Showing papers in "Israel Journal of Mathematics in 2010"

Graph norms and Sidorenko’s conjecture

Abstract: Let H and G be two finite graphs. Define h H (G) to be the number of homomorphisms from H to G. The function h H (·) extends in a natural way to a function from the set of symmetric matrices to ℝ such that for A G , the adjacency matrix of a graph G, we have h H (A G ) = h H (G). Let m be the number of edges of H. It is easy to see that when H is the cycle of length 2n, then h H (·)1/m is the 2n-th Schatten-von Neumann norm. We investigate a question of Lovasz that asks for a characterization of graphs H for which the function h H (·)1/m is a norm. We prove that h H (·)1/m is a norm if and only if a Holder type inequality holds for H. We use this inequality to prove both positive and negative results, showing that h H (·)1/m is a norm for certain classes of graphs, and giving some necessary conditions on the structure of H when h H (·)1/m is a norm. As an application we use the inequality to verify a conjecture of Sidorenko for certain graphs including hypercubes. In fact, for such graphs we can prove statements that are much stronger than the assertion of Sidorenko’s conjecture. We also investigate the h H (·)1/m norms from a Banach space theoretic point of view, determining their moduli of smoothness and convexity. This generalizes the previously known result for the 2n-th Schatten-von Neumann norms.

Base sizes for sporadic simple groups

Abstract: Let G be a permutation group acting on a set . A subset of is a base for G if its pointwise stabilizer in G is trivial. We write b(G) for the minimal size of a base for G. We determine the precise value of b(G) for every primitive almost simple sporadic group G, with the exception of two cases involving the Baby Monster group. As a corollary, we deduce that b(G) 6 7, with equality if and only if G is the Mathieu group M24 in its natural action on 24 points. This settles a conjecture of Cameron.

Testing properties of graphs and functions

Abstract: We define an analytic version of the graph property testing problem, which can be formulated as studying an unknown 2-variable symmetric function through sampling from its domain and studying the random graph obtained when using the function values as edge probabilities. We give a characterization of properties testable this way, and extend a number of results about “large graphs” to this setting.

Optimal transportation on non-compact manifolds

Abstract: In this work, we show how to obtain for non-compact manifolds the results that have already been done for Monge Transport Problem for costs coming from Tonelli Lagrangians on compact manifolds. In particular, the already known results for a cost of the type d r , r > 1, where d is the Riemannian distance of a complete Riemannian manifold, hold without any curvature restriction.

An extension theorem for slice monogenic functions and some of its consequences

Abstract: Slice monogenic functions were introduced by the authors in [6]. The central result of this paper is an extension theorem, which shows that every holomorphic function defined on a suitable domain D of a complex plane can be uniquely extended to a slice monogenic function defined on a domain UD, determined by D, in a Euclidean space of appropriate dimension. Two important consequences of the result are a structure theorem for the zero set of a slice monogenic function (with a related corollary for polynomials with coefficients in Clifford algebras), and the possibility to construct a multiplicative theory for such functions. Slice monogenic functions have a very important application in the definition of a functional calculus for n-tuples of noncommuting operators.

Quaternionic Monge-Ampere equation and Calabi problem for HKT-manifolds

Abstract: A quaternionic version of the Calabi problem on the Monge-Ampere equation is introduced, namely a quaternionic Monge-Ampere equation on a compact hypercomplex manifold with an HKT-metric. The equation is non-linear elliptic of second order. For a hypercomplex manifold with holonomy in SL(n,ℍ), uniqueness (up to a constant) of a solution is proven, aas well as the zero order a priori estimate. The existence of a solution is conjectured, similar to the Calabi-Yau theorem. We reformulate this quaternionic equation as a special case of the complex Hessian equation, making sense on any complex manifold.

Patching over fields

Abstract: We develop a new form of patching that is both far-reaching and more elementary than the previous versions that have been used in inverse Galois theory for function fields of curves. A key point of our approach is to work with fields and vector spaces, rather than rings and modules. After presenting a self-contained development of this form of patching, we obtain applications to other structures such as Brauer groups and differential modules.

The critical random graph, with martingales

Abstract: We give a short proof that the largest component C 1 of the random graph G(n, 1/n) is of size approximately n 2/3. The proof gives explicit bounds for the probability that the ratio is very large or very small. In particular, the probability that n −2/3|C 1| exceeds A is at most $${e^{ - c{A^3}}}$$ for some c > 0.

Final group topologies, Kac-Moody groups and Pontryagin duality

Abstract: We study final group topologies and their relations to compactness properties. In particular, we are interested in situations where a colimit or direct limit is locally compact, a kω-space, or locally kω. As a first application, we show that unitary forms of complex Kac-Moody groups can be described as the colimit of an amalgam of subgroups (in the category of Hausdorff topological groups, and the category of kω-groups). Our second application concerns Pontryagin duality theory for the classes of almost metrizable topological abelian groups, resp., locally kω topological abelian groups, which are dual to each other. In particular, we explore the relations between countable projective limits of almost metrizable abelian groups and countable direct limits of locally kω abelian groups.

A gelfand model for wreath products

Abstract: A Gelafand model for wreath products ℤr ≀ Sn is constructed. The proof relies on a combinatorial interpretation of the characters of the model, extending a classical result of Frobenius and Schur.

Multifractal analysis of non-uniformly hyperbolic systems

Abstract: We prove a multifractal formalismfor Birkhoff averages of continuous functions in the case of some non-uniformly hyperbolic maps, which includes interval examples such as the Manneville-Pomeau map.

Integral points on quadrics in three variables whose coordinates have few prime factors

Abstract: The main theorem states that if f(x1, x2, x3) is an indefinite anisotropic integral quadratic form with determinant d(f), and t a non-zero integer such that d(f)t is square-free, then as long as there is one integer solution to f(x1, x2, x3) = t there are infinitely many such solutions for which the product x1x2x3 has at most 26 prime factors. The proof relies on the affine linear sieve and in particular automorphic spectral methods to obtain a sharp level of distribution in the associated counting problem. The 26 comes from applying the sharpest known bounds towards Selberg’s eigenvalue conjecture. Assuming the latter the number 26 may be reduced to 22.

Strong wavefront lemma and counting lattice points in sectors

Abstract: We compute the asymptotics of the number of integral quadratic forms with prescribed orthogonal decompositions and more generally, the asymptotics of the number of lattice points lying in sectors of affine symmetric spaces. A new key ingredient in this article is the strong wavefront lemma, which shows that the generalized Cartan decomposition associated to a symmetric space is uniformly Lipschitz.

Seventh power moments of kloosterman sums

Abstract: Evaluations of the n-th power moments Sn of Kloosterman sums are known only for n ⩽ 6. We present here substantial evidence for an evaluation of S7 in terms of Hecke eigenvalues for a weight 3 newform on ΓO(525) with quartic nebentypus of conductor 105. We also prove some congruences modulo 3, 5 and 7 for the closely related quantity T7, where Tn is a sum of traces of n-th symmetric powers of the Kloosterman sheaf.

On the set of hypercyclic vectors for the differentiation operator

Abstract: Let D be the differentiation operator Df = f′ acting on the Frechet space H of all entire functions in one variable with the standard (compact-open) topology. It is known since the 1950’s that the set H(D) of hypercyclic vectors for the operator D is non-empty. We treat two questions raised by Aron, Conejero, Peris and Seoane-Sepulveda whether the set H(D) contains (up to the zero function) a non-trivial subalgebra of H or an infinite-dimensional closed linear subspace of H. In the present article both questions are answered affirmatively.

The Dixmier-Moeglin equivalence for twisted homogeneous coordinate rings

Abstract: Given a projective scheme X over a field k, an automorphism σ: X → X, and a σ-ample invertible sheaf L, one may form the twisted homogeneous coordinate ring B = B(X, L, σ), one of the most fundamental constructions in noncommutative projective algebraic geometry. We study the primitive spectrum of B, as well as that of other closely related algebras such as skew and skew-Laurent extensions of commutative algebras. Over an algebraically closed, uncountable field k of characteristic zero, we prove that the primitive ideals of B are characterized by the usual Dixmier-Moeglin conditions whenever dim X ≤ 2.

Distinction by the quasi-split unitary group

Abstract: In earlier work, we proved that any quadratic base change automorphic cuspidal representation of GL(n) is distinguished by a unitary group. Here we prove that we can take the unitary group to be quasi-split

Random sampling of bandlimited functions

Abstract: We consider the problem of random sampling for bandlimited functions. When can a bandlimited function f be recovered from randomly chosen samples f(x j ), j ∈ J ⊂ ℕ? We estimate the probability that a sampling inequality of the form $$ A\left\| f \right\|_2^2 \leqslant \sum\limits_{j \in J} {|f(x_j )|^2 \leqslant B\left\| f \right\|_2^2 } $$ hold uniformly for all functions f ∈ L 2(ℝ d ) with supp $$ \hat f \subseteq [ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern- ulldelimiterspace} 2},{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern- ulldelimiterspace} 2}]^d $$ or for some subset of bandlimited functions. In contrast to discrete models, the space of bandlimited functions is infinite-dimensional and its functions “live“ on the unbounded set ℝ d . These facts raise new problems and leads to both negative and positive results.

Characterization of weak convergence of Birkhoff sums for Gibbs-Markov maps

Abstract: We investigate limit theorems for Birkhoff sums of locally Holder functions under the iteration of Gibbs-Markov maps Aaronson and Denker have given sufficient conditions to have limit theorems in this setting We show that these conditions are also necessary: there is no exotic limit theorem for Gibbs-Markov maps Our proofs, valid under very weak regularity assumptions, involve weak perturbation theory and interpolation spaces For L 2 observables, we also obtain necessary and sufficient conditions to control the speed of convergence in the central limit theorem

The zeta functions of complexes from PGL(3): A representation-theoretic approach

Abstract: The zeta function attached to a finite complex XΓ arising from the Bruhat-Tits building for PGL3(F) was studied in [KL], where a closed form expression was obtained by a combinatorial argument. This identity can be rephrased using operators on vertices, edges, and directed chambers of XΓ. In this paper we re-establish the zeta identity from a different aspect by analyzing the eigenvalues of these operators using representation theory. As a byproduct, we obtain equivalent criteria for a Ramanujan complex in terms of the eigenvalues of the operators on vertices, edges, and directed chambers, respectively.

On the nonself-adjoint ordinary differential operators with periodic boundary conditions

Abstract: In this article we obtain asymptotic formulas for eigenvalues and eigenfunctions of the nonself-adjoint ordinary differential operator with periodic and antiperiodic boundary conditions, when coefficients are arbitrary summable complex-valued functions. Then using these asymptotic formulas, we obtain necessary and sufficient conditions on the coefficient for which the root functions of these operators form a Riesz basis.

Turán-type results for partial orders and intersection graphs of convex sets

Abstract: We prove Ramsey-type results for intersection graphs of geometric objects. In particular, we prove the following bounds, all of which are tight apart from the constant c. There is a constant c > 0 such that for every family F of n convex sets in the plane, the intersection graph of F or its complement contains a balanced complete bipartite graph of size at least cn. There is a constant c > 0 such that for every family F of n x-monotone curves in the plane, the intersection graph G of F contains a balanced complete bipartite graph of size at least cn/log n or the complement of G contains a balanced complete bipartite graph of size at least cn. Our bounds rely on new Turan-type results on incomparability graphs of partially ordered sets.

Enumeration formulas for young tableaux in a diagonal strip

Abstract: We derive combinatorial identities, involving the Bernoulli and Euler numbers, for the numbers of standard Young tableaux of certain skew shapes. This generalizes the classical formulas of D. Andre on the number of up-down permutations. The analysis uses a transfer operator approach extending the method of Elkies, combined with an identity expressing the volume of a certain polytope in terms of a Schur function.

Counting derangements, involutions and unimodal elements in the wreath product Cr ≀ Sn

Abstract: We count derangements, involutions and unimodal elements in the wreath product Cr ≀ Sn by the numbers of excedances, fixed points and 2-cycles. Properties of the generating functions, including combinatorial formulas, recurrence relations and real-rootedness are studied. The results obtained specialize to those on the symmetric group Sn and on the hyperoctahedral group Bn when r = 1, 2, respectively.

The de-Rham theorem and Shapiro lemma for Schwartz functions on Nash manifolds

Abstract: In this paper we continue our work on Schwartz functions and generalized Schwartz functions on Nash (i.e. smooth semi-algebraic) manifolds. Our first goal is to prove analogs of the de-Rham theorem for de-Rham complexes with coefficients in Schwartz functions and generalized Schwartz functions. Using that we compute the cohomologies of the Lie algebra g of an algebraic group G with coefficients in the space of generalized Schwartz sections of G-equivariant bundle over a G-transitive variety M. We do it under some assumptions on topological properties of G and M. This computation for the classical case is known as the Shapiro lemma.

The Colin de Verdière number and graphs of polytopes

Abstract: The Colin de Verdiere number µ(G) of a graph G is the maximum corank of a Colin de Verdiere matrix for G (that is, of a Schrodinger operator on G with a single negative eigenvalue). In 2001, Lovasz gave a construction that associated to every convex 3-polytope a Colin de Verdiere matrix of corank 3 for its 1-skeleton.

Integration in Hilbert generated Banach spaces

Abstract: We prove that McShane and Pettis integrability are equivalent for functions taking values in a subspace of a Hilbert generated Banach space. This generalizes simultaneously all previous results on such equivalence. On the other hand, for any super-reflexive generated Banach space having density character greater than or equal to the continuum, we show that Birkhoff integrability lies strictly between Bochner and McShane integrability. Finally, we give a ZFC example of a scalarly null Banach space-valued function (defined on a Radon probability space) which is not McShane integrable.

Complex algebraic plane curves via Poincare-Hopf formula. II Annuli

Abstract: We give a complete classification of algebraic curves in ℂ2 which are homeomorphic with ℂ* and which satisfy a certain natural condition about codimensions of its singularities. In the proof we use the method developed in [BZI]. It relies on estimation of certain invariants of the curve, the so-called numbers of double points hidden at singularities and at infinity. The sum of these invariants is given by the Poincare-Hopf formula applied to a suitable vector field.

Alternating groups and moduli space lifting Invariants

Abstract: The genus of a curve discretely separates decidely different alge- braic relations in two variables to focus us on the connected moduli space Mg. Yet, modern applications also require a data variable (function) on the curve. The resulting spaces are versions, depending on our needs for this data variable, of Hurwitz spaces. A Nielsen class (§1.1) consists of r � 3 conjugacy classes C in the data variable monodromy G. It generalizes the genus. Some Nielsen classes define connected spaces. To detect, however, the components of others requires further subtler invariants. We regard our Main Result (MR) as level 0 of Spin invariant information on moduli spaces. In the MR, G = An (the alternating group), r counts the data variable branch points and C = C3r is r repetitions of the 3-cycle conjugacy class. This Nielsen class defines two spaces called absolute and inner: H(An, C3r)abs of degree n, genus g = r (n 1) > 0 covers and H(An, C3r)in parametrizing Galois closures of such covers. The parity of a spin invariant precisely identifies the two components of each space. The inner result is the deeper. We examine the effect of combining the MR, (ArP05) and 1 -canonical classes on Mg. First: §5.2 considers an analog of a famous conjecture of Sha- farevich: With H the composite group of all Galois extensions K/Q with group some alternating group, does the canonical map GQ ! H have pro-free kernel. Second: Thm. 6.15 produces nonzero automorphic (�-null power) functions on the reduced Hurwitz spaces H+(An, C3r) abs,rd (resp. H (An, C3r) abs,rd ) when r is even (resp. odd), for either g = 1 or n � 12g + 4 .

Generic measures for hyperbolic flows on non-compact spaces

Abstract: We consider the geodesic flow on a complete connected negatively curved manifold. We show that the set of invariant borel probability measures contains a dense G δ -subset consisting of ergodic measures fully supported on the non-wandering set. We also treat the case of non-positively curved manifolds and provide general tools to deal with hyperbolic systems defined on non-compact spaces.