Seven papers have been published by Geometry & Topology in Volume 12
(2008). Papers (1) to (4) complete Issue 3 and papers (5) to (7)
open Issue 4.

(1) Geometry & Topology 12 (2008) 1711-1727
Small values of the Lusternik-Schnirelman category for manifolds
by Alexander N Dranishnikov, Mikhail G Katz and Yuli B Rudyak
URL: http://www.msp.warwick.ac.uk/gt/2008/12-03/p038.xhtml
DOI: 10.2140/gt.2008.12.1711

(2) Geometry & Topology 12 (2008) 1729-1776
Geodesible contact structures on 3-manifolds
by Patrick Massot
URL: http://www.msp.warwick.ac.uk/gt/2008/12-03/p039.xhtml
DOI: 10.2140/gt.2008.12.1729

(3) Geometry & Topology 12 (2008) 1777-1798
A combination theorem for strong relative hyperbolicity
by Mahan Mj and Lawrence Reeves
URL: http://www.msp.warwick.ac.uk/gt/2008/12-03/p040.xhtml
DOI: 10.2140/gt.2008.12.1777

The set of N-smooth integers is defined as the integers whose largest
prime factor is <=N. I conjecture (and I'm sure I'm not the first to
do so) that for any N and k, the N-smooth integers have only finitely
many "gaps" of size <=k.

(I suppose you could call a sequence with this property an "anti-
Cauchy sequence", since it is defined exactly like a Cauchy sequence
with the inequality reversed, so my conjecture can be stated "for all
N, the N-smooth integers are anti-Cauchy".)

However, that's not my question. My question is, has anyone ever come
up with a plausible heuristic bound for how high up, in terms of N and
k, you have to go for the N-smooth integers to always be more than k
apart from then on.

If that's too hard, I'd like a heuristic bound for how high you have
to go to get past all adjacent products of small primes. For example,

123200=64*25*7*11 = 2^6 * 5^2 * 7 * 11;
123201=729*169 = 3^6 * 13^2

Is this the largest example for N=13 and k=1?

-- Joe Shipman

I have a question on Berry phase for this very simple example:

Define D as the symmetric 4X4 diagonal 0-1 matrix

[1000]
[0000]
[0010]
[0000]

Define M as the symmetric 4X4 0-1 matrix

[0001]
[0011]
[0100]
[1100]

Define the parameterized Hamiltonian

H(x,y) = (x^2+y^2)*M + D (x,y real)

(I believe) The maximum eigenvalue of H(x,y) has only one
degeneracy at (x,y) = (0,0)

Let P(t) be the circular path (sin(t),cos(t)) [0<=t<=2*pi].

The maximum eigenvalue of H on P(t) is always 2 and has an
eigenvector = transpose of |1,1,1,1>/2 independent of t.

While looking at a paper by Zivkovic on 0-1 matrices
<http://arxiv.org/PS_cache/math/pdf/0511/0511636v1.pdf>
which uses SNF (Smith normal form) to aid in computing determinants
and do other classifications, I saw that "interesting" examples had
determinants whose absolute values were not squarefree integers.
("Interesting" here means the last two components of the SNF are d
and f*d for integers d and f with d > 1 , f > 0, so the determinant
is divisible by d*f*d.) This led me to consider positive integers
which were multiples of a square of a prime (S), and positive
integers
that were not (F) .

It is clear that F has no sequence of consecutive integers that has
more than 3 terms. Are there infinitely many such 3 term sequences?
Are there infinitely many such with 2 terms? (A conjecture of
Schinzel and other weaker conjectures imply that yes, there are
infinitely many such pairs of squarefree consecutive numbers.)