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.

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.

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.

Let p, q be distinct odd primes, and let a, b be positive integers. In this paper we prove that if S(pa, qb) is a Storer difference set with the parameters ν = paqb, k = (ν−1)/4 and λ =(ν−5)/16, then we have a = b = 1, p =(ρ3r+ρ−3r−1)/3 and q = ρ3r+ρ−3r+1, where ρ = 2+√3, ρ− = 2 − √3, and r is a positive integer.

We prove that a finite solvable group whose order is divisible by a prime p has at least 2√p−1 conjugacy classes.

We provide a new lower bound for the greatest prime factor of the norm of algebraic numbers of the form axm+byn. The improvement concerns the dependence on the number field containing a, b, x and y.

We consider the error term P3(R) = #{n ∈ ℤ3 :|n| [les ] R} − 4|3πR3 which occurs in the counting of lattice points in a sphere of radius R. By considering second and third power moments, we prove that P3(R) = Ω±(R√log R). An upper bound for the gap between the sign changes of P3(R) is also proved.

In the analysis of hyperbolic boundary value problems, the construction of Kreiss' symmetrizers relies on a suitable block structure decomposition of the symbol of the system. In this paper, we show that this block structure condition is satisfied by all symmetrizable hyperbolic systems of constant multiplicity.

We prove that if A is a prime non-commutative JB*-algebra which is neither quadratic nor commutative, then there exist a prime C*-algebra B and a real number λ with ½ < λ [les ] 1 such that A = B as involutive Banach spaces, and the product of A is related to that of B (denoted by ∘, say) by means of the equality xy = λx ∘ y + (1 − λ)y ∘ x.

Let 1 [les ] p [les ] ∞. For each n-dimensional Banach space E = (E, ∥ · ∥), we define a norm ∥ · ∥p on E × ℝ as follows:

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It is shown that the correspondence (E, ∥ · ∥) [map ] (E × ℝ, ∥ · ∥p) defines a topological embedding of one Banach–Mazur compactum, BM(n), into another, BM(n + 1), and hence we obtain a tower of Banach–Mazur compacta: BM(1) ⊂ BM(2) ⊂ BM(3) ⊂ ···. Let BMp be the direct limit of this tower. We prove that BMp is homeomorphic to Q∞ = dir lim Qn, where Q = [0, 1]ω is the Hilbert cube.

Mather has shown that almost all linear projections of a smooth submanifold of a vector space to a subspace yield a generic mapping (so in the nice dimensions, most linear projections give rise to ‘stable’ mappings). In this paper we show that the same is true if we replace the submanifold by the image of a stable mapping.

In this note, we show that a homotopy class of [[ ]3, [ ]2] admits an [ ]1-invariant harmonic map if and only if its Hopf invariant is ±k2 for some integer k, where [ ]3 is the unit 3-sphere and [ ]2 is a 2-sphere with an arbitrary metric.

In a further generalisation of the classical Ito formula, we show that if B = (At + A[dagger]t [ratio ] t ∈ [0, 1]) is the Brownian quantum martingale and J = (Jt [ratio ] t ∈ [0, 1]) is a bounded quantum semimartingale, then M = (Bt + Jt [ratio ] t ∈ [0, 1]) satisfies the functional quantum Ito formula [10, Section 6]

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The proof, which relies on the classical theory of unbounded self-adjoint operators, may be adapted to the case where B is replaced by a classical Brownian martingale.