This paper is a continuation of [15]. In that paper, I introduced a general framework which allows one to produce ‘weak’ counterexamples to Torelli for Calabi–Yau threefolds: deformation families containing non-isomorphic varieties Yt, Y+t with isomorphic Hodge theory on the third cohomology. The varieties arise as deformations of threefolds Y that are resolutions of singular varieties X with rather special properties (cf. Section 1). In [15], I discussed two families containing suitable X that do provide a counterexample and a third family with remarkably similar properties where however the existence of a nontrivial generic automorphism destroys the counterexample. This shows that explicit examples are necessary; there is a precise set of conditions one needs to check carefully in order to obtain counterexamples to Torelli (Theorem 1·1). In fact, there exist several families with renitent automorphisms (Remark 4·5).

A Lie subalgebra L of [gfr ][lfr ][ ](V) is said to be finitary if it consists of elements of finite rank. We study the situation when L acts irreducibly on the infinite-dimensional vector space V and show: if Char [ ] > 7, then L has a unique minimal ideal I. Moreover I is simple and L/I is solvable.

This is a paper about upper bounds for Dirichlet's L-function, L(s, χ), on its critical line (s + s¯ = 1). It is to be assumed throughout that, unless otherwise stated, the Dirichlet character, χ, is periodic modulo a prime, r, and is not the principal character mod r. Our main theorem below shows that, if ε > 0, then

formula here

(where A is an absolute constant), for 0 < α = (log r)/(log t) [les ] 2/753 − ε. Somewhat weaker bounds are obtained for other cases where 0 < α [les ] 11/180 − ε. Note that in [13] it was shown that, for 0 < α [les ] 2/57,

formula here

Our main theorem is a corollary of the new bounds we prove for certain exponential sums, S, with a Dirichlet character factor:

formula here

where M2 [les ] 2M and f(x) is a real function whose derivatives satisfy certain conditions restricting their size.

Given compact sets E and F in [Copf ]n (n [ges ] 1) related by F = q−1(E) where q is a polynomial map, we are interested in the general problem of comparing minimal polynomials for E with minimal polynomials for F. Let α be an n-multi-index of length d. We define the classes of polynomials ℙ(α) := zα + [Copf ]d−1[z] and [Pscr ](α) := {p: p(z) = zα + [sum ]β[pr ]αaβzβ} where [pr ] denotes the usual graded lexicographic order. Polynomials in ℙ(α) or in [Pscr ](α) of least deviation from zero on E (with respect to the supremum norm) are called minimal polynomials for E. We prove that if q is a simple (i.e. qˆi(z) = zmi) polynomial mapping of degree m and if p is minimal polynomial for E then p ∘ q is a minimal polynomial for F = q−1(E) and, using some algebraic machinery, we can also construct minimal polynomials for E from minimal polynomials for F. The result seems to be new even in the one-dimensional case.

In this paper, we study involutions of Godeaux surfaces whose bicanonical system has no base components. We classify all possible fixed loci and find corresponding examples to the classification.

The point of this note is to make an observation concerning the variety M(E) parametrizing line subbundles of maximal degree in a generic stable vector bundle E over an algebraic curve C. M(E) is smooth and projective and its dimension is known in terms of the rank and degree of E and the genus of C (see Section 1). Our observation (Theorem 3·1) is that it has exactly the Chern numbers of an étale cover of the symmetric product SδC where δ = dim M(E).

This suggests looking for a natural map M(E) → SδC; however, it is not clear what such a map should be. Indeed, we exhibit an example in which M(E) is connected and deforms non-trivially with E, while there are only finitely many isomorphism classes of étale cover of the symmetric product. This shows that for a general deformation in the family M(E) cannot be such a cover (see Section 4).

One may conjecture that M(E) is always connected. This would follow from ampleness of a certain Picard-type bundle on the Jacobian and there seems to be some evidence for expecting this, though we do not pursue this question here.

Note that by forgetting the inclusion of a maximal line subbundle in E we get a natural map from M(E) to the Jacobian whose image W(E) is analogous to the classical (Brill–Noether) varieties of special line bundles. (In this sense M(E) is precisely a generalization of the symmetric products of C.) In Section 2 we give some results on W(E) which generalise standard Brill–Noether properties. These are due largely to Laumon, to whom the author is grateful for the reference [9].

This paper continues our study started in [5] of the group G defined by the presentation

formula here

where m [ges ] 2, n [ges ] 2 and ∈i = ±1 for 1 [les ] i [les ] 2. The problem under consideration can be stated as: precisely when does both a have order m and b have order n in the group G? This question arose as a result of our investigation into equations over groups and the connection is explained in Section 2.

Let Γ be a finite G-symmetric graph whose vertex set admits a non-trivial G-invariant partition [Bscr ] with block size v. A framework for studying such graphs Γ was developed by Gardiner and Praeger which involved an analysis of the quotient graph Γ[Bscr ] relative to [Bscr ], the bipartite subgraph Γ[B, C] of Γ induced by adjacent blocks B, C of Γ[Bscr ] and a certain 1-design [Dscr ](B) induced by a block B ∈ [Bscr ]. The present paper studies the case where the size k of the blocks of [Dscr ](B) satisfies k = v − 1. In the general case, where k = v − 1 [ges ] 2, the setwise stabilizer GB is doubly transitive on B and G is faithful on [Bscr ]. We prove that [Dscr ](B) contains no repeated blocks if and only if Γ[Bscr ] is (G, 2)-arc transitive and give a method for constructing such a graph from a 2-arc transitive graph with a self-paired orbit on 3-arcs. We show further that each such graph may be constructed by this method. In particular every 3-arc transitive graph, and every 2-arc transitive graph of even valency, may occur as Γ[Bscr ] for some graph Γ with these properties. We prove further that Γ[B, C] ≅ Kv−1,v−1 if and only if Γ[Bscr ] is (G, 3)-arc transitive.

The Young–Fibonacci graph [ ][ ] is an important example (along with the Young lattice) of differential posets studied by Fomin and Stanley. For every differential poset there is a distinguished central measure called the Plancherel measure. We study the Plancherel measure and the associated Markov chain, the Plancherel process, on the Young–Fibonacci graph.

We establish a law of large numbers which implies that the Plancherel measure cannot be represented as a nontrivial mixture of central measures, i.e. is ergodic. Our second result claims the convergence of the level distributions of the Plancherel measure to the GEM(½) probability law in the space of nonnegative series with unit sum, which is a particular example of distribution from the class of Residual Allocation Models.

In order to obtain the Plancherel process as an image of a sequence of independent uniformly distributed random variables, we establish a new version of the Robinson–Schensted type correspondence between permutations and pairs of paths in the Young–Fibonacci graph. This correspondence is used to demonstrate a recurrence property of the Plancherel process.

Let S denote a compact, connected, orientable surface with genus g and h boundary components. We refer to S as a surface of genus g with h holes. Let [Mscr ]S denote the mapping class group of S, the group of isotopy classes of orientation-preserving homeomorphisms S → S.

Let G be a group. G is hopfian if every homomorphism from G onto itself is an automorphism. G is residually finite if for every g ∈ G with g ≠ 1 there exists a normal subgroup of finite index in G which does not contain g. Every finitely generated residually finite group is hopfian ([11, 12]). A group G is hyperhopfian ([2, 3]) if every homomorphism ψ G → G with ψ(G) normal in G and G/ψ(G) cyclic is an automorphism. As observed in [14], examples of hopfian groups which are not hyperhopfian are afforded by the fundamental groups of torus knots.

By a result of Grossman [5], [Mscr ]S is residually finite. Since [Mscr ]S is also finitely generated, it is hopfian. It is a natural question to ask whether [Mscr ]S is hyperhopfian. In this paper, we shall answer a more general question. We say that a group G is ultrahopfian if every homomorphism ψ: G → G with ψ(G) normal in G and G/ψ(G) abelian is an automorphism. Note that an ultrahopfian group is hyperhopfian. We shall prove the following result.

If a Quillen model category can be specified using a certain logical syntax (intuitively, ‘is algebraic/combinatorial enough’), so that it can be defined in any category of sheaves, then the satisfaction of Quillen's axioms over any site is a purely formal consequence of their being satisfied over the category of sets. Such data give rise to a functor from the category of topoi and geometric morphisms to Quillen model categories and Quillen adjunctions.

We consider the bond percolation model on the three-dimensional cubic lattice, in which individual edges are retained independently with probability p. We shall describe a subgraph of the lattice as ‘entangled’ if, roughly speaking, it cannot be ‘pulled apart’ in three dimensions. We shall discuss possible ways of turning this into a rigorous definition of entanglement. For a broad class of such definitions, we shall prove that for p sufficiently close to zero, the graph of retained edges has no infinite entangled subgraph almost surely, thereby establishing that there is a phase transition for entanglement at some value of p strictly between zero and unity.

For a hyperbolic knot, the excellent component of the character curve is the one containing the complete hyperbolic structure on the complement of the knot. In this paper we explain a method to compute the excellent component of the character variety of periodic knots. We apply the method to those knots obtained as the preimage of one component of a 2-bridge link by a cyclic covering of S3 branched on the other component. We call these knots periodic knots with rational quotient. Among this class of knots are the ‘Turk's head knots’. Finally we give some invariants deduced from the excellent component of the character curve, such as the h-polynomial and the limit of hyperbolicity for all the periodic knots with rational quotient, up to 10 crossings, which are not 2-bridge or toroidal.

Let p be a prime number. Given a p-group G, denote by H*(G) the mod-p cohomology algebra of G. For every subgroup K of G, let ResGK be the restriction map from H*(G) to H*(K). Consider the case where K = Ker u, with 0 ≠ u ∈ H1 (G) = Hom(G,[ ]p). It was known that, for p = 2, Ker ResGK is the principal ideal (u) generated by u. However, for p > 2, a corresponding statement does not hold: while an argument used by Quillen-Venkov in [14] shows that (Ker ResGK)2 ⊂ (βu) (with β the Bockstein homomorphism), it turns out that, in general, (u, βu) is strictly smaller that Ker ResGK; an example is the case where G is any extraspecial p-group of order p3 and K is maximal elementary abelian in G (see [5, 8]).

It is a theorem of Reidemeister [4] and Singer [6] that any two Heegaard splittings of a closed 3-manifold are stably equivalent. In this short note, we give this theorem a simple proof.

In [7], Hitchin showed that the data (∇, Φ), comprising an SU(2) Yang–Mills–Higgs monopole in the Prasad–Sommerfeld limit on ℝ3, encodes faithfully into an auxiliary rank 2 holomorphic vector bundle E˜ over T, the total space of the holomorphic tangent bundle of ℙ1. In this construction ℝ3 is viewed as a subset of H0(ℙ1, [Oscr ](T)) ≅ [Copf ]3.

Generically, the restriction of E˜ to a line is trivial. (The image of a global section ℙz ⊂ T, for z ∈ [Copf ]3, is referred to here as a line on T.) Hence c1(E˜) = 0 and, for all z ∈ [Copf ]3, there exists m ∈ {0} ∪ ℕ such that E˜[mid ]ℙz ≅ [Oscr ](m) [oplus ] [Oscr ](−m). If m [ges ] 1 then ℙz is a jumping line of E˜ of height m. The jumping lines are parameterized by an analytic set J ⊂ [Copf ]3, which is stratified by height. When the monopole has charge k, the height is bounded above by k. In this case we write J = J1 ∪ … ∪ Jk, where Ji parameterizes jumping lines of height i. A priori, some Ji may be empty.

The analytic continuation of the monopole to [Copf ]3 has singularities over J. To see this recall how the monopole data are recovered from E˜: very briefly, E˜ induces a sheaf [Escr ] = π2*ε*E˜ over [Copf ]3 which is locally free away from J2 ∪ … ∪ Jk, (π2 and ε are defined in Section 2). A holomorphic connection and Higgs field are defined in [Escr ] over [Copf ]3 null planes that cut out a given direction (see [1, 7, 9]). On restriction to ℝ3, [Escr ] gives a rank 2, SU(2) bundle and the holomorphic connection and Higgs field give the monopole data. It is easy to see that the flat connections are singular at points of J: for example, an analogous situation is described in [10].

In 1996, Cromwell and Nutt [7] found an upper bound on the arc index which is related to the minimal crossing number and conjectured that the upper bound achieves the lowest possible index for alternating links.

CONJECTURE. Let L be any prime link. Then α(L) [les ] c(L)+2. Moreover this inequality is strict if and only if L is not alternating.

In this paper, we define a new diagram, called a wheel diagram, of a link and use it to prove this conjecture.

The Dickson algebra Wn+1 of invariants in a polynomial algebra over [ ]2 is an unstable algebra over the mod 2 Steenrod algebra [Ascr ], or equivalently, over the Kudo–Araki–May algebra [Kscr ] of ‘lower’ operations. We prove that Wn+1 is a free unstable algebra on a certain cyclic module, modulo just one additional relation. To achieve this, we analyse the interplay of actions over [Ascr ] and [Kscr ] to characterize unstable cyclic modules with trivial action by the subalgebra [Ascr ]n−2 on a fundamental class in degree 2n – a. This involves a new family of left ideals [Iscr ]a in [Kscr ], which play the role filled by the ideals [Ascr ][Ascr ]n−2 in the Steenrod algebra.

L2 estimates of the Kohn–Hörmander type are obtained for the ∂u operator on relatively compact Stein subdomains of complex manifolds. This is done by using a result of Hörmander locally, patching with cohomology of coherent analytic sheaves with bounds to get global solutions of the ∂u equation and then finishing with standard Hilbert space methods to obtain weak elliptic estimates.

This note is intended to correct the proof of a lemma in the article ‘Hit polynomials and conjugation in the dual Steenrod algebra’, which appeared in 1998 in Volume 123 of these Proceedings.