Showing papers in "Annals of Mathematics in 1999"

Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model

Abstract: We derive semiclassical asymptotics for the orthogonal polynomials Pn(z) on the line with respect to the exponential weight exp(iNV(z)), where V (z) is a double-well quartic polynomial, in the limit when n;N!1. We assume that "• (n=N)• ‚cri" for some "> 0, where ‚cr is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut. Simultaneously we derive semiclassical asymptotics for the recursive coe‐cients of the orthogonal polynomials, and we show that these coe‐cients form a cycle of period two which drifts slowly with the change of the ratio n=N. The proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix RiemannHilbert problem. As an application of the semiclassical asymptotics of the orthogonal polynomials, we prove the universality of the local distribution of eigenvalues in the matrix model with the double-well quartic interaction in the presence of two cuts. Contents

Metal-insulator transition for the almost Mathieu operator

Abstract: We prove that for Diophantine ! and almost every µ; the almost Mathieu operator, (H!;‚;µ“)(n )=“ (n +1 ) +“ (ni 1) +‚ cos 2…(!n+µ)“(n), exhibits localization for ‚> 2 and purely absolutely continuous spectrum for ‚< 2:

Stable laws and domains of attraction in free probability theory

Abstract: In this paper we determine the distributional behavior of sums of free (in the sense of Voiculescu) identically distributed, inflnitesimal random variables. The theory is shown to parallel the classical theory of independent random variables, though the limit laws are usually quite difierent. Our work subsumes all previously known instances of weak convergence of sums of free, identically distributed random variables. In particular, we determine the domains of attraction of stable distributions in the free theory. These freely stable distributions are studied in detail in the appendix, where their unimodality and duality properties are demonstrated.

On Calderon s conjecture

Abstract: with constants Cfi;p1;p2 depending only on fi;p1;p2 and p := p1p2 p1+p2 hold. The flrst result of this type is proved in [4], and the purpose of the current paper is to extend the range of exponents p1 and p2 for which (2) is known. In particular, the case p1 =2 ,p2 = 1 is solved to the a‐rmative. This was

The instability of naked singularities in the gravitational collapse of a scalar field

Abstract: One of the fundamental unanswered questions in the general theory of relativity is whether aked" singularities, that is singular events which are visible from inflnity, may form with positive probability in the process of gravitational collapse. The conjecture that the answer to this question is in the negative has been called \cosmic censorship." The present paper, which is a continuation of the work in [1], [2], [3] and [4] addresses this question in the context of the spherical gravitational collapse of a scalar fleld. The problem of a spherically symmetric self-gravitating scalar fleld is formulated in terms of a 2-dimensional quotient space-time manifold Q with boundary (see [1]). The boundary ofQ corresponds to the set of flxed points of the group action, the center of symmetry, which is a timelike geodesic i. The manifoldQ is endowed with a Lorentzian metric gab, an area radius function i‚ @ i‚ @ @r

An analytic solution to the Busemann-Petty problem on sections of convex bodies

Abstract: We derive a formula connecting the derivatives of parallel section functions of an origin-symmetric star body in R n with the Fourier transform of powers of the radial function of the body. A parallel section function (or (ni 1)dimensional X-ray) gives the ((ni 1)-dimensional) volumes of all hyperplane sections of the body orthogonal to a given direction. This formula provides a new characterization of intersection bodies in R n and leads to a unifled analytic solution to the Busemann-Petty problem: Suppose that K and L are two origin-symmetric convex bodies inR n such that the ((ni 1)-dimensional) volume of each central hyperplane section of K is smaller than the volume of the corresponding section of L; is the (n-dimensional) volume of K smaller than the volume of L? In conjunction with earlier established connections between the Busemann-Petty problem, intersection bodies, and positive deflnite distributions, our formula shows that the answer to the problem depends on the behavior of the (ni 2)-nd derivative of the parallel section functions. The a‐rmative answer to the Busemann-Petty problem for n• 4 and the negative answer for n‚ 5 now follow from the fact that convexity controls the second derivatives, but does not control the derivatives of higher orders.

The Cassels-Tate pairing on polarized abelian varieties

Abstract: Let (A, e) be a principally polarized abelian variety defined over a global field k, and let ..(A) be its Shafarevich-Tate group. Let ..(A)nd denote the quotient of ..(A) by its maximal divisible subgroup. Cassels and Tate constructed a nondegenerate pairing ..(A)nd . ..(A)nd > Q/Z. If A is an elliptic curve, then by a result of Cassels the pairing is alternating. But in general it is only antisymmetric. Using some new but equivalent definitions of the pairing, we derive general criteria deciding whether it is alternating and whether there exists some alternating nondegenerate pairing on ..(A)nd. These criteria are expressed in terms of an element c . ..(A)nd that is canonically associated to the polarization e. In the case that A is the Jacobian of some curve, a down-to-earth version of the result allows us to determine effectively whether #..(A) (if finite) is a square or twice a square. We then apply this to prove that a positive proportion (in some precise sense) of all hyperelliptic curves of even genus g . 2 over Q have a Jacobian with nonsquare #.. (if finite). For example, it appears that this density is about 13% for curves of genus 2. The proof makes use of a general result relating global and local densities; this result can be applied in other situations.

Permanents, pfaffian orientations, and even directed circuits

Abstract: Given a 0-1 square matrix A, when can some of the 1’s be changed toi1’s in such a way that the permanent of A equals the determinant of the modifled matrix? When does a real square matrix have the property that every real matrix with the same sign pattern (that is, the corresponding entries either have the same sign or are both zero) is nonsingular? When is a hypergraph with n vertices and n hyperedges minimally nonbipartite? When does a bipartite graph have a \Pfa‐an orientation"? Given a digraph, does it have no directed circuit of even length? Given a digraph, does it have a subdivision with no even directed circuit? It is known that all of the above problems are equivalent. We prove a structural characterization of the feasible instances, which implies a polynomial-time algorithm to solve all of the above problems. The structural characterization says, roughly speaking, that a bipartite graph has a Pfa‐an orientation if and only if it can be obtained by piecing together (in a specifled way) planar bipartite graphs and one sporadic nonplanar bipartite graph.

Continuous rotation invariant valuations on convex sets

Abstract: The notion of valuation on convex sets can be considered as a generalization of the notion of measure, which is deflned only on the class of convex compact sets. It is well-known that there are important and interesting examples of valuations on convex sets, which are not measures in the usual sense as, for example, the mixed volumes. Basic deflnitions and some classical examples are discussed in Section 2 of this paper. For more detailed information we refer to the surveys [Mc-Sch] and [Mc3]. Throughout this paper all the valuations are assumed to be continuous with respect to the Hausdorfi metric. Note that the theory of valuations which are invariant or covariant with respect to translations belongs to the classical part of convex geometry. There exists an explicit description of translation invariant continuous valuations on 1 and 2 due to Hadwiger [H1] (the case of 2 is nontrivial). Continuous rigid motion invariant valuations on d are completely classifled by the remarkable Hadwiger theorem as linear combinations of the quermassintegrals (cf. [H2] or for a simpler proof [K]).

Orbit equivalence rigidity

Abstract: Consider a countable group acting ergodically by measure p reserving transformations on a probability space (X,µ), and let R be the corresponding orbit equivalence relation on X. The following rigidity phenomenon is shown: there exist group actions such that the equivalence relation R on X determines the group and the action ( X,µ,) uniquely, up to finite groups. The natural action of SLn(Z) on the n-torus R n /Z n , for n > 2, is one of such examples. The interpretation of these results in the context of von Neumann algebras provides some support to the conjecture of Connes on rigidity of group algebras for groups with property T. Our rigidity results also give examples of countable equivalence relations of type II1, which cannot be generated (mod 0) by a free action of any group. This gives a negative answer to a long standing problem of Feldman and Moore.

Dimension and product structure of hyperbolic measures

Abstract: We prove that every hyperbolic measure invariant under a C 1+fi difieomorphism of a smooth Riemannian manifold possesses asymptotically \almost" local product structure, i.e., its density can be approximated by the product of the densities on stable and unstable manifolds up to small exponentials. This has not been known even for measures supported on locally maximal hyperbolic sets. Using this property of hyperbolic measures we prove the long-standing Eckmann-Ruelle conjecture in dimension theory of smooth dynamical systems: the pointwise dimension of every hyperbolic measure invariant under a C 1+fi difieomorphism exists almost everywhere. This implies the crucial fact that virtually all the characteristics of dimension type of the measure (including the Hausdorfi dimension, box dimension, and information dimension) coincide. This provides the rigorous mathematical justiflcation of the concept of fractal dimension for hyperbolic measures.

A positive solution to the Busemann-Petty problem in R 4

Abstract: H. Busemann and C. M. Petty posed the following problem in 1956: If K and L are origin-symmetric convex bodies in R^n and for each hyperplane H through the origin the volumes of their central slices satisfy vol(K cap H) < vol(L cap H), does it follow that the volumes of the bodies themselves satisfy vol(K) < vol(L)? The problem is trivially positive in R^2. However, a surprising negative answer for n = 5 by a number of authors. That is, the problem has a negative answer for n >= 5. It was proved by Gardner that the problem has a positive answer for n=3. The case of n=4 was considered in [Ann. of Math. (2) 140 (1994), 331-346], but the answer given there is not correct. This paper presents the correct solution, namely, the Busemann-Petty problem has a positive solution in R^4, which, together with results of other cases, brings the Busemann-Petty problem to a conclusion.

Hessian measures II

Abstract: In our previous paper on this topic, we introduced the notion of k-Hessian measure associated with a continuous k-convex function in a domain Q in Euclidean n-space, k = 1,... , n, and proved a weak continuity result with respect to local uniform convergence. In this paper, we consider k-convex functions, not necessarily continuous, and prove the weak continuity of the associated k-Hessian measure with respect to convergence in measure. The proof depends upon local integral estimates for the gradients of k-convex functions.

Feigenbaum-Coullet-Tresser universality and Milnor's hairiness conjecture.

Abstract: We prove the Feigenbaum-Coullet-Tresser conjecture on the hyperbolicity of the renormalization transformation of bounded type. This gives the first computer-free proof of the original Feigenbaum observation of the universal parameter scaling laws. We use the Hyperbolicity Theorem to prove Milnor's conjectures on self-similarity and "hairiness" of the Mandelbrot set near the corresponding parameter values. We also conclude that the set of real infinitely renormalizable quadratics of type bounded by some N > 1 has Hausdorff dimension strictly between 0 and 1. In the course of getting these results we supply the space of quadratic-like germs with a complex analytic structure and demonstrate that the hybrid classes form a complex codimension-one foliation of the connectedness locus.

Surgery and duality

Abstract: Surgery, as developed by Browder, Kervaire, Milnor, Novikov, Sullivan, Wall and others is a method for comparing homotopy types of topological spaces with difieomorphism or homeomorphism types of manifolds of dimension‚ 5. In this paper, a modiflcation of this theory is presented, where instead of flxing a homotopy type one considers a weaker information. Roughly speaking, one compares n-dimensional compact manifolds with topological spaces whose k-skeletons are flxed, where k is at least [n=2]. A particularly attractive example which illustrates the concept is given by complete intersections. By the Lefschetz hyperplane theorem, a complete intersection of complex dimension n has the same n-skeleton as n and one can use the modifled theory to obtain information about their difieomorphism type although the homotopy classiflcation is not known. The theory reduces this classiflcation result to the determination of complete intersections in a certain bordism group. This was under certain restrictions carried out in [Tr]. The restrictions are: If d = d1¢:::¢dr is the total degree of a complete intersection X n1;:::;dr of complex dimension n, then the assumption is, that for all primes p with p(pi1)• n+1, the total degree d is divisible by p [(2n+1)=(2pi1)]+1 . Theorem A. Two complete intersections X n

Gradient estimates and blow-up analysis for stationary harmonic maps

Abstract: For stationary harmonic maps between Riemannian manifolds, we provide a necessary and sufficient condition for the uniform interior and boundary gradient estimates in terms of the total energy of maps. We also show that if analytic target manifolds do not carry any harmonic S^2, then the singular sets of stationary maps are m \leq n - 4 rectifiable. Both of these results follow from a general analysis on the defect measures and energy concentration sets associated with a weakly converging sequence of stationary harmonic maps.

Singular and maximal Radon transforms: analysis and geometry

Abstract: Part 2. Geometric theory 8. Curvature: Introduction 8.1. Three notions of curvature 8.2. Theorems 8.3. Examples 9. Curvature: Some details 9.1. The exponential representation 9.2. Diffeomorphism invariance 9.3. Curvature condition (CY ) 9.4. Two lemmas 9.5. Double fibration formulation 10. Equivalence of curvature conditions 10.1. Invariant submanifolds and deficient Lie algebras 10.2. Vanishing Jacobians 10.3. Construction of invariant submanifolds

Nonsymmetric Koornwinder polynomials and duality

Abstract: In the fundamental work of Lusztig [L] on affine Hecke algebras, a special role is played by the root system of type Cn. The affine Hecke algebra is a deformation of the group algebra of an affine Weyl group which usually depends on as many parameters as there are distinct root lengths, i.e. one or two for an irreducible root system. However in the Cn case, the Hecke algebra H has three parameters, corresponding to the fact that there is a simple coroot which is divisible by 2. Recently, Cherednik [C1]-[C3] has introduced the notion of a double affine Hecke algebra, and has used it to prove several conjectures on Macdonald polynomials. These polynomials, and Cherednik's double affine Hecke algebra, involve two or three parameters, i.e., one more than the number of root lengths.

The classification of punctured-torus groups.

Abstract: Thurston’s ending lamination conjecture proposes that a flnitely generated Kleinian group is uniquely determined (up to isometry) by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends. We present a proof of this conjecture for punctured-torus groups. These are free two-generator Kleinian groups with parabolic commutator, which should be thought of as representations of the fundamental group of a punctured torus. As a consequence we verify the conjectural topological description of the deformation space of punctured-torus groups (including Bers’ conjecture that the quasi-Fuchsian groups are dense in this space) and prove a rigidity theorem: two punctured-torus groups are quasi-conformally conjugate if and only if they are topologically conjugate.

Free boundary regularity for harmonic measures and Poisson kernels

Abstract: One of the basic aims of this paper is to study the relationship between the geometry of \hypersurface like" subsets of Euclidean space and the properties of the measures they support. In this context we show that certain doubling properties of a measure determine the geometry of its support. A Radon measure is said to be doubling with constant C if C times the measure of the ball of radius r centered on the support is greater than the measure of the ball of radius 2r and the same center. We prove that if the doubling constant of a measure on R n+1 is close to the doubling constant of the ndimensional Lebesgue measure then its support is well approximated by ndimensional a‐ne spaces, provided that the support is relatively ∞at to start with. Primarily we consider sets which are boundaries of domains in R n+1 . The n-dimensional Hausdorfi measure may not be deflned on the boundary of a domain in R n+1 . Thus we turn our attention to the harmonic measure which is well behaved under minor assumptions (see Section 3). We obtain a new characterization of locally ∞at domains in terms of the doubling properties of their harmonic measure (see Section 3). Along these lines we investigate how the \weak" regularity of the Poisson kernel of a domain determines the geometry of its boundary. Sections 5 and 6 pursue this goal, as in Alt and Cafiarelli’s work (see [AC], [C1], [C2]), and also Jerison’s [J]. In both cases the goal is to prove that, under the appropriate technical conditions at \∞at points" of the boundary, the oscillation of the Poisson kernel controls the oscillation of the unit normal vector. The difierence between our work and the work in [AC] is that we measure the oscillation in an integral sense (BMO estimates)

Basepoint freeness for nef and big line bundles in positive characteristic

Abstract: A necessary and sufficient condition is given for semi-ampleness of a numerically effective (nef) and big line bundle in positive characteristic. One application is to the geometry of the universal stable curve over Mg, specifically, the semi-ampleness of the relative dualizing sheaf, in positive characteristic. An example is given which shows this and the semi-ampleness criterion fail in characteristic zero. A second application is to Mori's program for minimal models of 3-folds in positive characteristic, namely, to the existence of birational extremal contractions.

A new approach to inverse spectral theory, I. Fundamental formalism

Abstract: We present a new approach (distinct from Gelfand-Levitan) to the the- orem of Borg-Marchenko that the m-function (equivalently, spectral measure) for a finite interval or half-line Schrodinger operator determines the poten- tial. Our approach is an analog of the continued fraction approach for the moment problem. We prove there is a representation for the m-function m(−κ 2 ) = −κ − R b

Gromov’s measure equivalence and rigidity of higher rank lattices

Abstract: In this paper the notion of Measure Equivalence (ME) of countable groups is studied. ME was introduced by Gromov as a measure-theoretic analog of quasi-isometries. All lattices in the same locally compact group are Measure Equivalent; this is one of the motivations for this notion. The main result of this paper is ME rigidity of higher rank lattices: any countable group which is ME to a lattice in a simple Lie group G of higher rank, is commensurable to a lattice in G.

Quantum cohomology of flag manifolds G/B and quantum Toda lattices

Abstract: Let G be a connected semi-simple complex Lie group, B its Borel subgroup, T a maximal complex torus contained in B, and Lie (T ) its Lie algebra. This setup gives rise to two constructions; the generalized nonperiodic Toda lattice ([28], [29]) and the flag manifold G/B. The Toda lattice for (G,B, T ) is the dynamical system on the cotangent bundle T ∗Lie (T ) endowed with the canonical holomorphic symplectic form and the holomorphic hamiltonian function we consider in this paper,

Menger curvature and rectifiability

Abstract: where 'HI is the 1-dimensional Hausdorff measure in Rn, c(x, y, z) is the inverse of the radius of the circumcircle of the triangle (x, y, z), that is, following the terminology of [6], the Menger curvature of the triple (x, y, z). A Borel set E C Rn is said to be "purely unrectifiable" if for any Lipschitz function y: R -t R, 1H l(E n -(R)) = 0 whereas it is said to be rectifiable if there exists a countable family of Lipschitz functions yi : R -E Rn such that Hi1(E\ Ui yi(IR)) = 0. It may be seen from this definition that any 1-set E (that is, E Borel and 0 < 7-(t(E) < oo) can be decomposed into two subsets

Rectifiability of flat chains

Abstract: We prove (without using Federer's structure theorem) that a finite-mass flat chain over any coefficient group is rectifiable if and only if almost all of its 0-dimensional slices are rectifiable. This implies that every flat chain of finite mass and finite size is rectifiable. It also leads to a simple necessary and sufficient condition on the coefficient group in order for every finite-mass flat

Solving moment problems by dimensional extension

Abstract: polynomial inequalities. The second part of the paper uses the representation results of positive functionals on certain spaces of rational functions developed in the flrst part, for decomposing a polynomial which is positive on such a semi-algebraic set into a canonical sum of squares of rational functions times explicit multipliers. Let n‚ 1 be a flxed integer. Due to the fact that for n> 1 not every nonnegative polynomial in R n can be written as a sum of squares of polynomials (see, for instance, [2,x6.3]), the moment problems in n variables are more di‐cult than the classical one variable problems. This very intriguing territory has been investigated by many authors (see [2], [7], [12] and their references), although characterizations for measures whose support lies in an arbitrary (generally unbounded) semi-algebraic set do not seem to exist. The present paper starts from an idea of the second author, see [19], about solving moment problems by a change of basis via an embedding of R n into a submanifold of a higher dimensional Euclidean space. Rougly speaking we prove that certain (n + 1)-dimensional extensions of a moment sequence are naturally characterized by positivity conditions and moreover, these extensions parametrize all possible solutions of the moment problem. To be more speciflc, let ∞fi = Z R n x fi d„(x) ;fi 2 Z n

The Spectrum of coupled random matrices

Abstract: The study of the spectrum of coupled random matrices has received rather little attention. To the best of our knowledge, coupled random matrices have been studied, to some extent, by Mehta. In this work, we explain how the integrable technology can be brought to bear to gain insight into the nature of the distribution of the spectrum of coupled Hermitean random matrices and the equations the associated probabilities satisfy. In particular, the two-Toda lattice, its algebra of symmetries and its vertex operators will play a prominent role in this interaction. Namely, the method is to introduce time parameters, in an artificial way, and to dress up a certain matrix integral with a vertex integral operator, for which we find Virasoro-like differential equations. These methods lead to very simple nonlinear third-order partial differential equations for the joint statistics of the spectra of two coupled Gaussian random matrices. Comment: 56 pages, published version, abstract added in migration

General linear and functor cohomology over finite fields

Abstract: In recent years, there has been considerable success in computing Extgroups of modular representations associated to the general linear group by relating this problem to one of computing Ext-groups in functor categories [F-L-S], [F-S]. In this paper, we extend our ability to make such Ext-group calculations by establishing several fundamental results. Throughout this paper, we work over fields of positive characteristic p. The reader familiar with the representation theory of algebraic objects will recognize the importance of an understanding of Ext-groups. For example, the existence of nonzero Ext-groups of positive degree is equivalent to the existence of objects which are not “direct sums” of simple objects. Indeed, a knowledge of Ext-groups provides considerable knowledge of compound objects. In the study of modular representation theory of finite Chevalley groups such as GLn(Fq), Ext-groups play an even more central role: it has been shown in [CPS] that a knowledge of certain Ext-groups is sufficient to prove Lusztig’s Conjecture concerning the dimension and characters of irreducible representations. We consider two different categories of functors, the category F(Fq) of all functors from finite dimensional Fq-vector spaces to Fq-vector spaces, where Fq is the finite field of cardinality q, and the category P(Fq) of strict polynomial functors of finite degree as defined in [F-S]. The category P(Fq) presents several advantages over the category F(Fq) from the point of view of computing Extgroups. These are the accessibility of injectives and projectives, the existence of a base change, and an even easier access to Ext-groups of tensor products. This explains the usefulness of our comparison in Theorem 3.10 of Ext-groups in the category P(Fq) with Ext-groups in the category F(Fq). Weaker forms of this theorem have been known to us since 1995 and to S. Betley independently

Analytic continuation of representations and estimates of automorphic forms

Abstract: 0.1. Analytic vectors and their analytic continuation. Let G be a Lie group and (�,G,V ) a continuous representation of G in a topological vector space V. A vector v ∈ V is called analytic if the functionv : g 7→ �(g)v is a real analytic function on G with values in V. This means that there exists a neighborhood U of G in its complexification GC such thatv extends to a holomorphic function on U. In other words, for each element g ∈ U we can unambiguously define the vector �(g)v asv(g), i.e., we can extend the action of G to a somewhat larger set. In this paper we will show that the possibility of such an extension sometimes allows one to prove some highly nontrivial estimates. Unless otherwise stated, G = SL(2, R), so GC = SL(2, C). We consider a typical representation ofG, i.e., a representation of the principal series. Namely, fix� ∈ C and consider the space Dof smooth homogeneous functions of degree � − 1 on R 2 \ 0, i.e., D� = {� ∈ C ∞ (R 2 \ 0) : �(ax,ay) = |a| �−1 �(x,y)}; we