By its very title, Groups and Analysis: The Legacy of Hermann Weyl, edited by Katrin Tent, holds out tremendous promise. After all, Hermann Weyl was one of the most ecumenical of twentieth century mathematicians, true to the example set him by Hilbert, his Göttingen advisor. Weyl’s contributions to mathematics are accordingly exceptionally diverse (and very, very deep), and the thirteen sections of the book under review reflect as much, ranging from harmonic analysis and spectral theory to index theory and Lie theory (i.e. affine buildings).
The quality of the articles is high: the promise of the title is met, and the material is presented at a very accessible level, but without courting triviality. With the material in the book being uncommonly informative and fascinating, I should nonetheless particularly like to draw attention to the following essays first (apologizing for my own parochial views): Chapters 1–5 concern symmetric spaces, eigenfunction expansions in such a setting, advanced Sturm-Liouville material, and then quantization and index theory and stuff on the Heisenberg Lapalacian. Kismet: these are all topics of great interest to me in my own work; for me the book is a Godsend.
But this material is obviously perched to lead anyone into the indicated area of (P)DE on more or less exotic (e.g. symmetric) spaces, eigenfunction analysis (the sine qua non for this sort of thing, after Hilbert’s own trailblazing work — with Weyl on the premises: see Reid’s beautiful biography, Hilbert), and then echoes of quantum mechanics (Heisenberg’s Nachlass). The fourth chapter, titled “From Weyl quantization to modern algebraic index theory,” is especially spellbinding. Along these lines I should also include Chapter 8, which deals with PDE: the regularity of solutions to second order elliptic equations.
Beyond this, Chapters 7 and 9, having to do with, respectively, automorphic forms and arithmetic geometry, are of particular interest to number theorists, of course. And then Chapters 10, 11, 12 turn in the direction of algebraic groups, representation theory, and Lie theory, if I may be forgiven a gross oversimplification of what is really going on here. Chapter 6 is about quadratic differentials (Weyl’s work in differential geometry is legendary, of course, being concerned in large part with both the theory of Riemann surfaces, as such, where Weyl was apparently the first to give a proper modern definition in his classic The Idea of a Riemann Surface, and with Einstein’s general relativity.
Chapter 13, tilted “Emmy Noether and Hermann Weyl,” is undoubtedly the article any one with an ounce of mathematical Kultur will read first, and, to be sure, it does not disappoint. For one thing it contains Weyl’s funeral speech over Noether’s grave in toto. In fact, the whole chapter is full of very moving material.
Finally, a word about the London Mathematical Society Lecture Note Series, in which Groups and Analysis is the 354th entry. My own favorite is no. 34, Representation Theory of Lie Groups, featuring Atiyah, Bott, Helgason, Kazhdan, Kostant, Lusztig, MacDonald, Mackey, Schmid, and Simms. In graduate school, in connection with my doctoral dissertation, I had to learn a large chunk of representation theory “of the right sort” as effectively and quickly as possible: LMS no. 34 did the job — like nothing before or since, really. I suspect that no. 354 shares this exceptional and laudable feature!
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.