Ramsey families of subtrees of the dyadic tree

We show that for every rooted, finitely branching, pruned tree of height there exists a family which consists of order isomorphic to subtrees of the dyadic tree with the following properties: (i) the family is a subset of ; (ii) every perfect subtree of contains a member of ; (iii) if is an analytic subset of , then for every perfect subtree of there exists a perfect subtree of such that the set either is contained in or is disjoint from .

     

Cut numbers of -manifolds

We investigate the relations between the cut number, and the first Betti number, of -manifolds We prove that the cut number of a ``generic' -manifold is at most This is a rather unexpected result since specific examples of -manifolds with large and are hard to construct. We also prove that for any complex semisimple Lie algebra there exists a -manifold with and Such manifolds can be explicitly constructed.

     

Asymptotic behaviour of arithmetically Cohen-Macaulay blow-ups

This paper addresses problems on arithmetic Macaulayfications of projective schemes. We give a surprising complete answer to a question poised by Cutkosky and Herzog. Let be the blow-up of a projective scheme along the ideal sheaf of . It is known that there are embeddings for , where denotes the maximal generating degree of , and that there exists a Cohen-Macaulay ring of the form (which gives an arithmetic Macaulayfication of ) if and only if , for , and is equidimensional and Cohen-Macaulay. We show that under these conditions, there are well-determined invariants and such that is Cohen-Macaulay for all d(I)e + \varepsilon$"> and e_0$">, and that these bounds are the best possible. We also investigate the existence of a Cohen-Macaulay Rees algebra of the form . If has negative -invariant, we prove that such a Cohen-Macaulay Rees algebra exists if and only if , for 0$">, and is equidimensional and Cohen-Macaulay. Moreover, these conditions imply the Cohen-Macaulayness of for all d(I)e + \varepsilon$"> and e_0$">.

     

Asymptotic properties of convolution operators and limits of triangular arrays on locally compact groups

We consider a locally compact group and a limiting measure of a commutative infinitesimal triangular system (c.i.t.s.) of probability measures on . We show, under some restrictions on , or , that belongs to a continuous one-parameter convolution semigroup. In particular, this result is valid for symmetric c.i.t.s. on any locally compact group . It is also valid for a limiting measure which has `full' support on a Zariski connected -algebraic group , where is a local field, and any one of the following conditions is satisfied: (1) is a compact extension of a closed solvable normal subgroup, in particular, is amenable, (2) has finite one-moment or (3) has density and in case the characteristic of is positive, the radical of is -defined. We also discuss the spectral radius of the convolution operator of a probability measure on a locally compact group , we show that it is always positive for any probability measure on , and it is also multiplicative in case of symmetric commuting measures.

     

Hausdorff measures, dimensions and mutual singularity

Let be a metric space. For a probability measure on a subset of and a Vitali cover of , we introduce the notion of a -Vitali subcover , and compare the Hausdorff measures of with respect to these two collections. As an application, we consider graph directed self-similar measures and in satisfying the open set condition. Using the notion of pointwise local dimension of with respect to , we show how the Hausdorff dimension of some general multifractal sets may be computed using an appropriate stochastic process. As another application, we show that Olsen's multifractal Hausdorff measures are mutually singular.

     

Threefolds with vanishing Hodge cohomology

We consider algebraic manifolds of dimension 3 over with for all and 0$">. Let be a smooth completion of with , an effective divisor on with normal crossings. If the -dimension of is not zero, then is a fibre space over a smooth affine curve (i.e., we have a surjective morphism from to such that the general fibre is smooth and irreducible) such that every fibre satisfies the same vanishing condition. If an irreducible smooth fibre is not affine, then the Kodaira dimension of is and the -dimension of is 1. We also discuss sufficient conditions from the behavior of fibres or higher direct images to guarantee the global vanishing of Hodge cohomology and the affineness of .

     

Prescribing analytic singularities for solutions of a class of vector fields on the torus

We consider the operator acting on distributions on the two-torus where and are real-valued, real analytic functions defined on the unit circle We prove, among other things, that when changes sign, given any subset of the set of the local extrema of the local primitives of there exists a singular solution of such that the projection of its analytic singular support is furthermore, for any and any closed subset of there exists such that and We also provide a microlocal result concerning the trace of at

     

Extension-orthogonal components of preprojective varieties

Let be a Dynkin quiver, and let be the corresponding preprojective algebra. Let be a set of pairwise different indecomposable irreducible components of varieties of -modules such that generically there are no extensions between and for all . We show that the number of elements in is at most the number of positive roots of . Furthermore, we give a module-theoretic interpretation of Leclerc's counterexample to a conjecture of Berenstein and Zelevinsky.

     

Plane Cremona maps, exceptional curves and roots

We address three different questions concerning exceptional and root divisors (of arithmetic genus zero and of self-intersection and , respectively) on a smooth complex projective surface which admits a birational morphism to . The first one is to find criteria for the properness of these divisors, that is, to characterize when the class of is in the -orbit of the class of the total transform of some point blown up by if is exceptional, or in the -orbit of a simple root if is root, where is the Weyl group acting on ; we give an arithmetical criterion, which adapts an analogous criterion suggested by Hudson for homaloidal divisors, and a geometrical one. Secondly, we prove that the irreducibility of the exceptional or root divisor is a necessary and sufficient condition in order that could be transformed into a line by some plane Cremona map, and in most cases for its contractibility. Finally, we provide irreducibility criteria for proper homaloidal, exceptional and effective root divisors.

     

The cyclic and simplicial cohomology of

Let be the unital semigroup algebra of . We show that the cyclic cohomology groups vanish when is odd and are one dimensional when is even (). Using Connes' exact sequence, these results are used to show that the simplicial cohomology groups vanish for . The results obtained are extended to unital algebras for some other semigroups of .

     

On the mod cohomology of

We study the mod  cohomology of the classifying space of the projective unitary group . We first prove that conjectures due to J.F. Adams and Kono and Yagita (1993) about the structure of the mod  cohomology of the classifying space of connected compact Lie groups hold in the case of . Finally, we prove that the classifying space of the projective unitary group is determined by its mod  cohomology as an unstable algebra over the Steenrod algebra for 3$">, completing previous work by Dwyer, Miller and Wilkerson (1992) and Broto and Viruel (1998) for the cases .

     

Minimal invariant tori in the resonant regions for nearly integrable Hamiltonian systems

Consider a real analytical Hamiltonian system of KAM type that has degrees of freedom (2$">) and is positive definite in . Let . In this paper we show that for most rotation vectors in , in the sense of ()-dimensional Lebesgue measure, there is at least one ()-dimensional invariant torus. These tori are the support of corresponding minimal measures. The Lebesgue measure estimate on this set is uniformly valid for any perturbation.

     

Cofinality of the nonstationary ideal

We show that the reduced cofinality of the nonstationary ideal on a regular uncountable cardinal may be less than its cofinality, where the reduced cofinality of is the least cardinality of any family of nonstationary subsets of such that every nonstationary subset of can be covered by less than many members of . For this we investigate connections of the various cofinalities of with other cardinal characteristics of and we also give a property of forcing notions (called manageability) which is preserved in -support iterations and which implies that the forcing notion preserves non-meagerness of subsets of (and does not collapse cardinals nor changes cofinalities).

     

Compactness of isospectral potentials

The Schrödinger operator , of a compact Riemannian manifold , has pure point spectrum. Suppose that is a smooth reference potential. Various criteria are given which guarantee the compactness of all satisfying . In particular, compactness is proved assuming an a priori bound on the norm of , where n/2-2$"> and . This improves earlier work of Brüning. An example involving singular potentials suggests that the condition n/2-2$"> is appropriate. Compactness is also proved for non-negative isospectral potentials in dimensions .

     

Double forms, curvature structures and the -curvatures

We introduce a natural extension of the metric tensor and the Hodge star operator to the algebra of double forms to study some aspects of the structure of this algebra. These properties are then used to study new Riemannian curvature invariants, called the -curvatures. They are a generalization of the -curvature obtained by substituting the Gauss-Kronecker tensor to the Riemann curvature tensor. In particular, for , the -curvatures coincide with the H. Weyl curvature invariants, for the -curvatures are the curvatures of generalized Einstein tensors, and for the -curvatures coincide with the -curvatures.

Also, we prove that the second H. Weyl curvature invariant is nonnegative for an Einstein manifold of dimension , and it is nonpositive for a conformally flat manifold with zero scalar curvature. A similar result is proved for the higher H. Weyl curvature invariants.

     

On a refinement of the generalized Catalan numbers for Weyl groups

Let be an irreducible crystallographic root system with Weyl group , coroot lattice and Coxeter number , spanning a Euclidean space , and let be a positive integer. It is known that the set of regions into which the fundamental chamber of is dissected by the hyperplanes in of the form for and is equinumerous to the set of orbits of the action of on the quotient . A bijection between these two sets, as well as a bijection to the set of certain chains of order ideals in the root poset of , are described and are shown to preserve certain natural statistics on these sets. The number of elements of these sets and their corresponding refinements generalize the classical Catalan and Narayana numbers, which occur in the special case and .

     

Blow-up examples for second order elliptic PDEs of critical Sobolev growth

Let be a smooth compact Riemannian manifold of dimension , and be the Laplace-Beltrami operator. Let also be the critical Sobolev exponent for the embedding of the Sobolev space into Lebesgue's spaces, and be a smooth function on . Elliptic equations of critical Sobolev growth such as

have been the target of investigation for decades. A very nice -theory for the asymptotic behaviour of solutions of such equations has been available since the 1980's. The -theory was recently developed by Druet-Hebey-Robert. Such a theory provides sharp pointwise estimates for the asymptotic behaviour of solutions of . It was used as a key point by Druet to prove compactness results for equations such as . An important issue in the field of blow-up analysis, in particular with respect to previous work by Druet and Druet-Hebey-Robert, is to get explicit nontrivial examples of blowing-up sequences of solutions of . We present such examples in this article.

     

Inequalities for finite group permutation modules

If is a nonzero complex-valued function defined on a finite abelian group and is its Fourier transform, then , where and are the supports of and . In this paper we generalize this known result in several directions. In particular, we prove an analogous inequality where the abelian group is replaced by a transitive right -set, where is an arbitrary finite group. We obtain stronger inequalities when the -set is primitive, and we determine the primitive groups for which equality holds. We also explore connections between inequalities of this type and a result of Chebotarëv on complex roots of unity, and we thereby obtain a new proof of Chebotarëv's theorem.

     

Anosov automorphisms on compact nilmanifolds associated with graphs

We associate with each graph a -step simply connected nilpotent Lie group and a lattice in . We determine the group of Lie automorphisms of and apply the result to describe a necessary and sufficient condition, in terms of the graph, for the compact nilmanifold to admit an Anosov automorphism. Using the criterion we obtain new examples of compact nilmanifolds admitting Anosov automorphisms, and conclude that for every there exist a -dimensional -step simply connected nilpotent Lie group which is indecomposable (not a direct product of lower dimensional nilpotent Lie groups), and a lattice in such that admits an Anosov automorphism; we give also a lower bound on the number of mutually nonisomorphic Lie groups of a given dimension, satisfying the condition. Necessary and sufficient conditions are also described for a compact nilmanifold as above to admit ergodic automorphisms.

     

An approximate universal coefficient theorem

An approximate Universal Coefficient Theorem (AUCT) for certain -algebras is established. We present a proof that Kirchberg-Phillips's classification theorem for separable nuclear purely infinite simple -algebras is valid for -algebras satisfying the AUCT instead of the UCT. It is proved that two versions of AUCT are in fact the same. We also show that -algebras that are locally approximated by -algebras satisfying the AUCT satisfy the AUCT. As an application, we prove that certain simple -algebras which are locally type I are in fact isomorphic to simple AH-algebras. As another application, we show that a sequence of residually finite-dimensional -algebras which are asymptotically nuclear and which asymptotically satisfies the AUCT can be embedded into the same simple AF-algebra.