Showing papers in "Duke Mathematical Journal in 2011"
TL;DR: In this article, a well-posedness theory for weak measure solutions of the Cauchy problem for a family of nonlocal interaction equations is provided, which enables the solution to form atomic parts of the measure in finite time.
Abstract: In this paper we provide a well-posedness theory for weak measure solutions of the Cauchy problem for a family of nonlocal interaction equations. These equations are continuum models for interacting particle systems with attractive/repulsive pairwise interaction potentials. The main phenomenon of interest is that, even with smooth initial data, the solutions can concentrate mass in finite time. We develop an existence theory that enables one to go beyond the blow-up time in classical norms and allows for solutions to form atomic parts of the measure in finite time. The weak measure solutions are shown to be unique and exist globally in time. Moreover, in the case of sufficiently attractive potentials, we show the finite-time total collapse of the solution onto a single point for compactly supported initial measures. Our approach is based on the theory of gradient flows in the space of probability measures endowed with the Wasserstein metric. In addition to classical tools, we exploit the stability of the flow with respect to the transportation distance to greatly simplify many problems by reducing them to questions about particle approximations.
327 citations
TL;DR: In this paper, a new explicit construction of n×N matrices satisfying the Restricted Isometry Property (RIP) was given, which overcomes the natural barrier k=O(n1/2) for proofs based on small coherence, which was used in all previous explicit constructions of RIP matrices.
Abstract: We give a new explicit construction of n×N matrices satisfying the Restricted Isometry Property (RIP). Namely, for some ϵ>0, large N, and any n satisfying N1−ϵ≤n≤N, we construct RIP matrices of order k≥n1/2+ϵ and constant δ=n−ϵ. This overcomes the natural barrier k=O(n1/2) for proofs based on small coherence, which are used in all previous explicit constructions of RIP matrices. Key ingredients in our proof are new estimates for sumsets in product sets and for exponential sums with the products of sets possessing special additive structure. We also give a construction of sets of n complex numbers whose kth moments are uniformly small for 1≤k≤N (Turan's power sum problem), which improves upon known explicit constructions when (logN)1+o(1)≤n≤(logN)4+o(1). This latter construction produces elementary explicit examples of n×N matrices that satisfy the RIP and whose columns constitute a new spherical code; for those problems the parameters closely match those of existing constructions in the range (logN)1+o(1)≤n≤(logN)5/2+o(1).
222 citations
TL;DR: In this article, a quasi-equivalence between matrix factorizations and differential graded (dg) derived categories of an explicitly computable dg algebra has been established, and the Hochschild chain and cochain complexes of these categories are derived.
Abstract: We study the category of matrix factorizations associated to the germ of an isolated hypersurface singularity. This category is shown to admit a compact generator which is given by the stabilization of the residue field. We deduce a quasi-equivalence between the category of matrix factorizations and the differential graded (dg) derived category of an explicitly computable dg algebra. Building on this result, we employ a variant of Toen's derived Morita theory to identify continuous functors between matrix factorization categories as integral transforms. This enables us to calculate the Hochschild chain and cochain complexes of these categories. Finally, we give interpretations of the results of this work in terms of noncommutative geometry based on dg categories.
193 citations
TL;DR: In this article, it was shown that fluctuations of linear statistics of eigenvalues of random normal matrices converge on compact subsets of the interior of the droplet to a Gaussian field.
Abstract: In this article, we consider a fairly general potential in the plane and the corresponding Boltzmann-Gibbs distribution of eigenvalues of random normal matrices. As the order of the matrices tends to infinity, the eigenvalues condensate on a certain compact subset of the plane—the “droplet.” We prove that fluctuations of linear statistics of eigenvalues of random normal matrices converge on compact subsets of the interior of the droplet to a Gaussian field, and we discuss various ramifications of this result.
186 citations
TL;DR: This article showed that the smoothing effect for the surface tension water waves proved by H. Christianson, V. M. Hur, and G. Staffilani is in fact a rather direct consequence of this reduction, which allows also to lower the regularity indexes of the initial data, and to obtain the natural weights in the estimates.
Abstract: The purpose of this article is to clarify the Cauchy theory of the water waves equations as well in terms of regularity indexes for the initial conditions as for the smoothness of the bottom of the domain (namely no regularity assumption is assumed on the bottom). Our main result is that, following the approach developped by T. Alazard and G. Metivier in [1], after suitable paralinearizations, the system can be arranged into an explicit symmetric system of Schrodinger type. We then show that the smoothing effect for the (one dimensional) surface tension water waves proved by H. Christianson, V. M. Hur, and G. Staffilani in [9], is in fact a rather direct consequence of this reduction, which allows also to lower the regularity indexes of the initial data, and to obtain the natural weights in the estimates.
184 citations
TL;DR: In this paper, a refined trilinear Strichartz estimate for solutions to the Schrodinger equation on the flat rational torus T3 is derived by a suitable modification of critical function space theory.
Abstract: A refined trilinear Strichartz estimate for solutions to the Schrodinger equation on the flat rational torus T3 is derived. By a suitable modification of critical function space theory this is applied to prove a small data global well-posedness result for the quintic nonlinear Schrodinger equation in Hs(T3) for all s≥1. This is the first energy-critical global well-posedness result in the setting of compact manifolds.
152 citations
TL;DR: In this paper, it was shown that the electromagnetic material parameters are uniquely determined by boundary measurements for the time-harmonic Maxwell equations in certain anisotropic settings, and the authors gave a uniqueness result in the inverse problem for Maxwell equations on an admissible Riemannian manifold.
Abstract: We prove that the electromagnetic material parameters are uniquely determined by boundary measurements for the time-harmonic Maxwell equations in certain anisotropic settings We give a uniqueness result in the inverse problem for Maxwell equations on an admissible Riemannian manifold and a uniqueness result for Maxwell equations in Euclidean space with admissible matrix coefficients The proofs are based on a new Fourier analytic construction of complex geometrical optics solutions on admissible manifolds and involve a proper notion of uniqueness for such solutions
110 citations
TL;DR: In this article, the existence of a metric of positive sectional curvature on the 3-Sasakian 7-manifold over Hitchin's self-dual Einstein orbifold with k = 5 was shown.
Abstract: We demonstrate the existence of a metric of positive sectional curvature on the 3-Sasakian 7-manifold over Hitchin's self-dual Einstein orbifold with k=5.
110 citations
TL;DR: In this article, the authors study the space of stability conditions on the total space of the canonical bundle over the projective plane and explicitly describe a chamber of geometric stability conditions, and show that its translates via autoequivalences cover a whole connected component.
Abstract: We study the space of stability conditions on the total space of the canonical bundle over the projective plane. We explicitly describe a chamber of geometric stability conditions, and show that its translates via autoequivalences cover a whole connected component. We prove that this connected component is simply-connected. We determine the group of autoequivalences preserving this connected component, which turns out to be closely related to 1(3). Finally, we show that there is a submanifold isomorphic to the univer- sal covering of a moduli space of elliptic curves with 1(3)-level struc- ture. The morphism is 1(3)-equivariant, and is given by solutions of Picard-Fuchs equations. This result is motivated by the notion of - stability and by mirror symmetry.
105 citations
TL;DR: In this article, a combinatorial formula for the stationary distribution of the asymmetric exclusion process with all parameters general, in terms of a new class of tableaux which are called staircase tableaux, was given.
Abstract: Introduced in the late 1960s, the asymmetric exclusion process (ASEP) is an important model from statistical mechanics that describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites. It has been cited as a model for traffic flow and protein synthesis. In the most general form of the ASEP with open boundaries, particles may enter and exit at the left with probabilities α and γ, and they may exit and enter at the right with probabilities β and δ. In the bulk, the probability of hopping left is q times the probability of hopping right. The first main result of this paper is a combinatorial formula for the stationary distribution of the ASEP with all parameters general, in terms of a new class of tableaux which we call staircase tableaux. This generalizes our previous work for the ASEP with parameters γ=δ=0. Using our first result and also results of Uchiyama, Sasamoto, and Wadati, we derive our second main result: a combinatorial formula for the moments of Askey-Wilson polynomials. Since the early 1980s there has been a great deal of work giving combinatorial formulas for moments of various other classical orthogonal polynomials (e.g., Hermite, Charlier, Laguerre, Meixner). However, this is the first such formula for the Askey-Wilson polynomials, which are at the top of the hierarchy of classical orthogonal polynomials.
103 citations
TL;DR: In this article, a general conjecture for the mixed Hodge polynomial of the generic character varieties of representations of the fundamental group of a Riemann surface of genus g to GLn(C) with fixed generic semisimple conjugacy classes at k punctures was proposed.
Abstract: We propose a general conjecture for the mixed Hodge polynomial of the generic character varieties of representations of the fundamental group of a Riemann surface of genus g to GLn(C) with fixed generic semisimple conjugacy classes at k punctures. This conjecture generalizes the Cauchy identity for Macdonald polynomials and is a common generalization of two formulas that we prove in this paper. The first is a formula for the E-polynomial of these character varieties which we obtain using the character table of GLn(Fq). We use this formula to compute the Euler characteristic of character varieties. The second formula gives the Poincare polynomial of certain associated quiver varieties which we obtain using the character table of gln(Fq). In the last main result we prove that the Poincare polynomials of the quiver varieties equal certain multiplicities in the tensor product of irreducible characters of GLn(Fq). As a consequence we find a curious connection between Kac-Moody algebras associated with comet-shaped, and typically wild, quivers and the representation theory of GLn(Fq).
TL;DR: In this article, it was shown that a random walk on the mapping class group of an orientable surface gives rise to a pseudo-Anosov element with asymptotic probability one.
Abstract: We show that a random walk on the mapping class group of an orientable surface gives rise to a pseudo-Anosov element with asymptotic probability one. Our methods apply to many subgroups of the mapping class group, including the Torelli group.
TL;DR: In this article, the authors combine complex-analytic and arithmetic tools to study the preperiodic points of 1-dimensional complex dynamical systems, and they show that for any fixed a,b ∈ C and any integer d ≥ 2, the set of c∈C for which both a and b are pre-periodic for zd+c is infinite if and only if ad=bd.
Abstract: In this article, we combine complex-analytic and arithmetic tools to study the preperiodic points of 1-dimensional complex dynamical systems. We show that for any fixed a,b∈C and any integer d≥2, the set of c∈C for which both a and b are preperiodic for zd+c is infinite if and only if ad=bd. This provides an affirmative answer to a question of Zannier, which itself arose from questions of Masser concerning simultaneous torsion sections on families of elliptic curves. Using similar techniques, we prove that if rational functions ϕ,ψ∈C(z) have infinitely many preperiodic points in common, then all of their preperiodic points coincide (and, in particular, the maps must have the same Julia set). This generalizes a theorem of Mimar, who established the same result assuming that ϕ and ψ are defined over Q¯. The main arithmetic ingredient in the proofs is an adelic equidistribution theorem for preperiodic points over number fields and function fields, with nonarchimedean Berkovich spaces playing an essential role.
TL;DR: In this article, the existence of extremal metrics on blowups at finitely many points of Kahler manifolds which already carry an extremal metric was shown to be provable.
Abstract: In this paper we provide conditions that are sufficient to guarantee the existence of extremal metrics on blowups at finitely many points of Kahler manifolds which already carry an extremal metric. As a particular case, we construct extremal metrics on $\mathbb {P}^2$ blown-up k points in general position, with $k \lt m+2$.
TL;DR: In this article, a conjecture of Premet was proved for finite dimensional irreducible modules for W-algebras of finite type and a relation between Harish- Chandra bimodules and BIMODULEs over Walgebra was studied.
Abstract: W-algebras of finite type are certain finitely generated associative algebras closely related to the universal enveloping algebras of semisimple Lie algebras. In this paper we prove a conjecture of Premet that gives an almost complete classification of finite dimensional irreducible modules for W-algebras. Also we study a relation between Harish- Chandra bimodules and bimodules over W-algebras.
TL;DR: In this article, it was shown that a free ergodic measure-preserving profinite action (i.e., an inverse limit of actions Γ↷Xn, with Xn finite) of a countable property group Γ is cohomologous to a cocycle w′ which factors through the map Γ×X→φ×Xn for some n.
Abstract: Consider a free ergodic measure-preserving profinite action Γ↷X (i.e., an inverse limit of actions Γ↷Xn, with Xn finite) of a countable property (T) group Γ (more generally, of a group Γ which admits an infinite normal subgroup Γ0 such that the inclusion Γ0⊂Γ has relative property (T) and Γ/Γ0 is finitely generated) on a standard probability space X. We prove that if w:Γ×X→Λ is a measurable cocycle with values in a countable group Λ, then w is cohomologous to a cocycle w′ which factors through the map Γ×X→Γ×Xn, for some n. As a corollary, we show that any orbit equivalence of Γ↷X with any free ergodic measure-preserving action Λ↷Y comes from a (virtual) conjugacy of actions.
TL;DR: For the Schrodinger operator Δg+V with potential V ∈ C1,α(M0) for some α>0, the Dirichlet-to-Neumann map N|Γ measured on an open set Γ⊂∂M0 determines uniquely the potential V as discussed by the authors.
Abstract: On a fixed smooth compact Riemann surface with boundary (M0,g), we show that, for the Schrodinger operator Δg+V with potential V∈C1,α(M0) for some α>0, the Dirichlet-to-Neumann map N|Γ measured on an open set Γ⊂∂M0 determines uniquely the potential V. We also discuss briefly the corresponding consequences for potential scattering at zero frequency on Riemann surfaces with either asymptotically Euclidean or asymptotically hyperbolic ends.
TL;DR: In this paper, the authors considered the Monge problem in a convex bounded subset of the road network and proved the existence of an optimal transport map under the classical assumption that the first marginal is absolutely continuous with respect to the Lebesgue measure.
Abstract: We first consider the Monge problem in a convex bounded subset of Rd. The cost is given by a general norm, and we prove the existence of an optimal transport map under the classical assumption that the first marginal is absolutely continuous with respect to the Lebesgue measure. In the final part of the paper we show how to extend this existence result to a general open subset of Rd.
TL;DR: In this paper, it was shown that for a finite set A of integers we have |A+A|≤K|A|, then A is contained in a generalized arithmetic progression of dimension at most K1+C(logK)−1/2 and size at most exp(K 1+C (logK)-1/ 2)|A| for some absolute constant C.
Abstract: We prove that if for a finite set A of integers we have |A+A|≤K|A|, then A is contained in a generalized arithmetic progression of dimension at most K1+C(logK)−1/2 and of size at most exp(K1+C(logK)-1/2)|A| for some absolute constant C. We also discuss a number of applications of this result.
TL;DR: In this paper, the authors studied Whittaker functions on nonlinear coverings of simple algebraic groups over a nonarchimedean local field and showed that these expressions agree with known formulae for the prime-power-supported coefficients of multiple Dirichlet series.
Abstract: We study Whittaker functions on nonlinear coverings of simple algebraic groups over a nonarchimedean local field. We produce a recipe for expressing such a Whittaker function as a weighted sum over a crystal graph and show that in type A, these expressions agree with known formulae for the prime-power-supported coefficients of multiple Dirichlet series.
TL;DR: In this article, the authors give examples of non-commutative Lp spaces without the completely bounded approximation property of Haagerup and Kraus for any p > 4 or p < 4/3 and r ≥ 3.
Abstract: For any 1\leq p \leq \infty different from 2, we give examples of non-commutative Lp spaces without the completely bounded approximation property. Let F be a non-archimedian local field. If p>4 or p<4/3 and r\geq 3 these examples are the non-commutative Lp-spaces of the von Neumann algebra of lattices in SL_r(F) or in SL_r(\R). For other values of p the examples are the non-commutative Lp-spaces of the von Neumann algebra of lattices in SL_r(F) for r large enough depending on p. We also prove that if r \geq 3 lattices in SL_r(F) or SL_r(\R) do not have the Approximation Property of Haagerup and Kraus. This provides examples of exact C^*-algebras without the operator space approximation property.
TL;DR: In this article, it was shown that if H is a group of polynomial growth whose growth rate is at least quadratic, then the Lp compression of the wreath product Z≀H equals max{1p,12}.
Abstract: We show that if H is a group of polynomial growth whose growth rate is at least quadratic, then the Lp compression of the wreath product Z≀H equals max{1p,12}. We also show that the Lp compression of Z≀Z equals max{p2p−1,23} and that the Lp compression of (Z≀Z)0 (the zero section of Z≀Z, equipped with the metric induced from Z≀Z) equals max{p+12p,34}. The fact that the Hilbert compression exponent of Z≀Z equals 2/3 while the Hilbert compression exponent of (Z≀Z)0 equals 3/4 is used to show that there exists a Lipschitz function f:(Z≀Z)0→L2 which cannot be extended to a Lipschitz function defined on all of Z≀Z.
TL;DR: In this paper, the authors obtain asymptotic expansions for Toeplitz determinants corresponding to a family of symbols depending on a parameter t. If t = 0, the symbol possesses a Fisher-Hartwig singularity and the symbols are regular so that the determinants obey Szegő's strong limit theorem.
Abstract: We obtain asymptotic expansions for Toeplitz determinants corresponding to a family of symbols depending on a parameter t. For t positive, the symbols are regular so that the determinants obey Szegő’s strong limit theorem. If t=0, the symbol possesses a Fisher-Hartwig singularity. Letting t→0 we analyze the emergence of a Fisher-Hartwig singularity and a transition between the two different types of asymptotic behavior for Toeplitz determinants. This transition is described by a special Painleve V transcendent. A particular case of our result complements the classical description of Wu, McCoy, Tracy, and Barouch of the behavior of a 2-spin correlation function for a large distance between spins in the two-dimensional Ising model as the phase transition occurs.
TL;DR: In this article, it was shown that Kontsevich's formality of the little disk operad is homotopic to Tamarkin's for a special choice of a Drinfeld associator given by parallel transport of the Alekseev-Torossian connection.
Abstract: We show that Kontsevich's formality of the little disk operad, obtained using graphs, is homotopic to Tamarkin's formality, for a special choice of a Drinfeld associator. The associator is given by parallel transport of the Alekseev-Torossian connection.
TL;DR: In this paper, it was shown that the Poisson deformation functor of an affine (singular) symplectic variety is unobstructed by smoothness by a smoothing by a deformation.
Abstract: We prove that the Poisson deformation functor of an affine (singular) symplectic variety is unobstructed. As a corollary, we prove the following result. For an affine symplectic variety X with a good C*-action (where its natural Poisson structure is positively weighted), the following are equivalent. (1) X has a crepant projective resolution. (2) X has a smoothing by a Poisson deformation. A typical example is (the normalization) of a nilpotent orbit closure in a complex simple Lie algebra. By the theorem, one can see which orbit closure has a smoothing by a Poisson deformation.
TL;DR: In this article, the authors developed linear and nonlinear harmonic analysis on compact Lie groups and homogeneous spaces relevant for the theory of evolutionary Hamiltonian PDEs and proved the existence of Cantor families of wave and Schrodinger equations with differentiable nonlinearities.
Abstract: We develop linear and nonlinear harmonic analysis on compact Lie groups and homogeneous spaces relevant for the theory of evolutionary Hamiltonian PDEs. A basic tool is the theory of the highest weight for irreducible representations of compact Lie groups. This theory provides an accurate description of the eigenvalues of the Laplace-Beltrami operator as well as the multiplication rules of its eigenfunctions. As an application, we prove the existence of Cantor families of small amplitude time-periodic solutions for wave and Schrodinger equations with differentiable nonlinearities. We apply an abstract Nash-Moser implicit function theorem to overcome the small divisors problem produced by the degenerate eigenvalues of the Laplace operator. We provide a new algebraic framework to prove the key tame estimates for the inverse linearized operators on Banach scales of Sobolev functions.
TL;DR: In this article, it was shown that the pushforward of the mass off to the modular curve of level 1 equidistributes with respect to the Poincare measure is independent of the weight of the holomorphic newform.
Abstract: Let f traverse a sequence of classical holomorphic newforms of fixed weight and increasing square-free level q -> infinity. We prove that the pushforward of the mass off to the modular curve of level 1 equidistributes with respect to the Poincare measure.
TL;DR: In this article, the authors give lower bounds on the energy of a mixed state ρ from its distribution in the partition and the spectral density of A. They imply an uncertainty principle, giving a lower bound on the sum of the spatial entropy of ρ, as measured from X and some spectral entropy, with respect to its energy distribution.
Abstract: Let A be a self-adjoint operator acting over a space X endowed with a partition. We give lower bounds on the energy of a mixed state ρ from its distribution in the partition and the spectral density of A. These bounds improve with the refinement of the partition, and generalize inequalities by Li and Yau and by Lieb and Thirring for the Laplacian in Rn. They imply an uncertainty principle, giving a lower bound on the sum of the spatial entropy of ρ, as measured from X, and some spectral entropy, with respect to its energy distribution. On Rn, this yields lower bounds on the sum of the entropy of the densities of ρ and its Fourier transform. A general log-Sobolev inequality is also shown. It holds on mixed states, without Markovian or positivity assumption on A.
TL;DR: In this paper, it was shown that 3-point Gromov-Witten invariants of genus zero on a Grassmann variety are equal to triple intersec- tions computed in the (equivariant) K-theory of a two-step flag manifold, thus generalizing an earlier result of Buch, Kresch and Tamvakis.
Abstract: We show that (equivariant) K-theoretic 3-point Gromov-Witten invariants of genus zero on a Grassmann variety are equal to triple intersec- tions computed in the (equivariant) K-theory of a two-step flag manifold, thus generalizing an earlier result of Buch, Kresch, and Tamvakis. In the process we show that the Gromov-Witten variety of curves passing through 3 general points is irreducible and rational. Our applications include Pieri and Giambelli formulas for the quantum K-theory ring of a Grassmannian, which determine the multiplication in this ring. We also compute the dual Schubert basis for this ring, and show that its structure constants satisfy S3-symmetry. Our for- mula for Gromov-Witten invariants can be partially generalized to cominuscule homogeneous spaces by using a construction of Chaput, Manivel, and Perrin.
TL;DR: In this article, the authors give non-density results for integral points on affine varieties, in the spirit of the Lang-Vojta conjecture, and show that any set of quasi-integral points (resp. any integral curve) on X − H is not Zariski dense.
Abstract: We give non-density results for integral points on affine varieties, inthe spirit of Lang-Vojta conjecture. In particular, let X be a projective variety of dimension d ≥ 2 over a number field K (resp. over C). Let H be the sum of 2d properly intersecting ample divisors on X. We show that any set of quasi-integral points (resp. any integral curve) on X − H is not Zariski dense. Resume : On donne des resultats de non-densite pour les points entiers sur des varietes affines, dans l'esprit de la conjecture de Lang-Vojta. En particulier,soit X une variete pro- jective de dimension d ≥ 2 sur un corps de nombres K (resp. sur C). Soit H la somme de 2d diviseurs amples sur X qui se coupent proprement. On montre que tout ensemble de points quasi-entiers (resp. toute courbe entiere) sur X − H est non Zariski-dense.