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

The entities A, B, X, Y in the title are operators, by which we mean either linear transformations on a finite-dimensional vector space (matrices) or bounded (= continuous) linear transformations on a Banach space. (All scalars will be complex numbers.) The definitions and statements below are valid in both the finite-dimensional and the infinite-dimensional cases, unless the contrary is stated.

1. Introduction 514

2. Eigenvalues and eigenfunctions 516

2.1 Spectral asymptotics 516

2.2 The isoperimetric problem 518

2.3 The Rellich inequality 519

2.4 Numerical studies of the rectangular plate 519

2.5 The ground state 521

2.6 Stability properties 522

2.7 The ideal column 523

3. Solving elliptic equations 524

3.1 One-dimensional results 524

3.2 Constant-coefficient operators 524

3.3 Green functions of bounded regions 525

3.4 The Poisson problem 526

3.5 The Dirichlet problem 526

3.6 Conical points 527

4. The semigroup e−Ht 528

4.1 Elliptic systems 528

4.2 Extensions to Lp 529

4.3 Generalized Schrödinger operators 530

4.4 Kato class potentials 530

5. Heat kernel bounds 531

5.1 General properties of heat kernels 532

5.2 Gaussian upper bounds 532

5.3 Sharp constants 533

5.4 Complex time bounds 534

5.5 Pointwise lower bounds 535

6. Lp spectral theory 536

6.1 Functional calculus 536

6.2 Helffer–Sjöstrand calculus 537

6.3 Spectral independence 538

7. Non-self-adjoint operators 538

7.1 Second-order operators 538

7.2 Lp multiplier theory 539

7.3 H∞ functional calculus 540

7.4 Applications of the H∞ functional calculus 540

References 541

It has been known since the last century that a single quadratic form in at least five variables has a nontrivial zero in any p-adic field, but the analogous question for systems of quadratic forms remains unanswered. It is plausible that the number of variables required for solubility of a system of quadratic forms simply is proportional to the number of forms; however, the best result to date, from an elementary argument of Leep [6], is that the number of variables needed is at most a quadratic function of the number of forms. The purpose of this paper is to show how these elementary arguments can be used, in a certain class of fields including the p-adic fields, to refine the upper bound for the number of variables needed to guarantee solubility of systems of quadratic forms. This result partially addresses Problem 6 of Lewis' survey article [7] on Diophantine problems.

This paper is based on a talk given at a joint meeting of the London Mathematical Society and the British Society for the History of Mathematics held in Oxford on 13 May 1995.

This paper encompasses a motley collection of ideas from several areas of mathematics, including, in no particular order, random walks, the Picard group, exchange rate networks, chip-firing games, cohomology, and the conductance of an electrical network. The linking threads are the discrete Laplacian on a graph and the solution of the associated Dirichlet problem. Thirty years ago, this subject was dismissed by many as a trivial specialisation of cohomology theory, but it has now been shown to have hidden depths. Plumbing these depths leads to new theoretical advances, many of which throw light on the diverse applications of the theory.

We consider the following strengthening of Hadwiger's Conjecture.

Let G be any graph of chromatic number k, S any subset of V(G) which takes all k colours in each proper k-colouring of G. Then there are k pairwise adjacent connected subgraphs of G, each of whose vertex sets has non-trivial intersection with S.

We show that the truth of this conjecture for all graphs of chromatic number [les ]k implies the truth of Hadwiger's Conjecture for all graphs of chromatic number [les ]k + 1. We also show that its truth implies the following statement (which is at first sight even stronger).

For any graph G of chromatic number k and any subset S of V(G), define χ(S; G) to be the least number of colours that can appear on S in any proper k-colouring of G, and h(S; G) to be the largest number of pairwise adjacent connected subgraphs of G each having non-trivial intersection with S. Then χ(S; G) [les ]h(S; G).

We define the number w(S; G) to be the largest cardinality of a subset T of S such that, however T is partitioned into pairs (possibly with one spare element), there are vertex-disjoint paths linking the elements in each pair, none passing through the spare element if it exists. We show that χ(S; G) [les ]([mid ]S[mid ]+w(S; G))/2 for any graph G and subset S of V(G).

Finally, we show that for any graph G, χ(S; G) [les ]h(S; G) whenever χ(S; G) [les ]3.

Let G be a finite group, and let IG be the augmentation ideal of ℤG. We denote by d(G) the minimum number of generators for the group G, and by d(IG) the minimum number of elements of IG needed to generate IG as a G-module. The connection between d(G) and d(IG) is given by the following result due to Roggenkamp ]14&]:

formula here

where pr(G) is a non-negative integer, called the presentation rank of G, whose definition comes from the study of relation modules (see [4&] for more details).

In [1] Almkvist and Meurman proved a result on the values of the Bernoulli polynomials (Theorem 5 below). Subsequently, Sury [5] and Bartz and Rutkowski [2] have given simpler proofs.

In this paper we show how this theorem can be obtained from classical results on the arithmetic of the Bernoulli numbers. The other ingredient is the remark that a polynomial with rational coefficients which is integer-valued on the integers is ℤ(p)-valued on ℤ(p). Here ℤ(p) denotes the ring of rational numbers whose denominator is not divisible by the prime p. An application is given in Section 3 to the arithmetic of generalised Bernoulli numbers.

Answering a long-standing question in the theory of torsion modules, we show that weakly productively bounded domains are necessarily productively bounded. (See the Introduction for definitions.) Moreover, we prove a twin result for the ideal lattice L of a domain equating weak and strong global intersection conditions for families (Xi)i∈I of subsets of L with the property that ∩i)i∈IAi≠0 whenever Ai∈Xi. Finally, we show that for domains with Krull dimension (and countably generated extensions thereof), these lattice-theoretic conditions are equivalent to productive boundedness.

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.

We prove that each interval of the lattice [Escr ] of c.e. sets under inclusion is either a boolean algebra or has an undecidable theory. This solves an open problem of Maass and Stob [11]. We develop a method to prove undecidability by interpreting ideal lattices, which can also be applied to degree structures from complexity theory. We also answer a question left open in [7] by giving an example of a non-definable subclass of [Escr ]* which has an arithmetical index set and is invariant under automorphisms.

A remarkable theorem of Birch [2] shows that a system of homogeneous polynomials with rational coefficients has a non-trivial zero, provided only that these polynomials are of odd degree, and the system has sufficiently many variables in terms of the number and degrees of these polynomials. Despite four decades of effort, the problem of obtaining a reasonable bound for the latter number of variables has proved to be one of great difficulty. When the system consists of a single cubic form, Davenport [4] has succeeded in showing that 16 variables suffice, and Schmidt [17, 18, 19, 20] has devoted a series of papers to systems of cubic forms, showing in particular that 5140 variables suffice for pairs of cubic forms, and that (10r)5 variables suffice for systems of r cubic forms. The current state of knowledge for forms of higher degree is, by comparison, extremely weak (but see [21, 22]), and so it seems worthwhile expending further effort on the case of systems of cubic forms. In this paper we improve on Schmidt's result for pairs of cubic forms. In contrast with the sophisticated versions of the Hardy–Littlewood method employed by Davenport and Schmidt, our approach is based on an elementary idea of Lewis [12], and is applicable in arbitrary number fields. This method also has consequences for the existence of linear spaces of rational solutions on cubic hypersurfaces, thereby improving on work of Lewis and Schulze-Pillot [14] on this topic.

Let (R, [mfr ]) be a local Noetherian ring. We show that if R is complete, then an R-module M satisfies local duality if and only if the Bass numbers μi([mfr ], M) are finite for all i. The class of modules with finite Bass numbers includes all finitely generated, all Artinian, and all Matlis reflexive R-modules. If the ring R is not complete, we show by example that modules with finite Bass numbers need not satisfy local duality. We prove that Matlis reflexive modules satisfy local duality in general, where R is any local ring with a dualizing complex.

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].

Let Γ⊂SL2(ℤ) be a subgroup of finite index, and let [Hscr ] denote the Hecke algebra of Γ. The aim of this note is to give some information about the action of [Hscr ] on spaces of modular forms for certain noncongruence subgroups Γ, which can be deduced from the geometric results of [9].

Every complex manifold carries a canonical orientation, and it is natural to wonder when the underlying topological or smooth manifold carries a complex structure compatible with the other orientation. In [2], Beauville raised this question for compact complex surfaces. He noted that there are a lot of examples, like products of curves, or Hopf surfaces, where the underlying manifold admits an orientation-reversing selfdiffeomorphism. This implies that the signature is zero. Beauville asked if there are any examples of non-zero signature.

Let Ω be the set {1, 2, [ctdot ], n}, and let [emptyv ] be the empty set. Let [Gscr ] be the family of all non-empty sets of subsets of Ω. For [Ascr ]∈[Gscr ] and X⊆Ω, put

formula here

An important discovery is the following.

For n a positive integer, a group G is called core-n if H/HG has order at most n for every subgroup H of G (where HG is the normal core of H, the largest normal subgroup of G contained in H). It is proved that a locally finite core-n group G has an abelian subgroup whose index in G is bounded in terms of n.

If G is a finitely presented group and [Kscr ] is any (G,2)-complex (that is, a finite 2-complex with fundamental group G), then it is well known that χ([Kscr ]) [ges ] ν(G), where ν(G) = 1−rk H1G+dH2G. We define χ(G) to be min{χ([Kscr ]): [Kscr ] a (G, 2)-complex}, and we say that G is efficient if χ(G) = ν(G). In this paper we give sufficient conditions for a Coxeter group to be efficient (Theorem 4.2). We also give examples of inefficient Coxeter groups (Theorem 5.1). In fact, we give an infinite family Gn(n = 2, 3, 4, . . . ) of Coxeter groups such that χ(Gn)−ν(Gn) [xrarr ] ∞ as n [xrarr ] ∞.