ar X iv : 0 71 0 . 03 28 v 1 [ m at h . M G ] 1 O ct 2 00 7 Hyperplane Arrangements with Large Average Diameter

ar X iv : 0 71 0 . 27 28 v 1 [ m at h . N T ] 1 5 O ct 2 00 7 PRIMES IN TUPLES II

We prove that lim inf n→∞ pn+1 − pn √ log pn(log log pn) < ∞, where pn denotes the n prime. Since on average pn+1 −pn is asymptotically log pn, this shows that we can always find pairs of primes much closer together than the average. We actually prove a more general result concerning the set of values taken on by the differences p− p between primes which includes the small gap result above.

ar X iv : 0 71 1 . 15 17 v 1 [ m at h . C O ] 9 N ov 2 00 7 COMBINATORIAL POLAR ORDERINGS AND FOLLOW - UP ARRANGEMENTS

Polar orderings arose in recent work of Salvetti and the second author on minimal CW-complexes for complexified hyperplane arrangements. We study the combinatorics of these orderings in the classical framework of oriented matroids, and reach thereby a weakening of the conditions required to actually determine such orderings. A class of arrangements for which the construction of the minimal comp...

ar X iv : 0 70 7 . 33 03 v 1 [ m at h . O A ] 2 3 Ju l 2 00 7 OPERATOR - VALUED FRAMES

ar X iv : 0 71 0 . 13 81 v 1 [ m at h . FA ] 6 O ct 2 00 7 On the symplectic phase space of KdV

We prove that the Birkhoff map Ω for KdV constructed on H 0 (T) can be interpolated between H 0 (T) and L20(T). In particular, the symplectic phase space H 1/2 0 (T) can be described in terms of Birkhoff coordinates.

ar X iv : 0 71 0 . 32 32 v 1 [ m at h . R T ] 1 7 O ct 2 00 7 REFLECTION GROUPS AND DIFFERENTIAL FORMS

We study differential forms invariant under a finite reflection group over a field of arbitrary characteristic. In particular, we prove an analogue of Saito’s freeness criterion for invariant differential 1-forms. We also discuss how twisted wedging endows the invariant forms with the structure of a free exterior algebra in certain cases. Some of the results are extended to the case of relative...