Showing papers in "Annals of Mathematics in 2006"
TL;DR: The strong perfect graph conjecture as discussed by the authors states that a graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced sub graph of G is an odd cycle of length at least five or the complement of one.
Abstract: A graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one. The ?strong perfect graph conjecture? (Berge, 1961) asserts that a graph is perfect if and only if it is Berge. A stronger conjecture was made recently by Conforti, Cornu?ejols and Vuiskovi?c ? that every Berge graph either falls into one of a few basic classes, or admits one of a few kinds of separation (designed so that a minimum counterexample to Berge?s conjecture cannot have either of these properties). In this paper we prove both of these conjectures.
1,161 citations
TL;DR: In this paper, it was shown that the Dirichlet to Neumann map for the equation ∇·σ∇u = 0 in a two-dimensional domain uniquely determines the bounded measurable constant.
Abstract: We show that the Dirichlet to Neumann map for the equation ∇·σ∇u =0 in a two-dimensional domain uniquely determines the bounded measurable �
620 citations
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TL;DR: In this paper, the Parisi formula was used to give the limiting value of the free energy per site for the Sherrington-Kirkpatrick model at each temperature, starting with the famous result of Guerra that the Parisian formula provided a bound for this free energy.
Abstract: In this chapter we prove the Parisi formula, which gives the limiting value of the free energy per site for the Sherrington-Kirkpatrick model at each temperature, starting with the famous result of Guerra that the Parisi formula provided a bound for this free energy. We make full use of Poisson-Dirichlet cascades. We then use the same techniques to prove in a strong sense that in presence of an external field, the overlaps are positive.
554 citations
TL;DR: In this article, the stochastic 2D Navier-Stokes equations on the torus driven by degenerate noise are studied and the smallest closed invariant subspace for this model and the dynamics restricted to that subspace is shown to be ergodic.
Abstract: The stochastic 2D Navier-Stokes equations on the torus driven by degenerate noise are studied. We characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic. In particular, our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in L 2 0 (T 2 ). Unlike previous works, this class is independent of the viscosity and the strength of the noise. The two main tools of our analysis are the asymptotic strong Feller property, introduced in this work, and an approximate integration by parts formula. The first, when combined with a weak type of irreducibility, is shown to ensure that the dynamics is ergodic. The second is used to show that the first holds un
526 citations
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TL;DR: In this paper, the spectral curve of the Kasteleyn operator of a bipartite, doubly periodic graph G embedded in the plane is shown to be a Harnack curve.
Abstract: We study random surfaces which arise as height functions of random perfect matchings (a.k.a. dimer configurations) on a weighted, bipartite, doubly periodic graph G embedded in the plane. We derive explicit formulas for the surface tension and local Gibbs measure probabilities of these models. The answers involve a certain plane algebraic curve, which is the spectral curve of the Kasteleyn operator of the graph. For example, the surface tension is the Legendre dual of the Ronkin function of the spectral curve. The amoeba of the spectral curve represents the phase diagram of the dimer model. Further, we prove that the spectral curve of a dimer model is always a real curve of special type, namely it is a Harnack curve. This implies many qualitative and quantitative statement about the behavior of the dimer model, such as existence of smooth phases, decay rate of correlations, growth rate of height function fluctuations, etc.
455 citations
TL;DR: In this article, it was shown that the Betti numbers of the standard equivalence relation associated with Cartan subalgebras A io M ([G2]) are invariants for the factors M, ¥â HT n (M), n iA 0.
Abstract: We prove that a type II1 factor M can have at most one Cartan subalgebra A satisfying a combination of rigidity and compact approximation properties. We use this result to show that within the class HT of factors M having such Cartan subalgebras A io M, the Betti numbers of the standard equivalence relation associated with A io M ([G2]), are in fact isomorphism invariants for the factors M, ¥â HT n (M), n iA 0. The class HT is closed under amplifications and tensor products, with the Betti numbers satisfying ¥â HT n (Mt) = ¥â HT n (M)/t, ¢£t > 0, and a Ki§unneth type formula. An example of a factor in the class HT is given by the group von Neumann factor M = L(Z2 SL(2,Z)), for which ¥â HT 1 (M) = ¥â1(SL(2,Z)) = 1/12. Thus, Mt M,¢£t = 1, showing that the fundamental group of M is trivial. This solves a long standing problem of R. V. Kadison. Also, our results bring some insight into a recent problem of A. Connes and answer a number of open questions on von Neumann algebras.
449 citations
TL;DR: In this paper, the authors classify measures on the locally homogeneous space SL(2,R) which are invariant and have positive entropy under the diagonal subgroup of SL(R) and recurrent under L. This classification can be used to show arithmetic quantum unique ergodicity for compact arithmetic surfaces, and a similar but slightly weaker result for the finite volume case.
Abstract: We classify measures on the locally homogeneous space ?i\ SL(2,R) ?~ L which are invariant and have positive entropy under the diagonal subgroup of SL(2,R) and recurrent under L. This classification can be used to show arithmetic quantum unique ergodicity for compact arithmetic surfaces, and a similar but slightly weaker result for the finite volume case. Other applications are also presented. In the appendix, joint with D. Rudolph, we present a maximal ergodic theorem, related to a theorem of Hurewicz, which is used in theproof of the main result.
437 citations
TL;DR: In this article, an explicit equivalence between the stationary sector of the Gromov-Witten theory of a target curve X and the enumeration of Hurwitz coverings of X in the basis of completed cycles was established.
Abstract: We establish an explicit equivalence between the stationary sector of the Gromov-Witten theory of a target curve X and the enumeration of Hurwitz coverings of X in the basis of completed cycles. The stationary sector is formed, by definition, by the descendents of the point class. Completed cycles arise naturally in the theory of shifted symmetric functions. Using this equivalence, we give a complete description of the stationary Gromov-Witten theory of the projective line and elliptic curve. Toda equations for the relative stationary theory of the projective line are derived.
360 citations
TL;DR: In this paper, the authors combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's last theorem.
Abstract: This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem. In this paper we give new improved bounds for linear forms in three logarithms. We also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the Fibonacci sequence are 0, 1, 8 and 144 and the only perfect powers in the Lucas sequence are 1 and 4.
249 citations
TL;DR: In this paper, the authors classify the measures on SL (k,R)/SL(k,Z) which are invariant and ergodic under the action of the group A of positive diagonal matrices with positive entropy, and apply this to prove that the set of exceptions to Littlewood's conjecture has Hausdorff dimension zero.
Abstract: We classify the measures on SL (k,R)/SL (k,Z) which are invariant and ergodic under the action of the group A of positive diagonal matrices with positive entropy. We apply this to prove that the set of exceptions to Littlewood's conjecture has Hausdorff dimension zero.
213 citations
TL;DR: In this article, the authors established new results and introduced new methods in the theory of measurable orbit equivalence, using bounded cohomology of group representations, for a wide class of groups arising from negative curvature geometry.
Abstract: We establish new results and introduce new methods in the theory of measurable orbit equivalence, using bounded cohomology of group representations. Our rigidity statements hold for a wide (uncountable) class of groups arising from negative curvature geometry. Amongst our applications are (a) measurable Mostow-type rigidity theorems for products of negatively curved groups; (b) prime factorization results for measure equivalence; (c) superrigidity for orbit equivalence; (d) the first examples of continua of type II1 equivalence relations with trivial outer automorphism group that are mutually not stably isomorphic.
TL;DR: In this article, a Hausdorff measure version of the Duffin-Schaeffer conjecture in metric number theory is introduced and discussed, and the general conjecture is established modulo the original conjecture.
Abstract: A Hausdorff measure version of the Duffin-Schaeffer conjecture in metric number theory is introduced and discussed. The general conjecture is established modulo the original conjecture. The key result is a Mass Transference Principle which allows us to transfer Lebesgue measure theoretic statements for lim sup subsets of R k to Hausdorff measure theoretic statements. In view of this, the Lebesgue theory of lim sup sets is shown to underpin the general Hausdorff theory. This is rather surprising since the latter theory is viewed to be a subtle refinement of the former.
TL;DR: In this article, it was shown that for almost every frequency and energy, the Schr?Nodinger cocycle is either reducible or non-uniformly hyperbolic.
Abstract: We show that for almost every frequency ?? ?? R\Q, for every C?O potential v : R/Z ?? R, and for almost every energy E the corresponding quasiperiodic Schr?Nodinger cocycle is either reducible or nonuniformly hyperbolic This result gives very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic Schr?Nodinger operator, and allows us to complete the proof of the Aubry-Andr?Le conjecture on the measure of the spectrum of the Almost Mathieu Operator
TL;DR: In this paper, all equivariant Gromov-Witten invariants of the projective line are expressed as matrix elements of explicit operators acting in the Fock space.
Abstract: We express all equivariant Gromov-Witten invariants of the projective line as matrix elements of explicit operators acting in the Fock space. As a consequence, we prove the equivariant theory is governed by the 2-Toda hierarchy of Ueno and Takasaki. This is the second in a sequence of three papers devoted to the Gromov-Witten theory of nonsingular target curves (the first paper of the series is math.AG/0204305).
TL;DR: In this article, the authors show how to decide whether a real-valued function on a compact set in Rn can be extended to a Cm function on Rn, where m is a positive integer.
Abstract: Let f be a real-valued function on a compact set in Rn, and let m be a positive integer. We show how to decide whether f extends to a Cm function on Rn.
TL;DR: In this paper, the authors prove that the sequence of projective quantum SU(n) representations of the mapping class group of a closed oriented surface, obtained from the projective flat SU (n)-Verlinde bundles over Teichmuller space, is asymptotically faithful.
Abstract: We prove that the sequence of projective quantum SU(n) representations of the mapping class group of a closed oriented surface, obtained from the projective flat SU(n)-Verlinde bundles over Teichm?uller space, is asymptotically faithful That is, the intersection over all levels of the kernels of these representations is trivial, whenever the genus is at least 3 For the genus 2 case, this intersection is exactly the order 2 subgroup, generated by the hyper-elliptic involution, in the case of even degree and n = 2 Otherwise the intersection is also trivial in the genus 2 case
TL;DR: In this article, a Voronoi-style summation formula for the Fourier coefficients of a cusp form on GL(3,Z) is given, and a treatment of the standard L-function on GL (3,R) focusing on the archimedean analysis as performed using distributions is given.
Abstract: This paper is third in a series of three, following "Summation Formulas, from Poisson and Voronoi to the Present" (math.NT/0304187) and "Distributions and Analytic Continuation of Dirichlet Series" (math.FA/0403030). The first is primarily an expository paper explaining the present one, whereas the second contains some distributional machinery used here as well. These papers concern the boundary distributions of automorphic forms, and how they can be applied to study questions about cusp forms on semisimple Lie groups. The main result of this paper is a Voronoi-style summation formula for the Fourier coefficients of a cusp form on GL(3,Z)\GL(3,R). We also give a treatment of the standard L-function on GL(3), focusing on the archimedean analysis as performed using distributions. Finally a new proof is given of the GL(3)xGL(1) converse theorem of Jacquet, Piatetski-Shapiro, and Shalika. This paper is also related to the later papers math.NT/0402382 and math.NT/0404521.
TL;DR: In this paper, a conformai discretization of polyhedral surfaces is proposed, which is based on quadrilateral meshes, i.e. discretizations in terms of circles and spheres and introduces a new discrete model for minimal surfaces.
Abstract: The theory of polyhedral surfaces and, more generally, the field of discrete differential geometry are presently emerging on the border of differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete geometry studies geometric shapes with a finite number of elements (polyhedra), the theory of polyhedral surfaces aims at a development of discrete equivalents of the geometric notions and methods of surface theory. The latter appears then as a limit of the refinement of the discretization. Current progress in this field is to a large extent stimulated by its relevance for computer graphics and visualization. One of the central problems of discrete differential geometry is to find proper discrete analogues of special classes of surfaces, such as minimal, con stant mean curvature, isothermic surfaces, etc. Usually, one can suggest vari ous discretizations with the same continuous limit which have quite different geometric properties. The goal of discrete differential geometry is to find a dis cretization which inherits as many essential properties of the smooth geometry as possible. Our discretizations are based on quadrilateral meshes, i.e. we discretize parametrized surfaces. For the discretization of a special class of surfaces, it is natural to choose an adapted parametrization. In this paper, we investigate conformai discretizations of surfaces, i.e. discretizations in terms of circles and spheres, and introduce a new discrete model for minimal surfaces. See Figures 1 and 2. In comparison with direct methods (see, in particular, [23]), leading
TL;DR: In this article, an upper bound on the number of extensions of a fixed number of fields of prescribed degree and discriminant ≤ X was given, which was improved on the lower bound of Schmidt.
Abstract: We give an upper bound on the number of extensions of a fixed number field of prescribed degree and discriminant ≤ X; these bounds improve on work of Schmidt. We also prove various related results, such as lower bounds for the number of extensions and upper bounds for Galois extensions.
TL;DR: In this paper, the descendent genus g Gromov-Witten invariants of a projective variety X in terms of genus 0 invariant of its symmetric product stack Sg+1(X) were derived.
Abstract: I prove a formula expressing the descendent genus g Gromov-Witten invariants of a projective variety X in terms of genus 0 invariants of its symmetric product stack Sg+1(X). When X is a point, the latter are structure constants of the symmetric group, and we obtain a new way of calculating the Gromov- Witten invariants of a point.
TL;DR: In this article, a geometric interpretation of the Littlewood-richardson rule is given, interpreted as deforming the intersection of two Schubert varieties into the union of Schuber varieties, where all multiplicities arising are 1.
Abstract: We describe a geometric Littlewood-Richardson rule, interpreted as deforming the intersection of two Schubert varieties into the union of Schubert varieties. There are no restrictions on the base eld, and all multiplicities arising are 1; this is important for applications. This rule should be seen as a generalization of Pieri’s rule to arbitrary Schubert classes, by way of explicit homotopies. It has straightforward bijections to other Littlewood-Richardson rules, such as tableaux, and Knutson and Tao’s puzzles. This gives the rst geometric proof and interpretation of the Littlewood-Richardson rule. Geometric consequences are described here and in [V2], [KV1], [KV2], [V3]. For example, the rule also has an interpretation in K-theory, suggested by Buch, which gives an extension of puzzles to K-theory.
TL;DR: For the complex parabolic Ginzburg-Landau equation, this paper proved that vorticity evolves according to motion by mean curvature in Brakke?s weak formulation.
Abstract: For the complex parabolic Ginzburg-Landau equation, we prove that, asymptotically, vorticity evolves according to motion by mean curvature in Brakke?s weak formulation. The only assumption is a natural energy bound on the initial data. In some cases, we also prove convergence to enhanced motion in the sense of Ilmanen.
TL;DR: In this article, it was shown that the renormalization operator is hyperbolic in a Banach space of real analytic maps with quadratic critical points, and that the set of parameters corresponding to infinitely renormalizable maps of bounded combinatorial type is a Cantor set with Hausdorff dimension less than one.
Abstract: In this paper we extend M. Lyubichi¯s recent results on the global hyperbolicity of renormalization of quadratic-like germs to the space of Cr unimodal maps with quadratic critical point. We show that in this space the boundedtype limit sets of the renormalization operator have an invariant hyperbolic structure provided r iÝ 2+?A with ?A close to one. As an intermediate step between Lyubichi¯s results and ours, we prove that the renormalization operator is hyperbolic in a Banach space of real analytic maps. We construct the local stable manifolds and prove that they form a continuous lamination whose leaves are C1 codimension one, Banach submanifolds of the ambient space, and whose holonomy is C1+? for some ? > 0. We also prove that the global stable sets are C1 immersed (codimension one) submanifolds as well, provided r iÝ 3 + ?A with ?A close to one. As a corollary, we deduce that in generic, oneparameter families of Cr unimodal maps, the set of parameters corresponding to infinitely renormalizable maps of bounded combinatorial type is a Cantor set with Hausdorff dimension less than one.
TL;DR: In this paper, the authors present an abstract mathematical formulation of propositional calcu lus (propositional logic) in which proofs are combinatorial (graph-theoretic) rather than syntactic.
Abstract: Proofs are traditionally syntactic, inductively generated objects. This paper presents an abstract mathematical formulation of propositional calcu lus (propositional logic) in which proofs are combinatorial (graph-theoretic), rather than syntactic. It defines a combinatorial proof of a proposition 0 as a graph homomorphism h : C ? G((f>), where G((j>) is a graph associated with and C is a coloured graph. The main theorem is soundness and completeness: (f) is true if and only if there exists a combinatorial proof h : C ?? G(
TL;DR: In this paper, the integrality of the ratiof, f, f� /� g, g� (outside an explicit finite set of primes), where g is an arithmetically normalized holomorphic newform on a Shimura curve, f is a normalized Hecke eigenform on GL(2) with the same HecKE eigenvalues as g and � denotes the Petersson inner product, was proved.
Abstract: We prove integrality of the ratiof, f� /� g, g� (outside an explicit finite set of primes), where g is an arithmetically normalized holomorphic newform on a Shimura curve, f is a normalized Hecke eigenform on GL(2) with the same Hecke eigenvalues as g and � , � denotes the Petersson inner product. The primes dividing this ratio are shown to be closely related to certain level-lowering con- gruences satisfied by f and to the central values of a family of Rankin-Selberg L-functions. Finally we give two applications, the first to proving the integral- ity of a certain triple product L-value and the second to the computation of the Faltings height of Jacobians of Shimura curves.
TL;DR: In this paper, it was shown that a positively curved, simply connected, compact manifold (M,g) is up to homotopy given by a rank onesymmetric space, provided that its isometry group Iso(M, g) is large.
Abstract: There are very few examples of Riemannian manifolds with positive sectionalcurvature known. In fact in dimensions above 24 all known examplesare diffeomorphic to locally rank one symmetric spaces. We give a partialexplanation of this phenomenon by showing that a positively curved, simplyconnected, compact manifold (M,g) is up to homotopy given by a rank onesymmetric space, provided that its isometry group Iso(M,g) is large. Moreprecisely we prove first that if dim(Iso(M,g)) iÝ 2 dim(M) . 6, then M is tangentially homotopically equivalent to a rank one symmetric space or M is homogeneous. Secondly, we show that in dimensions above 18(k +1)2 each M is tangentially homotopically equivalent to a rank one symmetric space, where k > 0 denotes the cohomogeneity, k = dim(M/Iso(M,g)).
TL;DR: In this article, it was shown that the class of uncountable linear orders has a five-element basis and that the existence of a supercompact cardinal is a strong form of the axiom of infinity.
Abstract: In this paper I will show that it is relatively consistent with the usual axioms of mathematics (ZFC) together with a strong form of the axiom of infinity (the existence of a supercompact cardinal) that the class of uncountable linear orders has a five element basis In fact such a basis follows from the Proper Forcing Axiom, a strong form of the Baire Category Theorem The elements are X, ¥o1, ¥o1, C, C where X is any suborder of the reals of cardinality 1 and C is any Countryman line This confirms a longstanding conjecture of Shelah
TL;DR: An improved version of the Siegel-Shidlovskii theorem is derived that gives a complete characterisation of algebraic relations over the algebraic numbers between values of E-functions at any nonzero algebraic point.
Abstract: Using Y. Andre's result on differential equations satisfied by E-functions, we derive an improved version of the Siegel-Shidlovskii theorem. It gives a complete characterisation of algebraic relations over the algebraic numbers between values of E-functions at any nonzero algebraic point.
TL;DR: A review of the classical setting of real quadratic fields can be found in this paper, where the Bruhat-Tits tree and the Dedekind sums are discussed.
Abstract: 1. A review of the classical setting 2. Elliptic units for real quadratic fields 2.1. p-adic measures 2.2. Double integrals 2.3. Splitting a two-cocycle 2.4. The main conjecture 2.5. Modular symbols and Dedekind sums 2.6. Measures and the Bruhat-Tits tree 2.7. Indefinite integrals 2.8. The action of complex conjugation and of Up 3. Special values of zeta functions 3.1. The zeta function 3.2. Values at negative integers 3.3. The p-adic valuation 3.4. The Brumer-Stark conjecture 3.5. Connection with the Gross-Stark conjecture 4. A Kronecker limit formula 4.1. Measures associated to Eisenstein series 4.2. Construction of the p-adic L-function 4.3. An explicit splitting of a two-cocycle 4.4. Generalized Dedekind sums 4.5. Measures on Zp × Zp 4.6. A partial modular symbol of measures on Zp × Zp 4.7. From Zp × Zp to X 4.8. The measures μ and Γ-invariance