Showing papers in "Annals of Mathematics in 2007"
TL;DR: In this paper, the authors introduce the notion of a stability condition on a triangulated category and prove a deformation result which shows that the space with its natural topology is a manifold, possibly infinite-dimensional.
Abstract: This paper introduces the notion of a stability condition on a triangulated category. The motivation comes from the study of Dirichlet branes in string theory, and especially from M.R. Douglas's notion of $\Pi$-stability. From a mathematical point of view, the most interesting feature of the definition is that the set of stability conditions $\Stab(\T)$ on a fixed category $\T$ has a natural topology, thus defining a new invariant of triangulated categories. After setting up the necessary definitions I prove a deformation result which shows that the space $\Stab(\T)$ with its natural topology is a manifold, possibly infinite-dimensional.
825 citations
TL;DR: In this article, the basic relationship between G and G is discussed, and a canonical construction of G, starting from G, is presented, which leads to a rather explicit construction of a Hopf algebra by Tannakian formalism.
Abstract: As such, it can be viewed as a first step in the geometric Langlands program. The connected complex reductive groups have a combinatorial classification by their root data. In the root datum the roots and the co-roots appear in a symmetric manner and so the connected reductive algebraic groups come in pairs. If G is a reductive group, we write G for its companion and call it the dual group G. The notion of the dual group itself does not appear in Satake's paper, but was introduced by Langlands, together with its various elaborations, in [LI], [L2] and is a cornerstone of the Langlands program. It also appeared later in physics [MO], [GNO]. In this paper we discuss the basic relationship between G and G. We begin with a reductive G and consider the affine Grassmannian Qx, the Grassmannian for the loop group of G. For technical reasons we work with formal algebraic loops. The affine Grassmannian is an infinite dimen sional complex space. We consider a certain category of sheaves, the spherical perverse sheaves, on ?r. These sheaves can be multiplied using a convolution product and this leads to a rather explicit construction of a Hopf algebra, by what has come to be known as Tannakian formalism. The resulting Hopf algebra turns out to be the ring of functions on G. In this interpretation, the spherical perverse sheaves on the affine Grassman nian correspond to finite dimensional complex representations of G. Thus, instead of defining G in terms of the classification of reductive groups, we pro vide a canonical construction of G, starting from G. We can carry out our construction over the integers. The spherical perverse sheaves are then those with integral coefficients, but the Grassmannian remains a complex algebraic object.
554 citations
TL;DR: In this paper, it was shown that the knowledge of the Cauchy data for the Schr?Nodinger equation measured on possibly very small subsets of the boundary determines uniquely the potential.
Abstract: In this paper we improve an earlier result by Bukhgeim and Uhlmann
[1], by showing that in dimension n . 3, the knowledge of the Cauchy data
for the Schr?Nodinger equation measured on possibly very small subsets of the
boundary determines uniquely the potential. We follow the general strategy
of [1] but use a richer set of solutions to the Dirichlet problem. This implies
a similar result for the problem of Electrical Impedance Tomography which
consists in determining the conductivity of a body by making voltage and
current measurements at the boundary.
449 citations
TL;DR: In this article, the authors prove the global existence and uniqueness of strong solutions to the three-dimensional viscous primitive equations, which model large scale ocean and atmosphere dynamics, and propose a global solution for the Navier-Stokes equation.
Abstract: In this paper we prove the global existence and uniqueness (regularity) of strong solutions to the three-dimensional viscous primitive equations, which model large scale ocean and atmosphere dynamics. 1. Introduction Large scale dynamics of oceans and atmosphere is governed by the primitive equations which are derived from the Navier-Stokes equations, with rotation, coupled to thermodynamics and salinity diffusion-transport equations, which account for the buoyancy forces and stratification effects under the Boussinesq approximation. Moreover, and due to the shallowness of the oceans and the atmosphere, i.e., the depth of the fluid layer is very small in comparison to the radius of the earth, the vertical large scale motion in the oceans and the atmosphere is much smaller than the horizontal one, which in turn leads to modeling the vertical motion by the hydrostatic balance. As a result
401 citations
TL;DR: In this article, it was shown that real projective three-space cannot be obtained from Dehn surgery on a nontrivial knot in the three-sphere, using a surgery long exact sequence for monopole Floer homology, together with a nonvanishing theorem.
Abstract: Monopole Floer homology is used to prove that real projective three-space cannot be obtained from Dehn surgery on a nontrivial knot in the three-sphere. To obtain this result, we use a surgery long exact sequence for monopole Floer homology, together with a nonvanishing theorem, which shows that monopole Floer homology detects the unknot. In addition, we apply these techniques to give information about knots which admit lens space surgeries, and to exhibit families of three-manifolds which do not admit taut foliations.
322 citations
TL;DR: In this article, the authors give the first combinatorial proof of the multidimensional Szemeredi theorem of Furstenberg and Katznelson, and the first proof that provides an explicit bound.
Abstract: We prove analogues for hypergraphs of Szemeredi's regularity lemma and the associated counting lemma for graphs. As an application, we give the first combinatorial proof of the multidimensional Szemeredi theorem of Furstenberg and Katznelson, and the first proof that provides an explicit bound. Similar results with the same consequences have been obtained independently by Nagle, Rodl, Schacht and Skokan.
318 citations
TL;DR: In this article, a stable homo- topy version of Mumford's conjecture was proved, relying on Harer's stability theorem (15), Vassiliev's theorem concerning spaces of functions with moderate singularities (43), and methods from homotopy theory.
Abstract: D.Mumford conjectured in (30) that the rational cohomology of the stable moduli space of Riemann surfaces is a polynomial algebra generated by certain classes i of di- mension 2i. For the purpose of calculating rational cohomology, one may replace the stable moduli space of Riemann surfaces by B 1, where 1 is the group of isotopy classes of automorphisms of a smooth oriented connected surface of "large" genus. Tillmann's insight (41) that the plus construction makes B 1 into an infinite loop space led to a stable homo- topy version of Mumford's conjecture, stronger than the original (22). We prove the stronger version, relying on Harer's stability theorem (15), Vassiliev's theorem concerning spaces of functions with moderate singularities (43), (42) and methods from homotopy theory.
316 citations
TL;DR: In this paper, Mumford's Grothendieck-Riemann-Roch theorem applied to the universal family over the moduli space of stable maps was extended to twisted Gromov-Witten invariants.
Abstract: Given a holomorphic vector bundle E over a compact Kahler manifold X, one defines twisted Gromov-Witten invariants of X to be intersection numbers in moduli spaces of stable maps f :Σ → X with the cap product of the virtual fundamental class and a chosen multiplicative invertible characteristic class of the virtual vector bundle H 0 (Σ ,f ∗ E) � H 1 (Σ ,f ∗ E). Using the formalism of quantized quadratic Hamiltonians (25), we express the descendant potential for the twisted theory in terms of that for X. This result (Theorem 1) is a consequence of Mumford's Grothendieck-Riemann-Roch theorem applied to the universal family over the moduli space of stable maps. It determines all twisted Gromov-Witten invariants, of all genera, in terms of untwisted invari- ants. When E is concave and the C × -equivariant inverse Euler class is chosen as the characteristic class, the twisted invariants of X give Gromov-Witten invariants of the total space of E. "Nonlinear Serre duality" (21), (23) expresses Gromov-Witten invariants of E in terms of those of the super-manifold ΠE: it relates Gromov-Witten invariants of X twisted by the inverse Euler class and E to Gromov-Witten invariants of X twisted by the Euler class and E ∗ . We derive from Theorem 1 nonlinear Serre duality in a very general form (Corollary 2). When the bundle E is convex and a submanifold Y ⊂ X is defined by a global section of E, the genus-zero Gromov-Witten invariants of ΠE coin- cide with those of Y. We establish a "quantum Lefschetz hyperplane section principle" (Theorem 2) expressing genus-zero Gromov-Witten invariants of a complete intersection Y in terms of those of X. This extends earlier results (4), (9), (18), (29), (33) and yields most of the known mirror formulas for toric complete intersections.
299 citations
220 citations
TL;DR: In this article, it was shown that irrational algebraic numbers are transcendent, for a wide class of morphisms, and that irrational morphic numbers also are transcendental for a general class of numbers.
Abstract: Let b ≥ 2 be an integer. We prove that the b-ary expansion of every irrational algebraic number cannot have low complexity. Furthermore, we establish that irrational morphic numbers are transcendental, for a wide class of morphisms. In particular, irrational automatic numbers are transcendental. Our main tool is a new, combinatorial transcendence criterion.
213 citations
TL;DR: In this paper, it was shown that in every finite profinite group, every subgroup of finite index is open, which implies that the topology on such groups is determined by the algebraic structure.
Abstract: We prove that in every finitely generated profinite group, every subgroup of finite index is open; this implies that the topology on such groups is determined by the algebraic structure. This is deduced from the main result about finite groups: let w be a 'locally finite' group word and d ∈ N. Then there exists f = f(w, d) such that in every d-generator finite group G, every element of the verbal subgroup w(G) is equal to a product of f w-values. An analogous theorem is proved for commutators; this implies that in every finitely generated profinite group, each term of the lower central series is closed. The proofs rely on some properties of the finite simple groups, to be established in Part II.
TL;DR: In this article, it was shown that if the multiplicities satisfy m2 ≥ 2m1 −1, then the isoparametric hypersurface M must be of FKM-type.
Abstract: Let M be an isoparametric hypersurface in the sphere S n with four distinct principal curvatures. Munzner showed that the four principal curvatures can have at most two distinct multiplicities m1 ,m 2, and Stolz showed that the pair (m1 ,m 2) must either be (2, 2), (4, 5), or be equal to the multiplicities of an isoparametric hypersurface of FKM-type, constructed by Ferus, Karcher and Munzner from orthogonal representations of Clifford algebras. In this paper, we prove that if the multiplicities satisfy m2 ≥ 2m1 −1, then the isoparametric hypersurface M must be of FKM-type. Together with known results of Takagi for the case m1 = 1, and Ozeki and Takeuchi for m1 = 2, this handles all possible pairs of multiplicities except for four cases, for which the classification problem remains open.
TL;DR: In this paper, the authors classify orbit closures and invariant measures for the natural-action of SL2(R) on UM2, the bundle of holomorphic 1-forms over the moduli space of Riemann surfaces of genus two.
Abstract: This paper classifies orbit closures and invariant measures for the natural
action of SL2(R) on UM2, the bundle of holomorphic 1-forms over the moduli
space of Riemann surfaces of genus two.
TL;DR: In this article, it was shown that a typical interval exchange transformation is either weakly mixing or it is an irrational rotation, and that the typical translation flow on a typical translation surface of genus g ≥ 2 (with prescribed singularity types) is weakly mixed.
Abstract: We prove that a typical interval exchange transformation is either weakly mixing or it is an irrational rotation. We also conclude that a typical translation flow on a typical translation surface of genus g ≥ 2 (with prescribed singularity types) is weakly mixing.
TL;DR: The Lovasz conjecture of as discussed by the authors states that if a cell complex Horn (C 2r+1, G) is k-connected, then Χ(G) > k + 4, where r, k ∈ Z, r > 1, k > -1, and C 2r-+1 denotes the cycle with 2 r+1 vertices.
Abstract: To any two graphs G and H one can associate a cell complex Horn (G, H) by taking all graph multihomomorphisms from G to H as cells. In this paper we prove the Lovasz conjecture which states that if Horn (C 2r+1 , G) is k-connected, then Χ(G) > k + 4, where r, k ∈ Z, r > 1, k > -1, and C 2r+1 denotes the cycle with 2r+1 vertices. The proof requires analysis of the complexes Horn (C 2r+1 , K n ). For even n, the obstructions to graph colorings are provided by the presence of torsion in H* (Hom (C 2r+1 , K n ); Z). For odd n, the obstructions are expressed as vanishing of certain powers of Stiefel-Whitney characteristic classes of Horn (C 2r+1 , K n ), where the latter are viewed as Z 2 -spaces with the involution induced by the reflection of C 2r+1 .
TL;DR: In this paper, it was shown that C is of Khintchine type for divergence, i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on C of C(ψ) is full.
Abstract: Let C be a nondegenerate planar curve and for a real, positive decreasing function ψ let C(ψ) denote the set of simultaneously ψ-approximable points lying on C. We show that C is of Khintchine type for divergence; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on C of C(ψ) is full. We also obtain the Hausdorff measure analogue of the divergent Khintchine type result. In the case that C is a rational quadric the convergence counterparts of the divergent results are also obtained. Furthermore, for functions ψ with lower order in a critical range we determine a general, exact formula for the Hausdorff dimension of C(ψ). These results constitute the first precise and general results in the theory of simultaneous Diophantine approximation on manifolds.
TL;DR: In this article, the authors studied the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Ricci tensor and proved an existence theorem for a wide class of symmetric functions on manifolds with positive Ricci curvature.
Abstract: We study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Ricci tensor. We prove an existence theorem for a wide class of symmetric functions on manifolds with positive Ricci curvature, provided the conformal class admits an admissible metric.
TL;DR: In this paper, the authors proved the topological rigidity property of real polynomials with all critical points real and non-degenerate, which completes the last step in solving the density of Axiom A conjecture in real one-dimensional dynamics.
Abstract: We prove the topological (or combinatorial) rigidity property for real polynomials with all critical points real and nondegenerate, which completes the last step in solving the density of Axiom A conjecture in real one-dimensional dynamics.
TL;DR: In this paper, the authors show that a real polynomial is hyperbolic if the real line is the union of a repelling hyperbola set and the basin of hyperbolas attracting periodic points is a basin of infinity.
Abstract: Here we say that a real polynomial is hyperbolic or Axiom A, if the real line is the union of a repelling hyperbolic set, the basin of hyperbolic attracting periodic points and the basin of infinity. We call a C1 endomorphism of the compact interval (or the circle) hyperbolic if it has finitely many hyperbolic attracting periodic points and the complement of the basin of attraction of these points is a hyperbolic set. By a theorem of Mañé for C2 maps, this is equivalent to the following conditions: all periodic points are hyperbolic and all critical points converge to periodic attractors. Note that the space of hyperbolic maps is an open subset in the space of real polynomials of fixed degree, and that every hyperbolic map satisfying the mild “no-cycle” condition (which states that orbits of critical points are disjoint) is structurally stable; see [dMvS93]. Theorem 1 solves the 2nd part of Smale’s eleventh problem for the 21st century [Sma00]:
TL;DR: In this article, lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields were obtained for all primes where the generator c of Gal(K/k) acts on Gal(κ-/K) by inversion, and it was shown that if the Zp-rank of the pro-p Selmer group is odd, then rank Zp S p (E/F) > [F: K] for every finite extension F of K in κ -.
Abstract: We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose K/k is a quadratic extension of number fields, E is an elliptic curve defined over k, and p is an odd prime. Let κ - denote the maximal abelian p-extension of K that is unramified at all primes where E has bad reduction and that is Galois over k with dihedral Galois group (i.e., the generator c of Gal(K/k) acts on Gal(κ-/K) by inversion). We prove (under mild hypotheses on p) that if the Zp-rank of the pro-p Selmer group Sp(E/K) is odd, then rank Zp S p (E/F) > [F: K] for every finite extension F of K in κ - .
TL;DR: In this article, the authors define and study an algebra Ψ ∞,0,V (M0) of pseudodifferential operators canonically associated to a non-compact, Riemannian manifold M0 whose geometry at infinity is described by a Lie algebra of vector fields V on a compactification M of M0 to a compact manifold with corners.
Abstract: We define and study an algebra Ψ ∞,0,V (M0) of pseudodifferential operators canonically associated to a noncompact, Riemannian manifold M0 whose geometry at infinity is described by a Lie algebra of vector fields V on a compactification M of M0 to a compact manifold with corners. We show that the basic properties of the usual algebra of pseudodifferential operators on a compact manifold extend to Ψ ∞,0,V (M0). We also consider the algebra Diff ∗ (M0) of differential operators on M0 generated by V and C ∞ (M), and show that Ψ ∞,0,V (M0) is a microlocalization of Diff ∗ (M0). Our construction solves a problem posed by Melrose in 1990. Finally, we introduce and study semi-classical and “suspended” versions of the algebra Ψ ∞,0,V (M0).
TL;DR: In this article, it was shown that a linear group over a local eld contains either a relatively open solvable subgroup or a relatively dense free subgroup, and this result has applications in dynamics, Riemannian foliations and pronite groups.
Abstract: Let k be a local eld, and GLn(k) a linear group over k. We prove that either contains a relatively open solvable subgroup, or it contains a relatively dense free subgroup. This result has applications in dynamics, Riemannian foliations and pronite groups.
TL;DR: In this article, the stability of mixing and rapid mixing for hyperbolic flows was studied and it was shown that there is a C 2 -open, C r -dense set of flows for which each nontrivial Hyperbolic basic set is rapid mixing.
Abstract: We obtain general results on the stability of mixing and rapid mixing (superpolynomial decay of correlations) for hyperbolic flows. Amongst C r Axiom A flows, r ≥ 2, we show that there is a C 2 -open, C r -dense set of flows for which each nontrivial hyperbolic basic set is rapid mixing. This is the first general result on the stability of rapid mixing (or even mixing) for Axiom A � � �
TL;DR: In this article, it was shown that there is an absolute constant D such that for any finite quasisimple group S, given 2D arbitrary automorphisms of S, every element of S is equal to a product of D twisted commutators defined by the given automomorphisms.
Abstract: We prove two results. (1) There is an absolute constant D such that for any finite quasisimple group S, given 2D arbitrary automorphisms of S, every element of S is equal to a product of D ‘twisted commutators’ defined by the given automorphisms. (2) Given a natural number q, there exist C = C(q) and M = M(q) such that: if S is a finite quasisimple group with |S/Z(S)| >C , βj (j =1 ,... , M) are any automorphisms of S, and qj (j =1 ,... , M) are any divisors of q, then there exist inner automorphisms αj of S such that S = � M [S, (αjβj) q j ]. These results, which rely on the classification of finite simple groups, are needed to complete the proofs of the main theorems of Part I.
TL;DR: For a transversal pair of closed Lagrangian submanifolds, the pages of the spectral sequence coincide with the terms of the Serre spectral sequence of the path-loop fibration as discussed by the authors.
Abstract: For a transversal pair of closed Lagrangian submanifolds L, Lof a sym- plectic manifold M such that π1(L )= π1(L � )=0= c1|π2(M) = ω|π2(M) and for a generic almost complex structure J, we construct an invariant with a high homotopical content which consists in the pages of order ≥ 2o f as pec- tral sequence whose differentials provide an algebraic measure of the high- dimensional moduli spaces of pseudo-holomorpic strips of finite energy that join L and L � . When L and Lare Hamiltonian isotopic, we show that the pages of the spectral sequence coincide (up to a horizontal translation) with the terms of the Serre spectral sequence of the path-loop fibration ΩL → PL → L and we deduce some applications.
TL;DR: In this article, the authors prove the existence of dynamical delocalization for random Landau Hamiltonians near each Landau level and show that for these two-dimensional Hamiltonians there exists at least one energy E near each landau level such that! (E)! 1, where!(E), the local transport exponent introduced in [GK5], is a measure of the rate of transport in wave packets with spectral support near E. Since typically there is dynamical localization at the edges of each disordered-broadened Landau band, this implies the existence
Abstract: We prove the existence of dynamical delocalization for random Landau Hamiltonians near each Landau level. Since typically there is dynamical localization at the edges of each disordered-broadened Landau band, this implies the existence of at least one dynamical mobility edge at each Landau band, namely a boundary point between the localization and delocalization regimes, which we prove to converge to the corresponding Landau level as either the magnetic field goes to infinity or the disorder goes to zero. In this article we prove the existence of dynamical delocalization for random Landau Hamiltonians near each Landau level. More precisely, we prove that for these two-dimensional Hamiltonians there exists at least one energy E near each Landau level such that ! (E) ! 1 , where ! (E), the local transport exponent introduced in [GK5], is a measure of the rate of transport in wave packets with spectral support near E. Since typically there is dynamical localization at the edges of each disordered-broadened Landau band, this implies the existence of at least one dynamical mobility edge at each Landau band, namely a boundary point between the localization and delocalization regimes, which we prove to converge to the corresponding Landau level as either the magnetic field goes to infinity or the disorder goes to zero. Random Landau Hamiltonians are the subject of intensive study due to their links with the integer quantum Hall e! ect [Kli], for which von Klitzing received the 1985 Nobel Prize in Physics. They describe an electron moving in a very thin flat conductor with impurities under the influence of a constant magnetic field perpendicular to the plane of the conductor, and play an important role in the understanding of the quantum Hall e! ect [L], [AoA], [T], [H], [NT], [Ku], [Be], [AvSS], [BeES]. Laughlin’s argument [L], as pointed out by Halperin [H], uses the assumption that under weak disorder and strong magnetic field the energy spectrum consists of bands of extended states separated
TL;DR: In this paper, it was shown that the counting function of the eigenvalues of the Casimir operator acting in the space of cusp forms for the modular group SL n (Z) satisfies Weyl's law.
Abstract: Let r be a principal congruence subgroup of SL n (Z) and let σ be an irreducible unitary representation of SO(n). Let N Γ cus (λ,σ) be the counting function of the eigenvalues of the Casimir operator acting in the space of cusp forms for Γ which transform under SO(n) according to a. In this paper we prove that the counting function N Γ cus (λ,σ) satisfies Weyl's law. Especially, this implies that there exist infinitely many cusp forms for the full modular group SL n (Z).
TL;DR: In this article, the authors studied the ac-conductivity in linear response theory in the general framework of ergodic magnetic Schrodinger operators and proved that the ac conductivity is bounded from above by Cν 2 (log 1 ) d+2 at small frequencies ν.
Abstract: We study the ac-conductivity in linear response theory in the general framework of ergodic magnetic Schrodinger operators. For the Anderson model, if the Fermi energy lies in the localization regime, we prove that the ac- conductivity is bounded from above by Cν 2 (log 1 ) d+2 at small frequencies ν. This is to be compared to Mott's formula, which predicts the leading term to be Cν 2 (log 1 ν ) d+1 .
TL;DR: In this paper, it was shown that the smallest Pisot number whose minimal polynomial has odd coefficients is a limit point, from both sides, of Salem [19] numbers with coef ficients in {?1,1}.
Abstract: We prove that if f(x) = YlkZo akxk is a polynomial with no cyclotomic factors whose coefficients satisfy ak = 1 mod 2 for 0 2. We also characterize the polynomials that appear as the noncyclotomic part of a polynomial whose coefficients satisfy ak = 1 mod p for each ft, for a fixed prime p. Last, we prove that the smallest Pisot number whose minimal polynomial has odd coefficients is a limit point, from both sides, of Salem [19] numbers whose minimal polynomials have coef ficients in {?1,1}.