On Artin's conjecture

Let $\text{F}$ be a family of number fields which are normal and of finite degree over a given number field K. Consider the lattice L(scF) spanned by all the elements of $\text{F}$. The generalized Artin problem is to determine the set of prime ideals of K which do not split completely in any element H of L(scF), H≠K. Assuming the generalized Riemann hypothesis and some mild restrictions on $\text{F}$, we solve this problem by giving an asymptotic formula for the number of such prime ideals below a given norm. The classical Artin conjecture on primitive roots appears as a special case. In another case, if $\text{F}$ is the family of fields obtained by adjoining to $\text{Q}$ the q-division points of an elliptic curve E over $\text{Q}$, the Artin problem determines how often E($\text{F}$_p) is cyclic. If E has complex multiplication, the generalized Riemann hypothesis can be removed by using the analogue of the Bombieri-Vinogradov prime number theorem for number fields.     

Finite group extensions and the Atiyah conjecture

The Atiyah conjecture for a discrete group states that the -Betti numbers of a finite CW-complex with fundamental group are integers if is torsion-free, and in general that they are rational numbers with denominators determined by the finite subgroups of .

Here we establish conditions under which the Atiyah conjecture for a torsion-free group implies the Atiyah conjecture for every finite extension of . The most important requirement is that is isomorphic to the cohomology of the -adic completion of for every prime number . An additional assumption is necessary e.g. that the quotients of the lower central series or of the derived series are torsion-free.

We prove that these conditions are fulfilled for a certain class of groups, which contains in particular Artin's pure braid groups (and more generally fundamental groups of fiber-type arrangements), free groups, fundamental groups of orientable compact surfaces, certain knot and link groups, certain primitive one-relator groups, and products of these. Therefore every finite, in fact every elementary amenable extension of these groups satisfies the Atiyah conjecture, provided the group does.

As a consequence, if such an extension is torsion-free, then the group ring contains no non-trivial zero divisors, i.e. fulfills the zero-divisor conjecture.

In the course of the proof we prove that if these extensions are torsion-free, then they have plenty of non-trivial torsion-free quotients which are virtually nilpotent. All of this applies in particular to Artin's full braid group, therefore answering question B6 on http://www.grouptheory.info.

Our methods also apply to the Baum-Connes conjecture. This is discussed by Thomas Schick in his preprint ``Finite group extensions and the Baum-Connes conjecture', where for example the Baum-Connes conjecture is proved for the full braid groups.

     

Fixing subgraphs and Ulam's conjecture

A spanning subgraph U of a graph G belongs to the set $\text{J}$(G) of fixing subgraphs (see [5]) of G if every embedding of U into G can be extended to an automorphism of G. Clearly G ∈ $\text{J}$(G). G is free if …$\text{J}$(G)… = 1. We establish a connection between Ulam's conjecture and free graphs and continue with an investigation of free graphs.     

On the torsion of group extensions and group actions

In this note, we study the torsion of extensions of finitely generated abelian by elementary abelian groups. When the action is trivial , we make a specific choice of a 1-cochain for a vanishing multiple of the cohomology class defining the extension and use it to completely describe the torsion of central extensions. As an application, one gets that, under the assumption of trivial action on homology, Z_p^r may act freely on (S¹)^k if and only if r?k, providing an alternative proof of the main theorem in [Trans. Amer. Math. Soc. 352 (6) (2000) 2689-2700] for central extensions.     

A proof of Wright's conjecture

Wright's conjecture states that the origin is the global attractor for the delay differential equation $y^{\prime }(t)=?\alpha y(t?1)[1+y(t)]$ for all $\alpha \in (0,\frac{\pi }{2}]$ when $y(t)>?1$. This has been proven to be true for a subset of parameter values α. We extend the result to the full parameter range $\alpha \in (0,\frac{\pi }{2}]$, and thus prove Wright's conjecture to be true. Our approach relies on a careful investigation of the neighborhood of the Hopf bifurcation occurring at $\alpha =\frac{\pi }{2}$. This analysis fills the gap left by complementary work on Wright's conjecture, which covers parameter values further away from the bifurcation point. Furthermore, we show that the branch of (slowly oscillating) periodic orbits originating from this Hopf bifurcation does not have any subsequent bifurcations (and in particular no folds) for $\alpha \in (\frac{\pi }{2},\frac{\pi }{2}+6.830\times {10}^{?3}]$. When combined with other results, this proves that the branch of slowly oscillating solutions that originates from the Hopf bifurcation at $\alpha =\frac{\pi }{2}$ is globally parametrized by $\alpha >\frac{\pi }{2}$.     

A case of hadwiger's conjecture

Sufficient conditions are established for Hadwiger's Conjecture to be valid in thecase r = 5.     

Carlsson's rank conjecture and a conjecture on square-zero upper triangular matrices

Let k be an algebraically closed field and A the polynomial algebra in r variables with coefficients in k. In case the characteristic of k is 2, Carlsson [9] conjectured that for any DG-A-module M of dimension N as a free A-module, if the homology of M is nontrivial and finite dimensional as a k-vector space, then $2^r\le N$. Here we state a stronger conjecture about varieties of square-zero upper triangular $N\times N$ matrices with entries in A. Using stratifications of these varieties via Borel orbits, we show that the stronger conjecture holds when $N<8$ or $r<3$ without any restriction on the characteristic of k. As a consequence, we obtain a new proof for many of the known cases of Carlsson's conjecture and give new results when $N>4$ and $r=2$.     

A combinatorial proof of Dyson's conjecture

Dyson's conjecture, already proved by Gunson, Wilson and Good, is given a direct combinatorial proof.     

Stillman's question for exterior algebras and Herzog's conjecture on Betti numbers of syzygy modules

Let K be a field of characteristic 0 and consider exterior algebras of finite dimensional K-vector spaces. In this short paper we exhibit principal quadric ideals in a family whose Castelnuovo–Mumford regularity is unbounded. This negatively answers the analogue of Stillman's Question for exterior algebras posed by I. Peeva. We show that, via the Bernstein–Gel'fand–Gel'fand correspondence, these examples also yields counterexamples to a conjecture of J. Herzog on the Betti numbers in the linear strand of syzygy modules over polynomial rings.     

On Doob's conjecture about fine limit and Julia point

In the 1960, professor Doob proved that if f(z) is a meromorphic function on a disk D, then at almost every point of the perimeter of D either (a) f has both an angular limit and an equal fine limit, (b) f has every point of the plane as cluster value in every angle opening into D with vertex at the point, and f has a fine limit at the point, or (c) f has every point of the plane both as cluster value in every angle opening into D with vertex at the point and as fine cluster value. He then mentioned that it would be interesting to find an example to show that case (b) can really occur on a perimeter set of strictly positive measure. In this article, we prove this conjecture to be true.     

The Alexander-Orbach conjecture holds in high dimensions

We examine the incipient infinite cluster (IIC) of critical percolation in regimes where mean-field behavior has been established, namely when the dimension d is large enough or when d>6 and the lattice is sufficiently spread out. We find that random walk on the IIC exhibits anomalous diffusion with the spectral dimension d_s=\frac43d_{s}=\frac{4}{3} , that is, p _t (x,x)=t ^−2/3+o(1). This establishes a conjecture of Alexander and Orbach (J. Phys. Lett. (Paris) 43:625–631, 1982). En route we calculate the one-arm exponent with respect to the intrinsic distance.     

Some results on Hu's conjecture concerning binary trees

Given n weights, w₁, w₂,…, w_n, such that 0?w₁?w₂???w₁, we examine a property of permutation π¹, where π¹=(w₁, w_n, w₂, w_n?1,…), concerning alphabetical binary trees.For each permutation π of these n weights, there is an optimal alphabetical binary tree corresponding to π, we denote it's cost by V(π). There is also an optimal almost uniform alphabetical binary tree, corresponding to π, we denote it's cost by V_u(π).This paper asserts that V_u(π¹)?V_u(π)?V(π) for all π. This is a preliminary result concerning the conjecture of T.C. Hu. Hu's conjecture is V(π¹)?V(π) for all π.     

A counterexample to kippenhahn's conjecture on hermitian pencils

A counterexample is constructed to a conjecture of Kippenhahn (Math. Nachr. 6:193–288 (1951–52)). A pair of Hermitian 8×8 matrices H, K is found such that (1) H, K generate $\text{M}{}_8\text{(}\text{C}\text{)}$ and (2) the minimal polynomial of the pencil xH + yK has degree 4. Recent work of H. Shapiro [e.g. Linear Algebra Appl. 43:201–221 (1982)] has established the conjecture for n×n matrices n?5.     

Some basic observations on Kelly's conjecture for graphs

P.J. Kelly first mentioned the possibility of determining a graph from subgraphs obtained by deleting several points. While such problems have received a great deal of attention in the case of deletions of single points, the problem for several points is virtually untouched. This paper contains some basic results on that problem, including the negative observation that for every k, there exist two non-isomorphic graphs with the same collection of k-point subgraphs.