$ad$-nilpotent $\mathfrak b$-ideals in $sl(n)$ having a fixed class of nilpotence: combinatorics and enumeration
George
E.
Andrews;
Christian
Krattenthaler;
Luigi
Orsina;
Paolo
Papi
3835-3853
Abstract: We study the combinatorics of $ad$-nilpotent ideals of a Borel subalgebra of $sl(n+1,\mathbb C)$. We provide an inductive method for calculating the class of nilpotence of these ideals and formulas for the number of ideals having a given class of nilpotence. We study the relationships between these results and the combinatorics of Dyck paths, based upon a remarkable bijection between $ad$-nilpotent ideals and Dyck paths. Finally, we propose a $(q,t)$-analogue of the Catalan number $C_n$. These $(q,t)$-Catalan numbers count, on the one hand, $ad$-nilpotent ideals with respect to dimension and class of nilpotence and, on the other hand, admit interpretations in terms of natural statistics on Dyck paths.
Inequalities for decomposable forms of degree $n+1$ in $n$ variables
Jeffrey
Lin
Thunder
3855-3868
Abstract: We consider the number of integral solutions to the inequality $\vert F(\mathbf{x}) \vert\le m$, where $F(\mathbf{X} )\in \mathbb{Z} [\mathbf{X} ]$ is a decomposable form of degree $n+1$ in $n$ variables. We show that the number of such solutions is finite for all $m$ only if the discriminant of $F$ is not zero. We get estimates for the number of such solutions that display appropriate behavior in terms of the discriminant. These estimates sharpen recent results of the author for the general case of arbitrary degree.
Hochschild homology criteria for trivial algebra structures
Micheline
Vigué-Poirrier
3869-3882
Abstract: We prove two similar results by quite different methods. The first one deals with augmented artinian algebras over a field: we characterize the trivial algebra structure on the augmentation ideal in terms of the maximality of the dimensions of the Hochschild homology (or cyclic homology) groups. For the second result, let $X$ be a 1-connected finite CW-complex. We characterize the trivial algebra structure on the cohomology algebra of $X$ with coefficients in a fixed field in terms of the maximality of the Betti numbers of the free loop space.
Differential operators on a polarized abelian variety
Indranil
Biswas
3883-3891
Abstract: Let $L$ be an ample line bundle over a complex abelian variety $A$. We show that the space of all global sections over $A$of ${Diff}^{n}_A(L,L)$ and $S^n({Diff}^1_A(L,L))$are both of dimension one. Using this it is shown that the moduli space $M_X$ of rank one holomorphic connections on a compact Riemann surface $X$ does not admit any nonconstant algebraic function. On the other hand, $M_X$ is biholomorphic to the moduli space of characters of $X$, which is an affine variety. So $M_X$ is algebraically distinct from the character variety if $X$ is of genus at least one.
Universal deformation rings and Klein four defect groups
Frauke
M.
Bleher
3893-3906
Abstract: In this paper, the universal deformation rings of certain modular representations of a finite group are determined. The representations under consideration are those which are associated to blocks with Klein four defect groups and whose stable endomorphisms are given by scalars. It turns out that these universal deformation rings are always subquotient rings of the group ring of a Klein four group over the ring of Witt vectors.
Character degrees and nilpotence class of finite $p$-groups: An approach via pro-$p$ groups
A.
Jaikin-Zapirain;
Alexander
Moretó
3907-3925
Abstract: Let $\mathcal{S}$ be a finite set of powers of $p$ containing 1. It is known that for some choices of $\mathcal{S}$, if $P$ is a finite $p$-group whose set of character degrees is $\mathcal{S}$, then the nilpotence class of $P$ is bounded by some integer that depends on $\mathcal{S}$, while for some other choices of $\mathcal{S}$ such an integer does not exist. The sets of the first type are called class bounding sets. The problem of determining the class bounding sets has been studied in several papers whose results made it tempting to conjecture that a set $\mathcal{S}$ is class bounding if and only if $p\notin\mathcal{S}$. In this article we provide a new approach to this problem. Our main result shows the relevance of certain $p$-adic space groups in this problem. With its help, we are able to prove some results that provide new class bounding sets. We also show that there exist non-class-bounding sets $\mathcal{S}$ such that $p\notin\mathcal{S}$.
On the crossing number of positive knots and braids and braid index criteria of Jones and Morton-Williams-Franks
A.
Stoimenow
3927-3954
Abstract: We give examples of knots with some unusual properties of the crossing number of positive diagrams or strand number of positive braid representations. In particular, we show that positive braid knots may not have positive minimal (strand number) braid representations, giving a counterpart to results of Franks-Williams and Murasugi. Other examples answer questions of Cromwell on homogeneous and (partially) of Adams on almost alternating knots. We give a counterexample to, and a corrected version of, a theorem of Jones on the Alexander polynomial of 4-braid knots. We also give an example of a knot on which all previously applied braid index criteria fail to estimate sharply (from below) the braid index. A relation between (generalizations of) such examples and a conjecture of Jones that a minimal braid representation has unique writhe is discussed. Finally, we give a counterexample to Morton's conjecture relating the genus and degree of the skein polynomial.
A theory of concordance for non-spherical 3-knots
Vincent
Blanloeil;
Osamu
Saeki
3955-3971
Abstract: Consider a closed connected oriented 3-manifold embedded in the $5$-sphere, which is called a $3$-knot in this paper. For two such knots, we say that their Seifert forms are spin concordant, if they are algebraically concordant with respect to a diffeomorphism between the 3-manifolds which preserves their spin structures. Then we show that two simple fibered 3-knots are geometrically concordant if and only if they have spin concordant Seifert forms, provided that they have torsion free first homology groups. Some related results are also obtained.
Embeddings up to homotopy of two-cones in euclidean space
Pascal
Lambrechts;
Don
Stanley;
Lucile
Vandembroucq
3973-4013
Abstract: We say that a finite CW-complex $X$ embeds up to homotopy in a sphere $S^{n+1}$ if there exists a subpolyhedron $K\subset S^{n+1}$ having the homotopy type of $X$. The main result of this paper is a sufficient condition for the existence of such a homotopy embedding in a given codimension when $X$ is a simply-connected two-cone (a two-cone is the homotopy cofibre of a map between two suspensions). We give different applications of this result: we prove that if $X$is a two-cone then there are no rational obstructions to embeddings up to homotopy in codimension 3. We give also a description of the homotopy type of the boundary of a regular neighborhood of the embedding of a two-cone in a sphere. This enables us to construct a closed manifold $M$ whose Lusternik-Schnirelmann category and cone-length are not affected by removing one point of $M$.
Critical Heegaard surfaces
David
Bachman
4015-4042
Abstract: In this paper we introduce critical surfaces, which are described via a 1-complex whose definition is reminiscent of the curve complex. Our main result is that if the minimal genus common stabilization of a pair of strongly irreducible Heegaard splittings of a 3-manifold is not critical, then the manifold contains an incompressible surface. Conversely, we also show that if a non-Haken 3-manifold admits at most one Heegaard splitting of each genus, then it does not contain a critical Heegaard surface. In the final section we discuss how this work leads to a natural metric on the space of strongly irreducible Heegaard splittings, as well as many new and interesting open questions.
Spectral asymptotics for Sturm-Liouville equations with indefinite weight
Paul
A.
Binding;
Patrick
J.
Browne;
Bruce
A.
Watson
4043-4065
Abstract: The Sturm-Liouville equation \begin{displaymath}-(py')' + qy =\lambda ry \text{\rm on} [0,l]\end{displaymath} is considered subject to the boundary conditions $O(1/\sqrt{n})$for $\sqrt{\lambda_n}$, or equivalently up to $O(\sqrt{n})$ for $\lambda_n$, the eigenvalues of the above boundary value problem.
The dynamics of expansive invertible onesided cellular automata
Masakazu
Nasu
4067-4084
Abstract: Using textile systems, we prove the conjecture of Boyle and Maass that the dynamical system defined by an expansive invertible onesided cellular automaton is topologically conjugate to a topological Markov shift. We also study expansive leftmost-permutive onesided cellular automata and bipermutive endomorphisms of mixing topological Markov shifts.
Trees and branches in Banach spaces
E.
Odell;
Th.
Schlumprecht
4085-4108
Abstract: An infinite dimensional notion of asymptotic structure is considered. This notion is developed in terms of trees and branches on Banach spaces. Every countably infinite countably branching tree $\mathcal{T}$of a certain type on a space $X$ is presumed to have a branch with some property. It is shown that then $X$ can be embedded into a space with an FDD $(E_i)$ so that all normalized sequences in $X$ which are almost a skipped blocking of $(E_i)$ have that property. As an application of our work we prove that if $X$ is a separable reflexive Banach space and for some $1<p<\infty$ and $C<\infty$ every weakly null tree $\mathcal{T}$ on the sphere of $X$ has a branch $C$-equivalent to the unit vector basis of $\ell_p$, then for all $\varepsilon>0$, there exists a subspace of $X$ having finite codimension which $C^2+\varepsilon$ embeds into the $\ell_p$ sum of finite dimensional spaces.
Functional Calculus in Hölder-Zygmund Spaces
G.
Bourdaud;
Massimo
Lanza de Cristoforis
4109-4129
Abstract: In this paper we characterize those functions $f$ of the real line to itself such that the nonlinear superposition operator $T_{f}$ defined by $T_{f}[ g]:= f\circ g$ maps the Hölder-Zygmund space ${\mathcal C}^{s}({\mathbf R}^{n})$ to itself, is continuous, and is $r$ times continuously differentiable. Our characterizations cover all cases in which $s$ is real and $s>0$, and seem to be novel when $s>0$ is an integer.
Weak amenability of module extensions of Banach algebras
Yong
Zhang
4131-4151
Abstract: We start by discussing general necessary and sufficient conditions for a module extension Banach algebra to be $n$-weakly amenable, for $n = 0,1,2,\cdots$. Then we investigate various special cases. All these case studies finally provide us with a way to construct an example of a weakly amenable Banach algebra which is not $3$-weakly amenable. This answers an open question raised by H. G. Dales, F. Ghahramani and N. Grønbæk.
Amenability and exactness for dynamical systems and their $C^{*}$-algebras
Claire
Anantharaman-Delaroche
4153-4178
Abstract: We study the relations between amenability (resp. amenability at infinity) of $C^{*}$-dynamical systems and equality or nuclearity (resp. exactness) of the corresponding crossed products.
Generalized pseudo-Riemannian geometry
Michael
Kunzinger;
Roland
Steinbauer
4179-4199
Abstract: Generalized tensor analysis in the sense of Colombeau's construction is employed to introduce a nonlinear distributional pseudo-Riemannian geometry. In particular, after deriving several characterizations of invertibility in the algebra of generalized functions, we define the notions of generalized pseudo-Riemannian metric, generalized connection and generalized curvature tensor. We prove a ``Fundamental Lemma of (pseudo-) Riemannian geometry'' in this setting and define the notion of geodesics of a generalized metric. Finally, we present applications of the resulting theory to general relativity.
Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains
Xing-Bin
Pan;
Keng-Huat
Kwek
4201-4227
Abstract: We establish an asymptotic estimate of the lowest eigenvalue $\mu (b\mathbf{F})$ of the Schrödinger operator $-\nabla _{b\mathbf{F}}^{2}$ with a magnetic field in a bounded $2$-dimensional domain, where curl $\mathbf{F}$ vanishes non-degenerately, and $b$is a large parameter. Our study is based on an analysis on an eigenvalue variation problem for the Sturm-Liouville problem. Using the estimate, we determine the value of the upper critical field for superconductors subject to non-homogeneous applied magnetic fields, and localize the nucleation of superconductivity.
Harmonic morphisms with one-dimensional fibres on Einstein manifolds
Radu
Pantilie;
John
C.
Wood
4229-4243
Abstract: We prove that, from an Einstein manifold of dimension greater than or equal to five, there are just two types of harmonic morphism with one-dimensional fibres. This generalizes a result of R.L. Bryant who obtained the same conclusion under the assumption that the domain has constant curvature.
Contact reduction
Christopher
Willett
4245-4260
Abstract: In this article I propose a new method for reducing co-oriented contact manifold $M$ equipped with an action of a Lie group $G$ by contact transformations. With a certain regularity and integrality assumption the contact quotient $M_\mu$ at $\mu \in \mathfrak{g}^*$ is a naturally co-oriented contact orbifold which is independent of the contact form used to represent the given contact structure. Removing the regularity and integrality assumptions and replacing them with one concerning the existence of a slice, which is satisfied for compact symmetry groups, results in a contact stratified space; i.e., a stratified space equipped with a line bundle which, when restricted to each stratum, defines a co-oriented contact structure. This extends the previous work of the author and E. Lerman.