Effective cones of quotients of moduli spaces of stable -pointed curves of genus zero

Let , the moduli space of -pointed stable genus zero curves, and let be the quotient of by the action of on the last marked points. The cones of effective divisors , , are calculated. Using this, upper bounds for the cones generated by divisors with moving linear systems are calculated, , along with the induced bounds on the cones of ample divisors of and . As an application, the cone is analyzed in detail.

     

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.

     

On the -compact groups corresponding to the -adic reflection groups

There exists an infinite family of -compact groups whose Weyl groups correspond to the finite -adic pseudoreflection groups of family 2a in the Clark-Ewing list. In this paper we study these -compact groups. In particular, we construct an analog of the classical Whitney sum map, a family of monomorphisms and a spherical fibration which produces an analog of the classical -homomorphism. Finally, we also describe a faithful complexification homomorphism from these -compact groups to the -completion of unitary compact Lie groups.

     

Functional distribution of with real characters and denseness of quadratic class numbers

We investigate the functional distribution of -functions with real primitive characters on the region as varies over fundamental discriminants. Actually we establish the so-called universality theorem for in the -aspect. From this theorem we can, of course, deduce some results concerning the value distribution and the non-vanishing. As another corollary, it follows that for any fixed with and positive integers , there exist infinitely many such that for every the -th derivative has at least zeros on the interval in the real axis. We also study the value distribution of for fixed with and variable , and obtain the denseness result concerning class numbers of quadratic fields.

     

Partitioning -large sets: Some lower bounds

Let denote the property: if is an -large set of natural numbers and is partitioned into parts, then there exists a -large subset of which is homogeneous for this partition. Here the notion of largeness is in the sense of the so-called Hardy hierarchy. We give a lower bound for in terms of for some specific .

     

The invariant factors of the incidence matrices of points and subspaces in and

We determine the Smith normal forms of the incidence matrices of points and projective -dimensional subspaces of and of the incidence matrices of points and -dimensional affine subspaces of for all , , and arbitrary prime power .

     

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.

     

Exponents for -stable ideals

Let be a simple algebraic group over the complex numbers containing a Borel subgroup . Given a -stable ideal in the nilradical of the Lie algebra of , we define natural numbers which we call ideal exponents. We then propose two conjectures where these exponents arise, proving these conjectures in types and some other types.

When , we recover the usual exponents of by Kostant (1959), and one of our conjectures reduces to a well-known factorization of the Poincaré polynomial of the Weyl group. The other conjecture reduces to a well-known result of Arnold-Brieskorn on the factorization of the characteristic polynomial of the corresponding Coxeter hyperplane arrangement.

     

-estimates for the linearized Monge-Ampère equation

Let be a strictly convex domain and let be a convex function such that    det in . The linearized Monge-Ampère equation is

where det is the matrix of cofactors of . We prove that there exist and depending only on , and such that

for all solutions to the equation .

     

Quivers with relations arising from clusters case)

Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. Let be a cluster algebra of type . We associate to each cluster of an abelian category such that the indecomposable objects of are in natural correspondence with the cluster variables of which are not in . We give an algebraic realization and a geometric realization of . Then, we generalize the ``denominator theorem' of Fomin and Zelevinsky to any cluster.

     

A version of Bourgain's theorem

Let satisfy We construct an orthonormal basis for such that and are both uniformly bounded in . Here . This generalizes a theorem of Bourgain and is closely related to recent results on the Balian-Low theorem.

     

The fourth power moment of automorphic -functions for over a short interval

In this paper we will prove bounds for the fourth power moment in the aspect over a short interval of automorphic -functions for on the central critical line Re. Here is a fixed holomorphic or Maass Hecke eigenform for the modular group , or in certain cases, for the Hecke congruence subgroup with . The short interval is from a large to . The proof is based on an estimate in the proof of subconvexity bounds for Rankin-Selberg -function for Maass forms by Jianya Liu and Yangbo Ye (2002) and Yuk-Kam Lau, Jianya Liu, and Yangbo Ye (2004), which in turn relies on the Kuznetsov formula (1981) and bounds for shifted convolution sums of Fourier coefficients of a cusp form proved by Sarnak (2001) and by Lau, Liu, and Ye (2004).

     

Quantum cohomology and -actions with isolated fixed points

This paper studies symplectic manifolds that admit semi-free circle actions with isolated fixed points. We prove, using results on the Seidel element, that the (small) quantum cohomology of a -dimensional manifold of this type is isomorphic to the (small) quantum cohomology of a product of copies of . This generalizes a result due to Tolman and Weitsman.

     

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 .

     

Constructive recognition of

Existing black box and other algorithms for explicitly recognising groups of Lie type over have asymptotic running times which are polynomial in , whereas the input size involves only . This has represented a serious obstruction to the efficient recognition of such groups. Recently, Brooksbank and Kantor devised new explicit recognition algorithms for classical groups; these run in time that is polynomial in the size of the input, given an oracle that recognises explicitly.

The present paper, in conjunction with an earlier paper by the first two authors, provides such an oracle. The earlier paper produced an algorithm for explicitly recognising in its natural representation in polynomial time, given a discrete logarithm oracle for . The algorithm presented here takes as input a generating set for a subgroup of that is isomorphic modulo scalars to , where is a finite field of the same characteristic as ; it returns the natural representation of modulo scalars. Since a faithful projective representation of in cross characteristic, or a faithful permutation representation of this group, is necessarily of size that is polynomial in rather than in , elementary algorithms will recognise explicitly in polynomial time in these cases. Given a discrete logarithm oracle for , our algorithm thus provides the required polynomial time oracle for recognising explicitly in the remaining case, namely for representations in the natural characteristic.

This leads to a partial solution of a question posed by Babai and Shalev: if is a matrix group in characteristic , determine in polynomial time whether or not is trivial.

     

Constant mean curvature surfaces in

The subject of this paper is properly embedded surfaces in Riemannian three manifolds of the form , where is a complete Riemannian surface. When , we are in the classical domain of surfaces in . In general, we will make some assumptions about in order to prove stronger results, or to show the effects of curvature bounds in on the behavior of surfaces in .

     

Generalized interpolation in with a complexity constraint

In a seminal paper, Sarason generalized some classical interpolation problems for functions on the unit disc to problems concerning lifting onto of an operator that is defined on ( is an inner function) and commutes with the (compressed) shift . In particular, he showed that interpolants (i.e., such that ) having norm equal to exist, and that in certain cases such an is unique and can be expressed as a fraction with . In this paper, we study interpolants that are such fractions of functions and are bounded in norm by (assuming that , in which case they always exist). We parameterize the collection of all such pairs and show that each interpolant of this type can be determined as the unique minimum of a convex functional. Our motivation stems from the relevance of classical interpolation to circuit theory, systems theory, and signal processing, where is typically a finite Blaschke product, and where the quotient representation is a physically meaningful complexity constraint.

     

-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.

     

Differentiability of spectral functions for symmetric -stable processes

Let be a signed Radon measure in the Kato class and define a Schrödinger type operator on . We show that its spectral bound is differentiable if and is Green-tight.