The Manin–Mumford conjecture asserts that if K is a field of characteristic zero, C a smooth proper geometrically irreducible curve over K, and J the Jacobian of C, then for any embedding of C in J, the set C(K) ∩ J(K)tors is finite. Although the conjecture was proved by Raynaud in 1983, and several other proofs have appeared since, a number of natural questions remain open, notably concerning bounds on the size of the intersection and the complete determination of C(K) ∩ J(K)tors for special families of curves C. The first half of this survey paper presents the Manin–Mumford conjecture and related general results, while the second describes recent work mostly dealing with the above questions.

It is well known that for a Noetherian ring R, an ideal I of R, and M a finitely generated R-module, the local cohomology modules HiI(M) are not always finitely generated. On the other hand, if R is local and m is its maximal ideal, then Him(M) are Artinian modules, which is equivalent to the following two properties:

(i) SuppR(Him(M)) ⊆{m};

(ii) the vector space HomR(k, Him(M)) has finite dimension over k, where k = R/m.

Taking these facts into account, Grothendieck [9] made the following conjecture.

In as elementary a way as possible, we place Wiles' proof of Fermat's last theorem into the context of a general description of reciprocity conjectured to obtain between algebraic varieties defined over Q and Hecke eigenvectors in the homology of the spaces of lattices in Rn.

We show that the halting times of infinite time Turing machines (considered as ordinals coded by sets of integers) are themselves all capable of being halting outputs of such machines. This gives a clarification of the nature of ‘supertasks’ or infinite time computations. The proof further yields that the class of sets coded by outputs of halting computations coincides with a level of Gödel's constructible hierarchy: namely that of Lλ where λ is the supremum of halting times. A number of other open questions are thereby answered.

Let q and N be integers, let a be an integer coprime to q, and let zN be defined implicitly by q = (log N)log22−ZN √(log2N). We show that for large N, an integer n has at least one divisor d with q [les ] d [les ] N and d ≡ a(mod q) with probability approximately Φ(zN), where Φ denotes the distribution function of the Gaussian Law. This solves a conjecture of Hall.

Examples are discussed of natural statements about irrational numbers that are equivalent, provably in ZFC, to strong set-theoretical hypotheses, and of apparently classical statements provable in ZFC of which the only known proofs use strong set-theoretical concepts.

Nous donnons une majoration de la somme d'exponentielles SM = [sum ]Mm=1e(f(m)) lorsque la dérivée cinquième de f est d'un ordre de grandeur constant petit, noté λ, en fonction de M et de λ, améliorant un résultat ancien de Van Der Corput. La démonstration utilise un théorème de moyenne des puissances sixièmes de sommes d'exponentielles qui fait l'objet d'un article indépendant [6].

We give a bound for the exponential sum SM = [sum ]Mm=1e(f(m)) where f is a real-valued function whose fifth derivative is of a constant small size, say λ, by means of M and λ, improving an old result of Van Der Corput. The proof relies on a mean value theorem for sixth powers of exponential sums which is treated independently in [6].

Let [Mscr ]n denote the space of complex n×n matrices, and let Ωn denote the spectral unit ball of [Mscr ]n, namely the set of matrices in [Mscr ]n whose eigenvalues lie within the open unit disc. As a step towards the eventual classification of the holomorphic automorphisms of Ωn, we prove that every such automorphism F satisfying F(0) = 0 and F′(0) = I has the property that F(x) is conjugate to x for each x∈Ωn. This result is obtained by combining earlier work of Ransford and White with a general theorem about spectrum-preserving maps proved in this paper.

The preceding paper ‘Strong statements of analysis’ by A. R. D. Mathias defends a so-called full-blooded set theory without full detail [3]. He again objects to a weak set theory which he calls ‘Mac’, in which the usual Zermelo–Fraenkel separation scheme is required only for formulas with suitably ‘restricted’ quantifiers. I had proposed that such separation is adequate for all standard uses of set theory in mathematics. But Mathias has not produced any counter examples of actual mathematics which requires the use of a stronger separation.

Infinitesimal conformal transformations of ℝn are always polynomial and finitely generated when n > 2. Here we prove that the Lie algebra of infinitesimal conformal polynomial transformations over ℝn, n > 2, is maximal in the Lie algebra of polynomial vector fields. When n is greater than 2 and p, q are such that p + q = n, this implies the maximality of an embedding of so(p + 1, q + 1, ℝ) into polynomial vector fields that was revisited in recent works about equivariant quantizations. It also refines a similar but weaker theorem by V. I. Ogievetsky.

We present a Borel reduction from a subset shift equivalence relation of a countable group to a subgroup conjugacy relation of a free product. The technique gives a much shorter proof of an earlier result of Thomas and Velickovic.

In 1968 Vizing conjectured that any independent vertex set of an edge-chromatic critical graph G contains at most half of the vertices of G, that is, α(G) [les ] ½[mid ]V(G)[mid ]. Let Δ be the maximum vertex degree in a critical graph. For each Δ, we determine c(Δ) such that α(G) [les ] c(Δ)[mid ]V)[mid ].

Let G be a residually-finite virtually soluble minimax group. We give upper and lower linear bounds for the degree of subgroup growth of G in terms of the Hirsch length of G.

An upper bound on operator norms of compound matrices is presented, and special cases that involve the l1, l2 and l∞ norms are investigated. The results are then used to obtain bounds on products of the largest or smallest eigenvalues of a matrix.

The purpose of this note is initially to present an elementary but surprising connectedness principle pertaining to the intersection of a fixed subvariety X of some ambient space Z with another subvariety Y which is ‘mobile’ (in the sense of being movable, rather than actually moving). It is via this mobility that monodromy enters the picture, permitting the crucial passage from ‘relative’ or total-space irreducibility to ‘absolute’ or fibrewise connectedness (and sometimes irreducibility). A general form of this principle is given in Theorem 2 below.

Given a group G, a G-set Ω and a graph Γ we present a construction for a family of graphs, the Ω-covers of Γ. A particular example of this construction gives a girth 17 cubic graph with 2530 vertices.

If s1, s2, …, st are integers such that n − 1 = s1 + s2 + ··· + st and such that for each i (1 [les ] i [les ] t), 2 [les ] si [les ] n − 1 and sin is even, then Kn can be expressed as the union G1 ∪ G2 ∪ ··· ∪ Gt of t edge-disjoint factors, where for each i, Gi is si-regular and si-connected. Moreover, whenever si = sj, Gi and Gj are isomorphic.

For a transcendental meromorphic function f, various properties of the set

formula here

were obtained in [8] and [9]. Here we establish analogous properties for the smaller sets

formula here

introduced in [5], and

formula here

We deduce a symmetry result for Julia sets J(f), and also indicate some techniques for showing that certain invariant curves lie in I′(f), Z(f) and J(f).

Let X be a smooth complex projective curve of genus g [ges ] 1. If g [ges ] 2, then assume further that X is either bielliptic or with general moduli. Fix integers r, s, a, b with r > 1, s > 1 and as [les ] br. Here we prove the existence of an exact sequence

formula here

of semistable vector bundles on X with rk(H) = r, rk(Q) = s, deg(H) = a and deg(Q) = b.

In this paper, branching rules for the fundamental representations of the symplectic groups in positive characteristic are found. The submodule structure of the restrictions of the fundamental modules for the group Sp2n(K) to the naturally embedded subgroup Sp2n−2(K) is determined. As a corollary, inductive systems of fundamental representations for Sp∞(K) are classified. The submodule structure of the fundamental Weyl modules is refined.