Algebraic topology is a young subject, and its foundations are not yet firmly in place. I shall give some history, examples and modern developments in that part of the subject called stable algebraic topology, or stable homotopy theory. This is by far the most calculationally accessible part of algebraic topology, although it is also the least intuitively grounded in visualizable geometric objects. It has a great many applications to other subjects such as algebraic geometry and geometric topology. Time will not allow me to say as much as I would like about that. Rather, I shall emphasize some foundational issues that have been central to this part of algebraic topology since the early 1960s, but that have been satisfactorily resolved only in the last few years.

We prove for a wide class of structures that if [Fscr ] is a family of substructures which are pairwise uniformly commensurable, then there is a commensurable substructure invariant under automorphisms stabilizing [Fscr ] setwise.

In this note we show that if an ideal I of a noetherian ring is principal, up to radical, then the local cohomology modules with support in V(I) are I-cofinite.

A dinilpotent group is a group that can be written as the product of two nilpotent subgroups. There is an extensive literature dealing with such groups (see, for example, the recent book of Amberg, Franciosi and de Giovanni [1]). In 1955, Itô proved in [6] that the product of two abelian groups is always metabelian, and in the following year, Hall and Higman proved, as a special case of [3, Theorem 1.2.4], that if G=AB is a finite soluble group with A and B nilpotent of coprime orders and classes c and d respectively, then G has derived length at most c+d. Wielandt [9] showed that if the finite group G=AB has A and B nilpotent and of coprime orders, then G is necessarily soluble. Kegel [7] then showed that the condition that the orders be coprime was unnecessary. A natural next question to ask is if the derived length of a finite dinilpotent group is bounded by some function of the classes of the factors, and, in the light of the Hall–Higman result and the result of Itô, the following conjecture seems natural. (The first explicit reference to this conjecture that we know of is in Kegel [8]; see also, for example, Problem 5 on page 36 of [1].)

Let A be an order integral over a valuation ring V in a central simple F-algebra, where F is the fraction field of V. We show that (a) if (Vh, Fh) is the Henselization of (V, F), then A is a semihereditary maximal order if and only if A[otimes ]VVh is a semihereditary maximal order, generalizing the result by Haile, Morandi and Wadsworth, and (b) if J(V) is a principal ideal of V, then a semihereditary V-order is an intersection of finitely many conjugate semihereditary maximal orders; if not, then there is only one maximal order containing the V-order.

Extending earlier work, we establish the finiteness of the number of two-generator arithmetic Kleinian groups with one generator parabolic and the other either parabolic or elliptic. We also identify all the arithmetic Kleinian groups generated by two parabolic elements. Surprisingly, there are exactly 4 of these, up to conjugacy, and they are all torsion free.

We study zeros of elliptic integrals I(h)=∫∫H[les ]hR(x, y)dx dy, where H(x, y) is a real cubic polynomial with a symmetry of order three, and R(x, y) is a real polynomial of degree at most n. It turns out that the vector space [Ascr ]n formed by such integrals is a Chebishev system: the number of zeros of each elliptic integral I(h)∈[Ascr ]n is less than the dimension of the vector space [Ascr ]n.

We introduce a game called Squares where the single player is presented with a pattern of black and white squares and has to reduce the pattern to white by making as few moves as possible. We present a method for solving the game, and show that the following problem is NP-complete.

Problem 1 (Squares-Solvability). Given a pattern X and k∈N, can X be solved in k or less moves?

We demonstrate a reduction to this problem from Not-All-Equal-3SAT. We also present another NP-complete problem that Squares-Solvability can be reduced to.

We prove that the spectrum of a suitably projected Dirac operator — sometimes called the no-pair operator — introduced by Brown and Ravenhall is bounded from below by some positive constant when the nuclear charge is below some critical charge. This bound improves a result obtained recently by Evans et al., and disproves a conjecture made by Hardekopf and Sucher.

Let 0<n1<n2<… <nN be N distinct integers, and let a1, a2, …, aN be complex numbers. We set

formula here

and

formula here

There are two well-known problems concerning the case a1=a2=…=aN=1.

Let M and N be closed non-positively curved manifolds, and let f[ratio ]M→N be a smooth map. Results of Eells and Sampson [1] show that f is homotopic to a harmonic map ϕ, and Hartman [6] showed that this ϕ is unique when N is negatively curved and f∗(π1M) is not cyclic. Lawson and Yau conjectured that if f[ratio ]M→N is a homotopy equivalence, where M and N are negatively curved, then the unique harmonic map ϕ homotopic to f would be a diffeomorphism. Counterexamples to this conjecture appeared in [2], and later in [7] and [5].

There remains the question of whether a ‘topological’ Lawson–Yau conjecture holds.

We consider a simple shift procedure which transforms any mapping f[ratio ]{0, …, n}→{0, …, n} into a permutation of 0, …, n. We study this shift under the assumption that the values f(0), …, f(n) are uniformly and independently chosen random numbers in {0, …, n}. The exact and asymptotic distributions of the required shift sizes are derived, and a connection to queueing theory is exhibited.

John Arthur Todd was one of the last survivors of the school of classical algebraic geometry that flourished in Cambridge under H. F. Baker, FRS. But, unlike most of his contemporaries, with the notable exception of W. V. D. Hodge, FRS, Todd made seminal contributions to the more modern theory, and his name is now enshrined in the literature and widely known.