Showing papers in "Annals of Mathematics in 2008"
TL;DR: In this paper, it was shown that there are arbitrarily long arithmetic progressions of primes and that a large fraction of the primes can be placed inside a pseudorandom set of almost primes with positive relative density.
Abstract: We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemeredi�s theorem, which asserts that any subset of the integers of positive density contains progressions of arbitrary length. The second, which is the main new ingredient of this paper, is a certain transference principle. This allows us to deduce from Szemeredi�s theorem that any subset of a suficiently pseudorandom set (or measure) of positive relative density contains progressions of arbitrary length. The third ingredient is a recent result of Goldston and Yildirim, which we reproduce here. Using this, one may place (a large fraction of) the primes inside a pseudorandom set of �almost primes� (or more precisely, a pseudorandom measure concentrated on almost primes) with positive relative density.
868 citations
TL;DR: In this article, the authors obtained global well-posedness, scattering, and global L 10 spacetime bounds for energy-class solutions to the quintic defocusing Schrodinger equa- tion in R 1+3, which is energy-critical.
Abstract: We obtain global well-posedness, scattering, and global L 10 spacetime bounds for energy-class solutions to the quintic defocusing Schrodinger equa- tion in R 1+3 , which is energy-critical. In particular, this establishes global existence of classical solutions. Our work extends the results of Bourgain (4) and Grillakis (20), which handled the radial case. The method is similar in spirit to the induction-on-energy strategy of Bourgain (4), but we perform the induction analysis in both frequency space and physical space simultaneously, and replace the Morawetz inequality by an interaction variant (first used in (12), (13)). The principal advantage of the interaction Morawetz estimate is that it is not localized to the spatial origin and so is better able to handle nonradial solutions. In particular, this interaction estimate, together with an almost-conservation argument controlling the movement of L 2 mass in fre- quency space, rules out the possibility of energy concentration.
485 citations
TL;DR: In this paper, a quantitative sharp form of the classical isoperimetric inequality is proved, thus giving a positive answer to a conjecture by Hall, which is a conjecture that was later disproved.
Abstract: A quantitative sharp form of the classical isoperimetric inequality is proved, thus giving a positive answer to a conjecture by Hall.
396 citations
TL;DR: In this article, the authors define and study sl2-categorifications on abelian categories and show that there is a self-derived (even homotopy) equivalence categorifying the adjoint action of the simple reflection.
Abstract: We define and study sl2-categorifications on abelian categories. We show in particular that there is a self-derived (even homotopy) equivalence categorifying the adjoint action of the simple reflection. We construct categorifications for blocks of symmetric groups and deduce that two blocks are splendidly Rickard equivalent whenever they have isomorphic defect groups and we show
389 citations
TL;DR: In this article, it was shown that Cayley graphs of SL2(Fp) are expanders with respect to the projection of any fixed elements in SL(2, Z) generating a non-elementary subgroup.
Abstract: We prove that Cayley graphs of SL2(Fp) are expanders with respect to the projection of any fixed elements in SL(2, Z) generating a non-elementary subgroup, and with respect to generators chosen at random in SL2(Fp).
287 citations
TL;DR: The Ricci flow was introduced by Hamilton in 1982 as discussed by the authors in order to prove that a compact three-manifold admitting a Riemannian metric of positive Ricci curvature is a spherical space form.
Abstract: The Ricci flow was introduced by Hamilton in 1982 [H1] in order to prove that a compact three-manifold admitting a Riemannian metric of positive Ricci curvature is a spherical space form. In dimension four Hamilton showed that compact four-manifolds with positive curvature operators are spherical space forms as well [H2]. More generally, the same conclusion holds for compact four-manifolds with 2-positive curvature operators [Che]. Recall that a curvature operator is called 2-positive, if the sum of its two smallest eigenvalues is positive. In arbitrary dimensions Huisken [Hu] described an explicit open cone in the space of curvature operators such that the normalized Ricci flow evolves metrics whose curvature operators are contained in that cone into metrics of constant positive sectional curvature. Hamilton conjectured that in all dimensions compact Riemannian manifolds with positive curvature operators must be space forms. In this paper we confirm this conjecture. More generally, we show the following
286 citations
TL;DR: In this article, the authors consider a complete metric measure space with µ Borel and doubling that admits a (1,p)-Poincare inequality for every q > p - e, quantitatively.
Abstract: Let p > 1 and let (X,d,µ) be a complete metric measure space with µ Borel and doubling that admits a (1,p)-Poincare inequality. Then there exists e > 0 such that (X,d,µ) admits a (1,q)-Poincare inequality for every q > p - e, quantitatively.
261 citations
TL;DR: Given a compact four dimensional manifold, it is proved existence of conformal metrics with constant Q-curvature under generic assumptions, jointly with the compactness result of [35].
Abstract: Given a compact four dimensional manifold, we prove existence of conformal metrics with constant Q-curvature under generic assumptions. The problem amounts to solving a fourth-order nonlinear elliptic equation with variational structure. Since the corresponding Euler functional is in general unbounded from above and from below, we employ topological methods and min-max schemes, jointly with the compactness result of [35].
253 citations
TL;DR: In this article, it was shown that every subset of SL2(Z/pZ) grows rapidly when it acts on itself by the group operation, and that for every set of generators A of SL 2(Z / pZ), every element of A can be expressed as a product of at most O((log p)c) elements of A? A-1, where c and the implied constant are absolute.
Abstract: We show that every subset of SL2(Z / pZ) grows rapidly when it acts on itself by the group operation. It follows readily that, for every set of generators A of SL2(Z / pZ), every element of SL2(Z / pZ) can be expressed as a product of at most O((log p)c) elements of A ? A-1, where c and the implied constant are absolute.
247 citations
TL;DR: In this paper, the derived categories of coherent sheaves of weighted projective spaces and their non-commutative deformations were studied and the derived category of Lagrangian vanishing cycles of their mirror Landau-Ginzburg models.
Abstract: We study the derived categories of coherent sheaves of weighted projective spaces and their noncommutative deformations, and the derived categories of Lagrangian vanishing cycles of their mirror Landau-Ginzburg models. In particular, we show that the derived category of coherent sheaves (B-branes) on the weighted projective plane $\CP^2(a,b,c)$ is equivalent to the derived category of vanishing cycles (A-branes) on the affine hypersurface $X=\{x^ay^bz^c=1\}\subset (\C^*)^3$ equipped with an exact symplectic form and the superpotential $W=x+y+z$. Hence, the homological mirror symmetry conjecture holds for weighted projective planes. Moreover, we also show that this mirror correspondence between derived categories can be extended to toric noncommutative deformations of $\CP^2(a,b,c)$ where B-branes are concerned, and their mirror counterparts, non-exact deformations of the symplectic structure of $X$ where A-branes are concerned. We also obtain similar results for other examples such as weighted projective lines or Hirzebruch surfaces.
202 citations
TL;DR: In this article, the authors study enhancement of diffusive mixing on a compact Riemannian manifold by a fast incompressible flow and show that weakly mixing flows always enhance dissipation in this sense.
Abstract: We study enhancement of diffusive mixing on a compact Riemannian manifold by a fast incompressible flow. Our main result is a sharp description of the class of flows that make the deviation of the solution from its average arbitrarily small in an arbitrarily short time, provided that the flow amplitude is large enough. The necessary and suficient condition on such flows is expressed naturally in terms of the spectral properties of the dynamical system associated with the flow. In particular, we find that weakly mixing flows always enhance dissipation in this sense. The proofs are based on a general criterion for the decay of the semigroup generated by an operator of the form G + iAL with a negative unbounded self-adjoint operator G, a self-adjoint operator L, and parameter A » 1. In particular, they employ the RAGE theorem describing evolution of a quantum state belonging to the continuous spectral subspace of the hamiltonian (related to a classical theorem of Wiener on Fourier transforms of measures). Applications to quenching in reaction-diffusion equations are also considered.
TL;DR: In this article, the order of magnitude of H(x;y;z), the number of integers n x having a divisor in (y,z), for all x;y and z, was established.
Abstract: We determine the order of magnitude of H(x;y;z), the number of integers n x having a divisor in (y;z], for all x;y and z. We also study Hr(x;y;z), the number of integers n x having exactly r divisors in (y;z]. Whenr = 1 we establish the order of magnitude ofH1(x;y;z) for allx;y;z satisfying z x 1=2 " . For every r 2, C > 1 and " > 0, we determine the order of magnitude of Hr(x;y;z) uniformly for y large and y +y=(logy) log 4 1 " z min(y C ;x 1=2 " ). As a consequence of these bounds, we settle a 1960 conjecture of Erd} os and some conjectures of Tenenbaum. One key element of the proofs is a new result on the distribution of uniform order statistics.
TL;DR: The proof of the Calabi-Yau conjecture for embedded surfaces is discussed in this article, where it is shown that injective immersions are in fact true for Riemannian surfaces.
Abstract: In this talk I will discuss the proof of the Calabi-Yau conjectures for embedded surfaces. This is joint work with Bill Minicozzi, [CM9]. The Calabi-Yau conjectures about surfaces date back to the 1960s. Much work has been done on them over the past four decades. In particular, examples of Jorge-Xavier from 1980 and Nadirashvili from 1996 showed that the immersed versions were false; we will show here that for embedded surfaces, i.e., injective immersions, they are in fact true. Their original form was given in 1965 in [Ca] where E. Calabi made the following two conjectures about minimal surfaces (see also S.S. Chern, page 212 of [Ch]): Conjecture 1. “Prove that a complete minimal hypersurface in R must be unbounded.” Calabi continued: “It is known that there are no compact minimal submanifolds of R (or of any simply connected complete Riemannian manifold with sectional curvature ≤ 0). A more ambitious conjecture is”: Conjecture 2. “A complete minimal hypersurface in R has an unbounded projection in every (n− 2)–dimensional flat subspace.”
TL;DR: In this paper, the authors studied the large eigenvalue limit for the eigenfunctions of the Laplacian on a compact Riemannian manifold M of dimension d > 2, and assumed that the geodesic flow (#*)teR> acting on the unit tangent bundle of M, has a chaotic behaviour.
Abstract: We study the large eigenvalue limit for the eigenfunctions of the Laplacian, on a compact manifold of negative curvature - in fact, we only assume that the geodesic flow has the Anosov property. In the semi-classical limit, we prove that the Wigner measures associated to eigenfunctions have positive metric entropy. In particular, they cannot concentrate entirely on closed geodesies. 1. Introduction, statement of results We consider a compact Riemannian manifold M of dimension d > 2, and assume that the geodesic flow (#*)teR> acting on the unit tangent bundle of M, has a "chaotic" behaviour. This refers to the asymptotic properties of the flow when time t tends to infinity: ergodicity, mixing, hyperbolicity. . . : we assume here that the geodesic flow has the Anosov property, the main example being the case of negatively curved manifolds. The words "quantum chaos" express the intuitive idea that the chaotic features of the geodesic flow should imply certain special features for the corresponding quantum dynamical system: that is, according to Schrodinger, the unitary flow (exp(iht^))t£R acting on the Hilbert space L2(M), where A stands for the Laplacian on M and h is proportional to the Planck constant. Recall that the quantum flow converges, in a sense, to the classical flow (g*) in the so-called semi-classical limit h - > 0; one can imagine that for small values ofh the quantum system will inherit certain qualitative properties of the classical flow. One expects, for instance, a very different behaviour of eigenfunctions of the Laplacian, or the distribution of its eigenvalues, if the geodesic flow is Anosov or, in the other extreme, completely integrable (see [Sa95]). The convergence of the quantum flow to the classical flow is stated in the Egorov theorem. Consider one of the usual quantization procedures Op^, which associates an operator Oph(a) acting on L2(M) to every smooth compactly supported function a e C%°(T*M) on the cotangent bundle T*M. According to the Egorov theorem, we have for any fixed t
TL;DR: In this article, a counterexample to Elliott's classification conjecture for simple, separable, and nuclear C ∗ -algebras whose construction is elementary is presented.
Abstract: We exhibit a counterexample to Elliott’s classification conjecture for simple, separable, and nuclear C ∗ -algebras whose construction is elementary, and demonstrate the necessity of extremely fine invariants in distinguishing both approximate unitary equivalence classes of automorphisms of such algebras and isomorphism classes of the algebras themselves. The consequences for the program to classify nuclear C ∗ -algebras are far-reaching: one has, among other things, that existing results on the classification of simple, unital AH algebras via the Elliott invariant of K-theoretic data are the best possible, and that these cannot be improved by the addition of continuous homotopy invariant functors to the Elliott invariant.
TL;DR: In this article, it was shown that the operator norm of A-1 does not exceed Cn3 / 2 with probability close to 1, where n is the number of entries in the n × n matrix.
Abstract: Let A be an n × n matrix, whose entries are independent copies of a centered random variable satisfying the subgaussian tail estimate. We prove that the operator norm of A-1 does not exceed Cn3 / 2 with probability close to 1.
TL;DR: In this paper, the authors present a solution of the problem of the kissing number in four dimensions, namely, the equality fc(4) = 24, which is based on a modification of Delsarte's method.
Abstract: The kissing number problem asks for the maximal number k(n) of equal size nonoverlapping spheres in n-dimensional space that can touch another sphere of the same size. This problem in dimension three was the subject of a famous discussion between Isaac Newton and David Gregory in 1694. In three dimensions the problem was finally solved only in 1953 by Schutte and van der Waerden. In this paper we present a solution of a long-standing problem about the kissing number in four dimensions. Namely, the equality fc(4) = 24 is proved. The proof is based on a modification of Delsarte's method.
TL;DR: In this paper, the existence problem is solved, and global pointwise estimates of solu- tions are obtained for quasilinear and Hessian equations of Lane-Emden type.
Abstract: The existence problem is solved, and global pointwise estimates of solu- tions are obtained for quasilinear and Hessian equations of Lane-Emden type, including the following two model problems: or viscosity sense has been an open problem even for good data fi E Z>s(fi), s > 1. Such results are deduced from our existence criteria with the sharp exponents s = n(q~p+1) for the first equation, and s = ^ for the second one. Furthermore, a complete characterization of removable singularities is given. Our methods are based on systematic use of Wolff's potentials, dyadic models, and nonlinear trace inequalities. We make use of recent advances in potential theory and PDE due to Kilpelainen and Maly, Trudinger and Wang, and Labutin. This enables us to treat singular solutions, nonlocal operators, and distributed singularities, and develop the theory simultaneously for quasi- linear equations and equations of Monge- Ampere type.
TL;DR: In this paper, the double scaling limits of the eigenvalue correlation kernel near the origin were computed using the Deift/Zhou steepest descent method applied to the Riemann-Hilbert problem for orthogonal polynomials on the real line with respect to the weight.
Abstract: We study unitary random matrix ensembles of the form Z-'N\detM\2«e-NTrVWdM, where a > -1/2 and V is such that the limiting mean eigenvalue density for n, N - > oc and n/N - ► 1 vanishes quadratically at the origin. In order to compute the double scaling limits of the eigenvalue correlation kernel near the origin, we use the Deift/Zhou steepest descent method applied to the Riemann-Hilbert problem for orthogonal polynomials on the real line with respect to the weight \x\2ae~NV^x\ Here the main focus is on the construction of a local parametrix near the origin with ^-functions associated with a special solution qa of the Painleve II equation q" = sq + 2q3 - a. We show that qa has no real poles for a > -1/2, by proving the solvability of the corresponding Riemann-Hilbert problem. We also show that the asymptotics of the recurrence coefficients of the orthogonal polynomials can be expressed in terms of qa in the double scaling limit.
TL;DR: For any s > 0, the majority of C s linear cocycles over any hyperbolic (uniformly or not) ergodic transformation exhibit some nonzero Lyapunov exponent as mentioned in this paper.
Abstract: We prove that for any s > 0 the majority of C s linear cocycles over any hyperbolic (uniformly or not) ergodic transformation exhibit some nonzero Lyapunov exponent: this is true for an open dense subset of cocycles and, actually, vanishing Lyapunov exponents correspond to codimension-1. This open dense subset is described in terms of a geometric condition involving the behavior of the cocycle over certain heteroclinic orbits of the transformation.
TL;DR: In this paper, the authors present tools for analyzing rank one maps and critical structure and orbits of critical points in one-dimensional space maps, and a set of good SRB measures.
Abstract: Introduction 1 Statement of results PART I PREPARATION 2 Relevant results from one dimension 3 Tools for analyzing rank one maps PART II PHASE-SPACE DYNAMICS 4 Critical structure and orbits 5 Properties of orbits controlled by critical set 6 Identification of hyperbolic behavior: formal inductive procedure 7 Global geometry via monotone branches 8 Completion of induction 9 Construction of SRB measures PART III PARAMETER ISSUES 10 Dependence of dynamical structures on parameter 11 Dynamics of curves of critical points 12 Derivative growth via statistics 13 Positive measure sets of good parameters APPENDICES
TL;DR: In this paper, the authors proved the existence and uniqueness of globally smooth solutions to the second boundary value problem for affine maximal surface equation and affine mean curvature equation. But they assumed that the inhomoge-neous term is only assumed to be Holder continuous.
Abstract: In this paper, we prove global second derivative estimates for solutions of the Dirichlet problem for the Monge-Ampere equation when the inhomoge- neous term is only assumed to be Holder continuous. As a consequence of our approach, we also establish the existence and uniqueness of globally smooth solutions to the second boundary value problem for the affine maximal surface equation and affine mean curvature equation.
TL;DR: In this paper, Chenciner and Montgomery showed that there are infinitely many solutions with a certain topological type, called retrograde orbits, that minimize the action functional on certain path spaces.
Abstract: Poincare made the first attempt in 1896 on applying variational calculus to the three-body problem and observed that collision orbits do not necessarily have higher values of action than classical solutions. Little progress had been made on resolving this difficulty until a recent breakthrough by Chenciner and Montgomery. Afterward, variational methods were successfully applied to the JV-body problem to construct new classes of solutions. In order to avoid collisions, the problem is confined to symmetric path spaces and all new planar solutions were constructed under the assumption that some masses are equal. A question for the variational approach on planar problems naturally arises: Are minimizing methods useful only when some masses are identical? This article addresses this question for the three-body problem. For various choices of masses, it is proved that there exist infinitely many solutions with a certain topological type, called retrograde orbits, that minimize the action functional on certain path spaces. Cases covered in our work include triple stars in retrograde motions, double stars with one outer planet, and some double stars with one planet orbiting around one primary mass. Our results largely complement the classical results by the Poincare continuation method and Conley's geometric approach.
TL;DR: In this article, it was shown that isoparametric hypersurfaces with four distinct principal curvatures in spheres are of Clifford type provided that the multiplicities m 1, m 2 of the principal curvature satisfy m 2 > 2m 1 - 1.
Abstract: In this paper we give a new proof for the classification result in [3]. We show that isoparametric hypersurfaces with four distinct principal curvatures in spheres are of Clifford type provided that the multiplicities m 1 , m 2 of the principal curvatures satisfy m 2 > 2m 1 - 1. This inequality is satisfied for all but five possible pairs (m 1 , m 2 ) with m 1 ≤ m 2 . Our proof implies that for (m 1 ,m 2 ) ≠ (1,1) the Clifford system may be chosen in such a way that the associated quadratic forms vanish on the higher-dimensional of the two focal manifolds. For the remaining five possible pairs (m 1 , m 2 ) with m 1 ≤ m 2 (see [13], [1], and [15]) this stronger form of our result is incorrect: for the three pairs (3,4), (6,9), and (7,8) there are examples of Clifford type such that the associated quadratic forms necessarily vanish on the lower-dimensional of the two focal manifolds, and for the two pairs (2,2) and (4,5) there exist homogeneous examples that are not of Clifford type; cf. [5, 4.3, 4.4].
TL;DR: In this article, it was shown that the derived functor of global sections provides an equivalence between the derived category of D-modules (with no divided powers) on the flag variety and the appropriate derived categories of modules over the corresponding Lie algebra.
Abstract: We show that, on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of finite characteristic with a given (generalized) regular central character are the same as coherent sheaves on the formal neighborhood of the corresponding (generalized) Springer fiber.
The first step is to observe that the derived functor of global sections provides an equivalence between the derived category of D-modules (with no divided powers) on the flag variety and the appropriate derived category of modules over the corresponding Lie algebra. Thus the “derived” version of the Beilinson-Bernstein localization theorem holds in sufficiently large positive characteristic. Next, one finds that for any smooth variety this algebra of differential operators is an Azumaya algebra on the cotangent bundle. In the case of the flag variety it splits on Springer fibers, and this allows us to pass from D-modules to coherent sheaves. The argument also generalizes to twisted D-modules. As an application we prove Lusztig’s conjecture on the number of irreducible modules with a fixed central character. We also give a formula for behavior of dimension of a module under translation functors and reprove the Kac-Weisfeiler conjecture.
The sequel to this paper [BMR2] treats singular infinitesimal characters.
TL;DR: In this paper, a blow-up formula for cyclic homology is used to show that infinitesimal K-theory satisfies cdh-descent, which is a conjecture of Weibel predicting the vanishing of algebraic Ktheory of a scheme in degrees less than the dimension of the scheme.
Abstract: We prove a blow-up formula for cyclic homology which we use to show that infinitesimal K-theory satisfies cdh-descent. Combining that result with some computations of the cdh-cohomology of the sheaf of regular functions, we verify a conjecture of Weibel predicting the vanishing of algebraic K-theory of a scheme in degrees less than minus the dimension of the scheme, for schemes essentially of finite type over a field of characteristic zero.
TL;DR: In this article, it was shown that the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of MCG(S) groups.
Abstract: We study the large scale geometry of the mapping class group, MCG(S). Our main result is that for any asymptotic cone of MCG(S), the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of MCG(S). An application is a proof of Brock-Farb’s Rank Conjecture which asserts that MCG(S) has quasi-flats of dimension N if and only if it has a rank N free abelian subgroup. (Hamenstadt has also given a proof of this conjecture, using different methods.) We also compute the maximum dimension of quasi-flats in Teichmuller space with the Weil-Petersson metric.
TL;DR: In this paper, it was shown that the Langlands fundamental lemma for unitary groups of rank < p implies a deformation of orbital integrals with the help of the Hitchin fibration.
Abstract: Let G be an unramified reductive group over a nonarchimedian local field F. The so-called Langlands Fundamental Lemma is a family of conjectural identities between orbital integrals for G(F) and orbital integrals for endoscopic groups of G. In this paper we prove the Langlands fundamental lemma in the particular case where F is a finite extension of Fp((t)), G is a unitary group and p > rank(G). Waldspurger has shown that this particular case implies the Langlands fundamental lemma for unitary groups of rank < p when F is any finite extension of Qp.
We follow in part a strategy initiated by Goresky, Kottwitz and MacPherson. Our main new tool is a deformation of orbital integrals which is constructed with the help of the Hitchin fibration for unitary groups over projective curves.
TL;DR: In this paper, the propagation of C°° and Sobolev singularities for the wave equation at hyperbolic points was studied on C°-folds with corners M equipped with a Riemannian metric g, and it was shown that WF b is a union of maximally extended generalized broken bicharacteristics.
Abstract: In this paper we describe the propagation of C°° and Sobolev singularities for the wave equation on C°° manifolds with corners M equipped with a Riemannian metric g. That is, for X = M x R t , P = D 2 t - Δ M , and u ∈ H 1 loc (X) solving Pu = 0 with homogeneous Dirichlet or Neumann boundary conditions, we show that WF b (u) is a union of maximally extended generalized broken bicharacteristics. This result is a C°° counterpart of Lebeau's results for the propagation of analytic singularities on real analytic manifolds with appropriately stratified boundary, [11]. Our methods rely on b-microlocal positive commutator estimates, thus providing a new proof for the propagation of singularities at hyperbolic points even if M has a smooth boundary (and no corners).