tables

Abstract: In this paper we show that for a generic strictly convex domain, one can recover the eigendata corresponding to Aubry–Mather periodic orbits of the induced billiard map from the (maximal) marked length spectrum of the domain.

Carathéodory’s metrics on Teichmüller spaces and L-shaped pillowcases

Abstract: One of the most important results in Teichmuller theory is Royden’s theorem, which says that the Teichmuller and Kobayashi metrics agree on the Teichmuller space of a given closed Riemann surface. The problem that remained open is whether the Caratheodory metric agrees with the Teichmuller metric as well. In this article, we prove that these two metrics disagree on each T_g, the Teichmuller space of a closed surface of genus g ≥ 2. The main step is to establish a criterion to decide when the Teichmuller and Caratheodory metrics agree on the Teichmuller disk corresponding to a rational Jenkins–Strebel differential φ. First, we construct a holomorphic embedding ℰ:H^k → Tg,n corresponding to φ. The criterion says that the two metrics agree on this disk if and only if a certain function Φ: ℰ (H^k) → H can be extended to a holomorphic function Φ : T_(g,n) → H. We then show by explicit computation that this is not the case for quadratic differentials arising from L-shaped pillowcases.