What the book is about

In 1976 I gave a new proof to the Grothendieck (two-dimensional) inequality. The proof, pushed a little further, yielded extensions of the inequality to higher dimensions. These extensions, in turn, revealed ‘Cartesian products in fractional dimensions’, and led in a setting of harmonic analysis to the solution of the (so-called) p-Sidon set problem. The solution subsequently gave rise to an index of combinatorial dimension, a general measurement of interdependence with connections to harmonic, functional, and stochastic analysis. In 1993 I was ready to tell the story, and began teaching topics courses about this work. The notes for these courses eventually became this book.

Broadly put, the book is about ‘dimensionality’. There are several interrelated themes, sub-themes, variations on themes. But at its very core, there is the notion that when we do mathematics – whatever mathematics we do – we start with independent building blocks, and build our constructs. Or, from an observer's viewpoint – not that of a builder – we assume existence of building blocks, and study structures we see. In either case, these are the questions: How are building blocks used, or put together? How complex are the constructs we build, or the structures we observe? How do we gauge, or detect, complexity? The answers involve notions of dimension.

The book is a mix of harmonic analysis, functional analysis, and probability theory.

Mise en Scène: The Last Chapter

I have come to the end, but believe it is only the beginning; there is hard work ahead, and much to be discovered. Questions I could not answer have been scattered throughout the book, and in this chapter I look back, assess what has been done, and try to point to future lines. I will recall various notions and results from previous chapters, and expect readers (if there are any left…) to be familiar with them.

In §2, we outline rudiments of measure theory in fractional dimensions. Grothendieck inequality-type issues surface naturally in this context. The relevant chapters are IV, V, VI, VIII, IX, XII, and XIII.

In §3, combinatorial dimension – a basic gauge of interdependence – is cast in general topological and measurable settings. There are other notions of dimension, and questions regarding connections between these and combinatorial dimension lead to interesting problems. Relevant chapters are XII and XIII.

In §4, we reexamine basic structures underlying classical harmonic analysis. Standard textbook harmonic analysis starts and continues with Borel measures and their transforms, but going further one could start with finitely additive set-functions, and, in the spirit of §2, follow a course where the space of measures is but a first stop. Relevant chapters are VII, XII, and XIII.

Riemann surfaces have played a central role in mathematics ever since their introduction by Riemann in his dissertation in 1851; for a biography of Riemann, see Riemann, topology and physics, Birkhäuser, 1987 by M. Monastyrsky. Following Riemann, we first consider a Riemann surface to be the natural maximal domain of some analytic function under analytic continuation, and this point of view enables one to put the theory of ‘manyvalued functions’ on a firm foundation. However, one soon realises that Riemann surfaces are the natural spaces on which one can study complex analysis and then an alternative definition presents itself, namely that a Riemann surface is a one dimensional complex manifold. This is the point of view developed by Weyl in his classic text (The concept of a Riemann surface, Addison-Wesley, 1964) and this idea leads eventually on to the general theory of manifolds. These two points of view raise interesting questions. If we start with a Riemann surface as an abstract manifold, how do we know that it supports analytic functions? On the other hand, if we develop Riemann surfaces from the point of view of analytic continuation, how do we know that in this way we get all complex manifolds of one (complex) dimension? Fortunately, it turns out that these two different views of a Riemann surface are indeed identical.

Introduction

This article is an extended version of the lecture given at UNED in July 1998 on “Topics on Riemann surfaces and Fuchsian groups” to mark the 25th anniversary of UNED.

The object of that lecture was to motivate the definition of arithmetic Fuchsian groups from the special and very familiar example of the classical modular group. This motivation proceeded via quaternion algebras and the lecture ended with the definition of arithmetic Fuchsian groups in these terms. This essay will go a little beyond that to indicate how the number theoretic data defining an arithmetic Fuchsian group can be used to determine geometric and group-theoretic information. No effort is made here to investigate other approaches to arithmetic Fuchsian groups via quadratic forms or to discuss and locate these groups in the general theory of discrete arithmetic subgroups of semi-simple Lie groups. Thus the horizons of this article are limited to giving one method of introducing an audience familiar with the ideas of Fuchsian groups and Riemann surfaces to the interesting special subclass of arithmetic Fuchsian groups.

Basics

A Fuchsian group is a discrete subgroup of SL(2,ℝ) or of PSL(2,ℝ) = SL(2,ℝ)/ < −I >. We will frequently employ the usual abuse of notation by writing elements of PSL(2,ℝ) as matrices, while strictly they are only determined up to sign.