Square -lattice paths and

The combinatorial -Catalan numbers are weighted sums of Dyck paths introduced by J. Haglund and studied extensively by Haglund, Haiman, Garsia, Loehr, and others. The -Catalan numbers, besides having many subtle combinatorial properties, are intimately connected to symmetric functions, algebraic geometry, and Macdonald polynomials. In particular, the 'th -Catalan number is the Hilbert series for the module of diagonal harmonic alternants in variables; it is also the coefficient of in the Schur expansion of . Using -analogues of labelled Dyck paths, Haglund et al. have proposed combinatorial conjectures for the monomial expansion of and the Hilbert series of the diagonal harmonics modules.

This article extends the combinatorial constructions of Haglund et al. to the case of lattice paths contained in squares. We define and study several -analogues of these lattice paths, proving combinatorial facts that closely parallel corresponding results for the -Catalan polynomials. We also conjecture an interpretation of our combinatorial polynomials in terms of the nabla operator. In particular, we conjecture combinatorial formulas for the monomial expansion of , the ``Hilbert series' , and the sign character .

     

-continuity properties of the symmetric -stable process

Let be a domain of finite Lebesgue measure in and let be the symmetric -stable process killed upon exiting . Each element of the set of eigenvalues associated to , regarded as a function of , is right continuous. In addition, if is Lipschitz and bounded, then each is continuous in and the set of associated eigenfunctions is precompact.

     

Surface symmetries and

We classify, up to conjugacy, all orientation-preserving actions of on closed connected orientable surfaces with spherical quotients. This classification is valid in the topological, PL, smooth, conformal, geometric and algebraic categories and is related to the Inverse Galois Problem.

     

On algebraic -groups

We introduce the categories of algebraic -varieties and -groups over a difference field . Under a ``linearly -closed" assumption on we prove an isotriviality theorem for -groups. This theorem immediately yields the key lemma in a proof of the Manin-Mumford conjecture. The present paper crucially uses ideas of Pilay and Ziegler (2003) but in a model theory free manner. The applications to Manin-Mumford are inspired by Hrushovski's work (2001) and are also closely related to papers of Pink and Roessler (2002 and 2004).

     

Surjectivity for Hamiltonian -spaces in -theory

Let be a compact connected Lie group, and a Hamiltonian -space with proper moment map . We give a surjectivity result which expresses the -theory of the symplectic quotient in terms of the equivariant -theory of the original manifold , under certain technical conditions on . This result is a natural -theoretic analogue of the Kirwan surjectivity theorem in symplectic geometry. The main technical tool is the -theoretic Atiyah-Bott lemma, which plays a fundamental role in the symplectic geometry of Hamiltonian -spaces. We discuss this lemma in detail and highlight the differences between the -theory and rational cohomology versions of this lemma.

We also introduce a -theoretic version of equivariant formality and prove that when the fundamental group of is torsion-free, every compact Hamiltonian -space is equivariantly formal. Under these conditions, the forgetful map is surjective, and thus every complex vector bundle admits a stable equivariant structure. Furthermore, by considering complex line bundles, we show that every integral cohomology class in admits an equivariant extension in .

     

Postnikov pieces and -homotopy theory

We present a constructive method to compute the cellularization with respect to for any integer of a large class of -spaces, namely all those which have a finite number of non-trivial -homotopy groups (the pointed mapping space is a Postnikov piece). We prove in particular that the -cellularization of an -space having a finite number of -homotopy groups is a -torsion Postnikov piece. Along the way, we characterize the -cellular classifying spaces of nilpotent groups.

     

Pseudofree -actions on surfaces

In this paper, we give a weak classification of locally linear pseudofree actions of the cyclic group of order on a surface, and prove the existence of such an action which cannot be realized as a smooth action on the standard smooth surface.

     

A minimal pair of -degrees

We construct a minimal pair of -degrees. We do this by showing the existence of an unbounded nondecreasing function which forces -triviality in the sense that is -trivial if and only if for all , .

     

Curves of genus 2 with group of automorphisms isomorphic to or

The classification of curves of genus 2 over an algebraically closed field was studied by Clebsch and Bolza using invariants of binary sextic forms, and completed by Igusa with the computation of the corresponding three-dimensional moduli variety . The locus of curves with group of automorphisms isomorphic to one of the dihedral groups or is a one-dimensional subvariety.

In this paper we classify these curves over an arbitrary perfect field of characteristic in the case and in the case. We first parameterize the -isomorphism classes of curves defined over by the -rational points of a quasi-affine one-dimensional subvariety of ; then, for every curve representing a point in that variety we compute all of its -twists, which is equivalent to the computation of the cohomology set .

The classification is always performed by explicitly describing the objects involved: the curves are given by hyperelliptic models and their groups of automorphisms represented as subgroups of . In particular, we give two generic hyperelliptic equations, depending on several parameters of , that by specialization produce all curves in every -isomorphism class.

     

estimates for overdetermined Radon transforms

We prove several variations on the results of F. Ricci and G. Travaglini (2001), concerning bounds for convolution with all rotations of arc length measure on a fixed convex curve in . Estimates are obtained for averages over higher-dimensional convex (nonsmooth) hypersurfaces, smooth -dimensional surfaces, and nontranslation-invariant families of surfaces. We compare Ricci and Travaglini's approach, based on average decay of the Fourier transform, with an approach based on boundedness of Fourier integral operators, and show that essentially the same geometric condition arises in proofs using the two techniques.

     

All -local finite groups of rank two for odd prime

In this paper we give a classification of the rank two -local finite groups for odd . This study requires the analysis of the possible saturated fusion systems in terms of the outer automorphism group of the possible -radical subgroups. Also, for each case in the classification, either we give a finite group with the corresponding fusion system or we check that it corresponds to an exotic -local finite group, getting some new examples of these for .

     

Uniformizable families of -motives

Abelian -modules and the dual notion of -motives were introduced by Anderson as a generalization of Drinfeld modules. For such Anderson defined and studied the important concept of uniformizability. It is an interesting question and the main objective of the present article to see how uniformizability behaves in families. Since uniformizability is an analytic notion, we have to work with families over a rigid analytic base. We provide many basic results, and in fact a large part of this article concentrates on laying foundations for studying the above question. Building on these, we obtain a generalization of a uniformizability criterion of Anderson and, among other things, we establish that the locus of uniformizability is Berkovich open.

     

A generalization of half-plane mappings to the ball in

Let be a normalized (, ) biholomorphic mapping of the unit ball onto a convex domain that is the union of lines parallel to some unit vector . We consider the situation in which there is one infinite singularity of on . In one case with a simple change-of-variables, we classify all convex mappings of that are half-plane mappings in the first coordinate. In the more complicated case, when is not in the span of the infinite singularity, we derive a form of the mappings in dimension .

     

A Pieri-type formula for the -theory of a flag manifold

We derive explicit Pieri-type multiplication formulas in the Grothendieck ring of a flag variety. These expand the product of an arbitrary Schubert class and a special Schubert class in the basis of Schubert classes. These special Schubert classes are indexed by a cycle which has either the form or the form , and are pulled back from a Grassmannian projection. Our formulas are in terms of certain labeled chains in the -Bruhat order on the symmetric group and are combinatorial in that they involve no cancellations. We also show that the multiplicities in the Pieri formula are naturally certain binomial coefficients.

     

The -stable pieces of the wonderful compactification

Let be a connected, simple algebraic group over an algebraically closed field. There is a partition of the wonderful compactification of into finite many -stable pieces, which was introduced by Lusztig. In this paper, we will investigate the closure of any -stable piece in . We will show that the closure is a disjoint union of some -stable pieces, which was first conjectured by Lusztig. We will also prove the existence of cellular decomposition if the closure contains finitely many -orbits.

     

Associahedra, cellular -construction and products of -algebras

The aim of this paper is to construct a functorial tensor product of -algebras or, equivalently, an explicit diagonal for the operad of cellular chains, over the integers, of the Stasheff associahedron. These constructions in fact already appeared (Saneblidze and Umble, 2000 and 2002); we will try to give a more conceptual presentation. We also prove that there does not exist a coassociative diagonal.

     

Strongly self-absorbing -algebras

Say that a separable, unital -algebra is strongly self-absorbing if there exists an isomorphism such that and are approximately unitarily equivalent -homomorphisms. We study this class of algebras, which includes the Cuntz algebras , , the UHF algebras of infinite type, the Jiang-Su algebra and tensor products of with UHF algebras of infinite type. Given a strongly self-absorbing -algebra we characterise when a separable -algebra absorbs tensorially (i.e., is -stable), and prove closure properties for the class of separable -stable -algebras. Finally, we compute the possible -groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing -algebras.

     

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 .

     

Torsion in coinvariants of certain Cantor minimal -systems

Let be a finite abelian group. We will consider a skew product extension of a product of two Cantor minimal -systems associated with a -valued cocycle. When is non-cyclic and the cocycle is non-degenerate, it will be shown that the skew product system has torsion in its coinvariants.