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Harmonic Analysis on Symmetric Spaces — Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane

Audrey Terras
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Michael Berg
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It’s hard to believe that it’s almost 30 years ago, but indeed it is: I finished my doctorate at UCSD in 1985, with the author of this book as my thesis advisor. During my years in La Jolla (possibly the most lovely part of the burgeoning metropolis that in those days called itself “America’s Finest City” — with some solid evidence to back up the claim), I was made the recipient of chapter after (photocopied) chapter of the first iteration of the book that we are now considering in its published and highly evolved stage. It’s a wonderful thing to have in my hands, now, after all these years, such a marvelous finished product.

And it’s all vintage Audrey Terras (now emerita but happily going strong). As the preface shows, there is a built-in dimension of iconoclasm in her approach — this is not a textbook in the usual sense at all. Audrey in point of fact makes no bones about her dislike of and disagreement with the prevalent dominance of texts that all but ignore the wide application of things mathematical beyond the boundaries set by narrow specialization. Says Audrey (in her Preface to the First Edition, appearing in 1985 and recommended by the MAA for undergraduate libraries):

…in the past 30 years there have been some really exciting discoveries in the field of harmonic analysis on symmetric spaces and their fundamental domains for discrete isometry groups … It is time that these ideas received an exposition comprehensible to the average applied mathematician, number theorist, etc. In particular … many of the results to be described have interesting applications for statistical physics and number theory.

Audrey notes in her Preface to the present edition that with

[a]lmost 30 years hav[ing] passed … [m]uch has changed … however, [b]asic mathematics is as it ever was [and] I tried not to change much in the first edition. I made corrections and added some updates on new developments on harmonic analysis on symmetric spaces, keeping myself to those developments that fit in with the spirit of the original.

Thus, in the book under review we encounter an updated version of an established work of scholarship that has both stood the test of time, and remains very relevant and well-worth reading.

The intended audience is very broad, and includes scientists outside of pure mathematics (and even cousins as distant as physicists) who wish to learn about the type of harmonic analysis on symmetric spaces (attached to a discrete group), as well as mathematicians who wish to enter the indicated field and desire a very well-drawn road-map. Well, the book is actually much more than that: it is filled with solid mathematics, theorems are either proved in full or sketches are given replete with references to other sources. There are many exercises, well chosen and meant to get the reader’s hands nice and dirty — as they should be when learning mathematics. Finally, there is a lot of depth to what is covered: the reader should be prepared to be tantalized and then moved to dig deeper and deeper (and before you know it, you have a mathematician (modulo said reader’s tenacity, of course)).

As the title indicates, the paradigms of symmetric spaces the book is concerned with are flat space, the sphere, and the complex upper half-plane. For the plane it’s largely classical Fourier analysis; heterodox highlights include the central limit theorem, some quantum mechanics (“Schrödinger eigenvalues”), crystallography, and — going finite — wavelets and quasicrystals. Next, the sphere is presented as an exemplar of a compact symmetric space, and here highlights include more QM (the hydrogen atom), some group representation theory, and then material on the Radon transform (including stuff on CAT scanners).

Finally, and most titanically: we encounter the Poincaré upper half plane — and given that, when all is said and done. Audrey Terras is a number theorist, it cannot be otherwise. Indeed, this third and final part of the book is the longest of the three and contains a graduate course in and of itself: hyperbolic geometry, the special linear group and its ever-so-important subgroups, especially the discrete groups favored by Hecke (and Klein, Siegel, Maass, &c.) — indeed the last sections of Audrey’s book take the reader from SL(2,Z) and the attendant theory of modular forms to Maass wave forms — Hecke theory proper (i.e. the interplay between modular forms and Dirichlet series already anticipated by, e.g., Riemann, what with his famous second proof of the functional equation for the zeta function), and trace formulas à la Selberg. It bears mentioning that in the spirit of the other two chapters we also encounter such unexpected themes as microwave engineering, connections between theta functions and coding theory, the equation of Korteweg and de Vries, and modular knots, as well as a lot of very interesting material on finite analogues of the, for lack of a better word, classical material, such as Eisenstein series and Selberg’s trace formula.

As I already said, this is truly vintage Audrey Terras, and it all makes for a pleasurable and enlightening reading experience as well as a tantalizing one. The book contains a great deal of serious mathematics, geared toward displaying deep connections as well as established and fecund methods from both harmonic analysis (broadly understood) and number theory. And doing the exercises is a solid pedagogical experience that will inaugurate the novice as well as the interested “visitor” into active areas of research.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

Chapter 1 Flat Space. Fourier Analysis on \(\mathbf{R}^m\}
1.1 Distributions or Generalized Functions
1.2 Fourier Integrals
1.3 Fourier Series and the Poisson Summation Formula
1.4 Mellin Transforms, Epstein and Dedekind Zeta Functions
1.5 Finite Symmetric Spaces, Wavelets, Quasicrystals, Weyl
’s Criterion for Uniform Distribution

Chapter 2 A Compact Symmetric Space — The Sphere
2.1 Fourier Analysis on the Sphere
O(3) and \(\mathbf{R}^3\). The Radon Transform

Chapter 3 The Poincaré Upper Half-Plane
3.1 Hyperbolic Geometry
3.2 Harmonic Analysis on
3.3 Fundamental Domains for Discrete Subgroups \(\Gamma\) of \(G=\mathrm{SL}_2(\mathbf{R})\)
3.4 Modular of Automorphic FormsClassical
3.5 Automorphic Forms
Not So ClassicalMaass Waveforms
3.6 Modular Forms and Dirichlet Series. Hecke Theory and Generalizations