-manifolds with planar presentations and the width of satellite knots

We consider compact -manifolds having a submersion to in which each generic point inverse is a planar surface. The standard height function on a submanifold of is a motivating example. To we associate a connectivity graph . For , is a tree if and only if there is a Fox reimbedding of which carries horizontal circles to a complete collection of complementary meridian circles. On the other hand, if the connectivity graph of is a tree, then there is a level-preserving reimbedding of so that is a connected sum of handlebodies.

Corollary.

The width of a satellite knot is no less than the width of its pattern knot and so

.

     

A density theorem on automorphic -functions and some applications

We establish a density theorem on automorphic -functions and give some applications on the extreme values of these -functions at and the distribution of the Hecke eigenvalue of holomorphic cusp forms.

     

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.

     

On the Andrews-Stanley refinement of Ramanujan's partition congruence modulo and generalizations

In a recent study of sign-balanced, labelled posets, Stanley introduced a new integral partition statistic

where denotes the number of odd parts of the partition and is the conjugate of . In a forthcoming paper, Andrews proved the following refinement of Ramanujan's partition congruence mod :

where () denotes the number of partitions of with and is the number of unrestricted partitions of . Andrews asked for a partition statistic that would divide the partitions enumerated by () into five equinumerous classes.

In this paper we discuss three such statistics: the ST-crank, the -quotient-rank and the -core-crank. The first one, while new, is intimately related to the Andrews-Garvan (1988) crank. The second one is in terms of the -quotient of a partition. The third one was introduced by Garvan, Kim and Stanton in 1990. We use it in our combinatorial proof of the Andrews refinement. Remarkably, the Andrews result is a simple consequence of a stronger refinement of Ramanujan's congruence mod . This more general refinement uses a new partition statistic which we term the BG-rank. We employ the BG-rank to prove new partition congruences modulo . Finally, we discuss some new formulas for partitions that are -cores and discuss an intriguing relation between -cores and the Andrews-Garvan crank.

     

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 .

     

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).

     

Automorphisms of fiber surfaces of genus , inducing the identity in cohomology

Let be a complex non-singular projective surface of general type with a genus fibration and . Let be a non-trivial subgroup of automorphisms of , inducing trivial actions on for all . Then , and . Examples of such surfaces are given.

     

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 .

     

On the -theory and topological cyclic homology of smooth schemes over a discrete valuation ring

We show that for a smooth and proper scheme over a henselian discrete valuation ring of mixed characteristic , the -adic étale -theory and -adic topological cyclic homology agree.

     

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.

     

Non-Moishezon twistor spaces of with non-trivial automorphism group

We show that a twistor space of a self-dual metric on with -isometry is not Moishezon iff there is a -orbit biholomorphic to a smooth elliptic curve, where the -action is the complexification of the -action on the twistor space. It follows that the -isometry has a two-sphere whose isotropy group is . We also prove the existence of such twistor spaces in a strong form to show that a problem of Campana and Kreußler is affirmative even though a twistor space is required to have a non-trivial automorphism group.

     

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.

     

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 .

     

Invariance in and

We define , a substructure of (the lattice of classes), and show that a quotient structure of , , is isomorphic to . The result builds on the isomorphism machinery, and allows us to transfer invariant classes from to , though not, in general, orbits. Further properties of and ramifications of the isomorphism are explored, including degrees of equivalence classes and degree invariance.

     

Polar sets on metric spaces

We show that if is a proper metric measure space equipped with a doubling measure supporting a Poincaré inequality, then subsets of with zero -capacity are precisely the -polar sets; that is, a relatively compact subset of a domain in is of zero -capacity if and only if there exists a -superharmonic function whose set of singularities contains the given set. In addition, we prove that if is a -hyperbolic metric space, then the -superharmonic function can be required to be -superharmonic on the entire space . We also study the the following question: If a set is of zero -capacity, does there exist a -superharmonic function whose set of singularities is precisely the given set?

     

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.

     

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.

     

Invariants, Boolean algebras and ACA

The sentences asserting the existence of invariants for mathematical structures are usually third order ones. We develop a general approach to analyzing the strength of such statements in second order arithmetic in the spirit of reverse mathematics. We discuss a number of simple examples that are equivalent to ACA. Our major results are that the existence of elementary equivalence invariants for Boolean algebras and isomorphism invariants for dense Boolean algebras are both of the same strength as ACA. This system corresponds to the assertion that (the arithmetic jump of ) exists for every set . These are essentially the first theorems known to be of this proof theoretic strength. The proof begins with an analogous result about these invariants on recursive (dense) Boolean algebras coding .

     

Unramified cohomology of classifying varieties for exceptional simply connected groups

Let be a classifying variety for an exceptional simple simply connected algebraic group . We compute the degree 3 unramified Galois cohomology of with values in over an arbitrary field . Combined with a paper by Merkurjev, this completes the computation of these cohomology groups for semisimple simply connected over all fields.

These computations provide another family of examples of simple simply connected groups such that is not stably rational.