This article is an extended version of a lecture given in Oxford on 12 May 1995 at the invitation of the London Mathematical Society and the British Society for the History of Mathematics.

Contents

1. A few figures

2. Taylor series before 1900. A strange statement of Borel

3. Fourier series before 1900. A strange field

4. Brownian motion around 1900. A rising subject

5. Fourier and Taylor series after 1900. A revival

6. Lacunarity and randomness

7. The appearance of random series of functions

8. The Wiener theory of Brownian motion

9. The merging of Brownian motion and random Fourier series

10. The non-differentiability and local behaviour of Brownian motion

11. Three ways to figure out the Brownian motion

12. The plane Brownian motion

13. Applications of Brownian motion to Taylor series and analytic functions

14. Applications of Brownian motion to Fourier series and harmonic analysis

We produce a family of algebraic curves (closed Riemann surfaces) Sλ admitting two cyclic groups H1 and H2 of conformal automorphisms, which are topologically (but not conformally) conjugate and such that S/Hi is the Riemann sphere [Copf ]ˆ. The relevance of this example is that it shows that the subvarieties of moduli space consisting of points parametrizing curves which occur as cyclic coverings (of a fixed topological type) of [Copf ]ˆ need not be normal.

1. Introduction

Let G be a group, and write an(G) for the number of subgroups of index n in G. If G is a profinite group, we count only open subgroups. When G is finitely generated (either abstractly or topologically), an(G) is finite for all n∈N. In this case we can encode the arithmetic function n[map ]an(G) in the following zeta function:

formula here

This function was introduced by Grunewald, Segal and Smith [9] and subsequently studied in [1–6, 7, 11].

1. Introduction

The classical circle method of Hardy and Littlewood (see Baker [1]) ensures that the diophantine equation

a1xk1+[ctdot ]+ asxks =0 (1.1)

has a nontrivial solution in nonnegative integers x1, [ctdot ], xs, provided s[ges ]C(k) and the coefficients a1, [ctdot ], as are all integers and not all the same sign.

In this paper we solve the embedding problem for partial transitive triple systems of order n and index λ, partial TTS (n, λ) s, showing that any partial TTS (n, λ) can be embedded in a TTS (v, λ) for all λ-admissible v[ges ]2n+1. This lower bound on v is the best possible. A simple proof is also given of a result of Colbourn and Harms which shows that every partial triple system of order n and index 2λ is the underlying triple system of a partial TTS (n, λ).

Necessary and sufficient conditions are obtained for a bipartite graph to admit a matching which includes a preassigned set of edges and excludes another preassigned set of edges.

Let G and A be finite groups with coprime orders, and suppose that A acts on G by automorphisms. Let π(G, A)[ratio ]IrrA(G)→Irr (CG(A)) be the Glauberman–Isaacs correspondence. Let B[les ]A and χ∈IrrA(G). We exhibit a counterexample to the conjecture that χπ(G, A) is an irreducible constituent of the restriction of χπ(G, B) to CG(A).

We give an explicit example of an exotic (non-simplicial), geometric free action of the free group [ ]3 on an ℝ-tree T. We begin by associating an interval translation mapping of the unit interval to an automorphism ψ of [ ]3. We use a result of D. Gaboriau and G. Levitt to obtain an [ ]3-action on an ℝ-tree T. We show that for our particular choice of ψ, the resulting [ ]3-action is minimal, free and exotic.

We give an elementary proof for the fact that the Julia set of a rational or entire function is the closure of the repelling periodic points.

1. Introduction

Let k be an algebraically closed field of any characteristic. A toric variety over k is a normal variety X containing the algebraic group T=(k*)n as an open dense subset, with a group action T × X→X extending the group law of T.

On any smooth variety X over a field k, we can define the sheaf of differential operators [Dscr ], which carries a natural structure as an [Oscr ]X-bisubalgebra of Endk([Oscr ]X). A [Dscr ]-module on X is a sheaf [Fscr ] of abelian groups having a structure as a left [Dscr ]-module, such that [Fscr ] is quasi-coherent as an [Oscr ]X-module. A smooth variety X is called [Dscr ]-affine if for every [Dscr ]-module [Fscr ] we have

[bull ] [Fscr ] is generated by global sections over [Dscr ],

[bull ] Hi(X, [Fscr ])=0, i>0.

Beilinson and Bernstein have shown [1] that every flag variety over a field of characteristic zero is [Dscr ]-affine, from which they deduced a conjecture of Kazhdan and Lusztig. In fact, flag varieties are the only known examples of [Dscr ]-affine projective varieties. In this paper we prove that the [Dscr ]-affinity of a smooth complete toric variety implies that it is a product of projective spaces. Part of the method will be to translate a proof of the non [Dscr ]-affinity of a 2-dimensional Schubert variety, given by Haastert in [4], into the language of toric varieties.

In this note we discuss the radial positive solutions of

formula here

where D is the annulus {s∈Rn [ratio ]b<|s|<1}. Here 0<b<1, n[ges ]2, and g[ratio ]R→R is a suitable C1 function with g(0)[ges ]0 but with g changing sign. We answer a question on page 223 of the original Gidas–Ni–Nirenberg paper [8], by showing that the maximum of these solutions occurs at a point sε, where |sε|→b as ε→0. This also shows that the non-negativity condition in the result on page 223 of [8] is necessary.

A Banach algebra [afr ] is AMNM if whenever a linear functional ϕ on [afr ] and a positive number δ satisfy [mid ]ϕ(ab)−ϕ(a)ϕ(b)[mid ] [les ]δ|a|·|b| for all a, b∈[afr ], there is a multiplicative linear functional ψ on [afr ] such that |ϕ−ψ|=o(1) as δ→0. K. Jarosz [1] asked whether every Banach algebra, or every uniform algebra, is AMNM. B. E. Johnson [3] studied the AMNM property and constructed a commutative semisimple Banach algebra that is not AMNM. In this note we construct uniform algebras that are not AMNM.

The well-known inequality of Hardy for Fourier coefficients of functions f(t)∼[sum ]∞n=−∞bneint in the real Hardy space is [sum ]∞n= −∞[mid ]bn[mid ]/([mid ]n[mid ]+1)<∞. We shall establish analogues of this inequality for the Hermite function expansions and also for the Laguerre function expansions.

We show that if X is a complex hereditarily indecomposable space, then every operator from a subspace Y of X to X is of the form λI+S, where I is the inclusion map and S is strictly singular.

Let S be a locally compact Hausdorff space, and let C0(S) be the Banach space of complex-valued continuous functions on S vanishing at infinity. It is known [1, p. 93] that every isometry U on C0(S) has the form

(Uf)(s)=θ(s)f(γ(s)) (s∈S), ([midast ])

where γ is an automorphism of S, and θ is a continuous function on S satisfying [mid ]θ(s)[mid ]=1 for all s∈S. Recall that a norm on a Banach space X is maximal if there is no equivalent norm for which the group of isometries is strictly larger. It is easy to see that the norm of C0(S) is not maximal if S contains a finite number of isolated points (S has at least two isolated points) [3, Remarks on Theorem 8.2]. In [3, 5, 6], Kalton and Wood proved the following theorem.

The notion of an asymptotic cycle is extended to flows on non-compact spaces, and it is proved that for a suitable class of flows on manifolds it is true that the set of points with asymptotic cycles defined has measure one with respect with every invariant probability measure.

1. Ordered spaces

Risking semantic incoherence, we propose in this paper to use the term ordered space to mean any totally ordered set X equipped with a topology [Tscr ], while retaining the term linearly ordered space for its usual role of denoting the special case in which [Tscr ] is the order topology: the topology generated by the order-open initial and final segments of X.

The categories that we shall consider are full subcategories of the category OrdTop of ordered spaces and order-preserving continuous maps. It is easy to see that — in contrast to the situation in the full subcategory LinOrdTop of linearly ordered spaces — every inverse spectrum in OrdTop has a limit and that OrdTop admits subspaces and quotient spaces. These limits, subspaces and quotient spaces are all constructed precisely as in the category Top and are given their naturally induced orderings: where, however, we note that, for the ordered space X to induce an ordering on a quotient space, the quotient space (construed as a decomposition) must partition X into intervals rather than into arbitrary subsets. (In the terminology of [1], these constructions define subspaces and quotient spaces in OrdTop as, respectively, the domains and codomains of extremal monomorphisms and epimorphisms.) The purpose of the paper is to identify the closures of LinOrdTop under these constructions.

1. Introduction

In 1978, Atiyah, Hitchin and Singer [1] introduced the twistor space as a ℙ1-fibration over a half-conformally flat 4-manifold, and thereby established a beautiful correspondence between Yang–Mills fields on 4-manifolds and holomorphic vector bundles on complex 3-folds. Hitchin gave the Penrose correspondence between solutions of anti-self-dual zero rest-mass field equations on a half-conformally flat manifold M and the sheaf cohomology groups H1(Z, F(k)), k[les ]−2, on its twistor space Z [4]. Here the holomorphic bundle F is the pull-back bundle of an anti-self-dual bundle over M, and we denote by [Oscr ](−1) the tautological line bundle associated to the half-spin bundle. Hitchin further showed how H1(Z, F(−1)) corresponds to solutions of the self-dual Dirac equations, and interpreted H1(Z, F(k)), k[ges ]0, in terms of the cohomology of an elliptic complex on M.

The first part of the proof of Lemma 3 given in the author's paper [1] is obviously incomplete. Namely, [Kfr ]v(ζ) in the fifth line from the bottom of page 245 of [1] is not necessarily a cyclic extension over [Kfr ]v−1(ζ) in the next line. To complete the proof, we shall prove an assertion after simple preliminaries, as follows. Let p be an odd prime. For any CM-field k of finite or infinite degree, let k+ denote the maximal real subfield of k, A(k) the p-class group of k (that is, the p-primary component of the ideal class group of k), A(k+) the p-class group of k+, and A(k)− the kernel of the norm map A(k)→A(k+). Since p>2, A(k)−={a∈A(k)[mid ] aj=a−1} where j denotes the complex conjugation of k. For any extension F′/F of CM-fields, let ϕF′/F denote the homomorphism A(F)−→ A(F′)− induced by the natural map A(F)→A(F′). Now, the following completes the proof in question.

The author thanks Martin Wursthorn, who pointed out that the supposedly distinct 2-groups H(m, 1) and H(m, 3), defined on page 588 of [1], are in fact isomorphic to each other. This is an error (rather than a misprint), but may be corrected by changing the presentations of H(m, q) and H˜ as follows. Replace the relation B4=1 in the presentation for H(m, q) by the relation B8=1, and replace the relation Y4=1 in the presentation for H˜ by the relation Y8=1. With these modifications, the paragraph concerning the groups H(m, q) becomes valid.