We prove uniqueness of ground state solutions Q = Q(|x|) ≥ 0 of the non-linear equation $$(-\Delta)^s Q+Q-Q^{\alpha+1}= 0 \quad {\rm in} \, \mathbb{R},$$ ( − Δ ) s Q + Q − Q α + 1 = 0 i n R , where 0 < s < 1 and 0 < α < 4s/(1−2s) for $${s<\frac{1}{2}}$$ s < 1 2 and 0 < α < ∞ for $${s\geq \frac{1}{2}}$$ s ≥ 1 2 . Here (−Δ) s denotes the fractional Laplacian in one dimension. In particular, we answer affirmatively an open question recently raised by Kenig–Martel–Robbiano and we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for $${s=\frac{1}{2}}$$ s = 1 2 and α = 1 in [5] for the Benjamin–Ono equation.As a technical key result in this paper, we show that the associated linearized operator L + = (−Δ) s +1−(α+1)Q α is non-degenerate; i.e., its kernel satisfies ker L + = span{Q′}. This result about L + proves a spectral assumption, which plays a central role for the stability of solitary waves and blowup analysis for non-linear dispersive PDEs with fractional Laplacians, such as the generalized Benjamin–Ono (BO) and Benjamin–Bona–Mahony (BBM) water wave equations.

We prove uniqueness of ground state solutions Q = Q(vertical bar x vertical bar) >= 0 of the non-linear equation (-Delta)(s)Q+Q-Q(alpha+1)=0 in R, (-Delta) s Q + Q - Q alpha + 1 = 0 in R, where 0 = 1/2 s >= 1 2. Here (-Delta)(s) denotes the fractional Laplacian in one dimension. In particular, we answer affirmatively an open question recently raised by Kenig-Martel-Robbiano and we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for s=1/2 s=1 2 and alpha = 1 in [5] for the Benjamin-Ono equation. As a technical key result in this paper, we show that the associated linearized operator L+ = (-Delta)(s) +1-(alpha+1)Q(alpha) is non-degenerate; i.e., its kernel satisfies ker L+=span{Q'}. This result about L+ proves a spectral assumption, which plays a central role for the stability of solitary waves and blowup analysis for non-linear dispersive PDEs with fractional Laplacians, such as the generalized Benjamin-Ono (BO) and Benjamin-Bona-Mahony (BBM) water wave equations.

We construct families of two-dimensional travelling water waves propagating under the influence of gravity in a flow of constant vorticity over a flat bed, in particular establishing the existence of waves of large amplitude. A Riemann–Hilbert problem approach is used to recast the governing equations as a one-dimensional elliptic pseudodifferential equation with a scalar constraint. The structural properties of this formulation, which arises as the Euler–Lagrange equation of an energy functional, enable us to develop a theory of analytic global bifurcation.

Going beyond the linearized study has been a longstanding problem in the theory of Landau damping. In this paper we establish exponential Landau damping in analytic regularity. The damping phenomenon is reinterpreted in terms of transfer of regularity between kinetic and spatial variables, rather than exchanges of energy; phase mixing is the driving mechanism. The analysis involves new families of analytic norms, measuring regularity by comparison with solutions of the free transport equation; new functional inequalities; a control of non-linear echoes; sharp “deflection” estimates; and a Newton approximation scheme. Our results hold for any potential no more singular than Coulomb or Newton interaction; the limit cases are included with specific technical effort. As a side result, the stability of homogeneous equilibria of the non-linear Vlasov equation is established under sharp assumptions. We point out the strong analogy with the KAM theory, and discuss physical implications. Finally, we extend these results to some Gevrey (non-analytic) distribution functions.

In this paper, we consider the universality of the local eigenvalue statistics of random matrices. Our main result shows that these statistics are determined by the first four moments of the distribution of the entries. As a consequence, we derive the universality of eigenvalue gap distribution and k-point correlation, and many other statistics (under some mild assumptions) for both Wigner Hermitian matrices and Wigner real symmetric matrices.

We answer Hubbard's question on determining the Thurston equivalence class of “twisted rabbits”, i.e. composita of the “rabbit” polynomial with nth powers of the Dehn twists about its ears. The answer is expressed in terms of the 4-adic expansion of n. We also answer the equivalent question for the other two families of degree-2 topological polynomials with three post-critical points. In the process, we rephrase the questions in group-theoretical language, in terms of wreath recursions.

We prove that uniform random quadrangulations of the sphere with n faces, endowed with the usual graph distance and renormalized by n −1/4, converge as n → ∞ in distribution for the Gromov–Hausdorff topology to a limiting metric space. We validate a conjecture by Le Gall, by showing that the limit is (up to a scale constant) the so-called Brownian map, which was introduced by Marckert–Mokkadem and Le Gall as the most natural candidate for the scaling limit of many models of random plane maps. The proof relies strongly on the concept of geodesic stars in the map, which are configurations made of several geodesics that only share a common endpoint and do not meet elsewhere.

A longstanding question in the dual Brunn–Minkowski theory is “What are the dual analogues of Federer’s curvature measures for convex bodies?” The answer to this is provided. This leads naturally to dual versions of Minkowski-type problems: What are necessary and sufficient conditions for a Borel measure to be a dual curvature measure of a convex body? Sufficient conditions, involving measure concentration, are established for the existence of solutions to these problems.

We study the energy-critical focusing non-linear wave equation, with data in the energy space, in dimensions 3, 4 and 5. We prove that for Cauchy data of energy smaller than the one of the static solution W which gives the best constant in the Sobolev embedding, the following alternative holds. If the initial data has smaller norm in the homogeneous Sobolev space H 1 than the one of W, then we have global well-posedness and scattering. If the norm is larger than the one of W, then we have break-down in finite time.

This is a continuation of our earlier paper (Tao and Vu, http://arxiv.org/abs/0908.1982v4[math.PR],2010) on the universality of the eigenvalues of Wigner random matrices. The main new results of this paper are an extension of the results in Tao and Vu (http://arxiv.org/abs/0908.1982v4[math.PR], 2010) from the bulk of the spectrum up to the edge. In particular, we prove a variant of the universality results of Soshni-kov (Commun Math Phys 207(3):697-733, 1999) for the largest eigenvalues, assuming moment conditions rather than symmetry conditions. The main new technical observation is that there is a significant bias in the Cauchy interlacing law near the edge of the spectrum which allows one to continue ensuring the delocalization of eigenvectors.

We introduce a curvature-dimension condition CD (K, N) for metric measure spaces. It is more restrictive than the curvature bound $\underline{{{\text{Curv}}} {\left( {M,{\text{d}},m} \right)} > K$ (introduced in [Sturm K-T (2006) On the geometry of metric measure spaces. I. Acta Math 196:65–131]) which is recovered as the borderline case CD(K, ∞). The additional real parameter N plays the role of a generalized upper bound for the dimension. For Riemannian manifolds, CD(K, N) is equivalent to $${\text{Ric}}_{M} {\left( {\xi ,\xi } \right)} > K{\left| \xi \right|}^{2} $$ and dim(M) ⩽ N.The curvature-dimension condition CD(K, N) is stable under convergence. For any triple of real numbers K, N, L the family of normalized metric measure spaces (M, d, m) with CD(K, N) and diameter ⩽ L is compact.Condition CD(K, N) implies sharp version of the Brunn–Minkowski inequality, of the Bishop–Gromov volume comparison theorem and of the Bonnet–Myers theorem. Moreover, it implies the doubling property and local, scale-invariant Poincaré inequalities on balls. In particular, it allows to construct canonical Dirichlet forms with Gaussian upper and lower bounds for the corresponding heat kernels.

We introduce and analyze lower (Ricci) curvature bounds $ \underline{{Curv}} {\left( {M,d,m} \right)} $ ⩾ K for metric measure spaces $ {\left( {M,d,m} \right)} $ . Our definition is based on convexity properties of the relative entropy $ Ent{\left( { \cdot \left| m \right.} \right)} $ regarded as a function on the L 2-Wasserstein space of probability measures on the metric space $ {\left( {M,d} \right)} $ . Among others, we show that $ \underline{{Curv}} {\left( {M,d,m} \right)} $ ⩾ K implies estimates for the volume growth of concentric balls. For Riemannian manifolds, $ \underline{{Curv}} {\left( {M,d,m} \right)} $ ⩾ K if and only if $ Ric_{M} {\left( {\xi ,\xi } \right)} $ ⩾ K $ {\left| \xi \right|}^{2} $ for all $ \xi \in TM $ .The crucial point is that our lower curvature bounds are stable under an appropriate notion of D-convergence of metric measure spaces. We define a complete and separable length metric D on the family of all isomorphism classes of normalized metric measure spaces. The metric D has a natural interpretation, based on the concept of optimal mass transportation.We also prove that the family of normalized metric measure spaces with doubling constant ⩽ C is closed under D-convergence. Moreover, the family of normalized metric measure spaces with doubling constant ⩽ C and diameter ⩽ L is compact under D-convergence.

We define non-pluripolar products of an arbitrary number of closed positive (1, 1)-currents on a compact Kähler manifold X. Given a big (1, 1)-cohomology class α on X (i.e. a class that can be represented by a strictly positive current) and a positive measure μ on X of total mass equal to the volume of α and putting no mass on pluripolar sets, we show that μ can be written in a unique way as the top-degree self-intersection in the non-pluripolar sense of a closed positive current in α. We then extend Kolodziedj’s approach to sup-norm estimates to show that the solution has minimal singularities in the sense of Demailly if μ has L 1+ε -density with respect to Lebesgue measure. If μ is smooth and positive everywhere, we prove that T is smooth on the ample locus of α provided α is nef. Using a fixed point theorem, we finally explain how to construct singular Kähler–Einstein volume forms with minimal singularities on varieties of general type.

We study Schrödinger operators with a one-frequency analytic potential, focusing on the transition between the two distinct local regimes characteristic respectively of large and small potentials. From the dynamical point of view, the transition signals the emergence of non-uniform hyperbolicity, so the dependence of the Lyapunov exponent with respect to parameters plays a central role in the analysis. Though often ill-behaved by conventional measures, we show that the Lyapunov exponent is in fact remarkably regular in a “stratified sense” which we define: the irregularity comes from the matching of nice (analytic or smooth) functions along sets with complicated geometry. This result allows us to establish that the “critical set” for the transition lies within countably many codimension one subvarieties of the (infinite-dimensional) parameter space. A more refined renormalization-based analysis shows that the critical set is rather thin within those subvarieties, and allows us to conclude that a typical potential has no critical energies. Such acritical potentials also form an open set and have several interesting properties: only finitely many “phase transitions” may happen, but never at any specific point in the spectrum, and the Lyapunov exponent is minorated in the region of the spectrum where it is positive. On the other hand, we do show that the number of phase transitions can be arbitrarily large.Key to our approach are two results about the dependence of the Lyapunov exponent of one-frequency SL $${(2,\mathbb{C})}$$ ( 2 , C ) cocycles with respect to perturbations in the imaginary direction: on one hand there is a severe “quantization” restriction, and on the other hand “regularity” of the dependence characterizes uniform hyperbolicity when the Lyapunov exponent is positive. Our method is independent of arithmetic conditions on the frequency.

We prove that if μ is a d-dimensional Ahlfors-David regular measure in $${\mathbb{R}^{d+1}}$$ R d + 1 , then the boundedness of the d-dimensional Riesz transform in L 2(μ) implies that the non-BAUP David–Semmes cells form a Carleson family. Combined with earlier results of David and Semmes, this yields the uniform rectifiability of μ.