The algebraic theory of group representations and the analytical theory of spectral analysis come together in the study of harmonic analysis. Symmetries and Laplacians provides a broad introduction to this elegant subject, weaving together many threads, creating a colorful and eclectic tapestry of beautiful mathematics.
A Dover edition of the original from 1992, the new printing unfortunately retains all of its original typos. It is somewhat amusing to read in the Introduction about how the author used a typesetting method which was rather revolutionary for its times: an actual word processor for the text and the good old PaintBrush of MS Windows for the images.
The reader who is able to move beyond the old-fashioned type and is willing to let go of the obsessive need for grammatically correct sentences will find a beautiful, if somewhat rough, diamond here. Leaving aside the typos, there are a couple instances where further clarification of hypotheses of statements is needed to make them correct. A good copyeditor, even a mediocre one, would have made this into a most wonderful text. The author seems to have moved on to greener pastures, changed his research focus, and perhaps lost interest.
Still, we have something quite good here. The book captures in a snapshot what a comprehensive set of lectures on the harmonic analysis would contain. So we can enjoy and benefit from reading it and still wish that it had gone through a full revision, or at least a careful sprinkling of definite and indefinite articles in the proper places.
We now pass beyond the superficial, and move to a discussion of the contents of the book. Gurarie begins with a standard, fast-paced introduction to basic representation theory. Regular and induced representations, irreducibility, decomposition, the relationship between Lie groups and Lie algebras, Haar measure, all of this and some more are covered in the first chapter, in approximately fifty pages. The second chapter, titled Commutative Harmonic Analysis, introduces Fourier series and Fourier transforms. After looking at n-dimensional Euclidean space, the n-dimensional torus, and Zn, we learn some of the basic properties and applications of classical Fourier analysis. The Laplacian is introduced for the first time in the text and studied in this context.
Next follow chapters on the representation theory of finite and compact groups. Chapter 3 lays the foundations of this theory by introducing Peter-Weyl theory, Frobenius reciprocity, and semi-direct products. Lie algebras and representations of SU(2) and SO(3) are studied in detail in Chapter 4. The classical theory is further discussed in Chapter 5 where we learn about simple and semisimple Lie algebras, Weyl’s unitary trick, root systems, highest weight representations, Young tableaux, Haar measure on compact semisimple Lie groups, and a lot more. This all happens within sixty-five pages; readers will run out of breath if they are not prepared for such a fast pace.
In Chapter 6 we see various instances of the semi-direct product construction, this time focusing on the infinite case. Representations of the Euclidean motion group, affine motion group and the Poincaré group are studied in detail. The Heisenberg group gets its own subsection, and the chapter is wrapped up with a brief discussion of Kirillov’s orbit method.
Next we study SL(2), in Chapter 7. Its representation theory is described in detail, as a fundamental example of a non-compact group. The eighth and final chapter of the book is somewhat different in flavor. It focuses solely on how Lie theory and hamiltonian mechanics are interrelated. Three appendices (on self-adjoint operators, integral operators and Riemannian geometry) conclude the book.
Gurarie’s book is dense with great examples, ripe with motivational discussions and almost musical in its structure. The material which is more or less standard and can be found in several other texts is interlaced with special sections focusing on the Laplacian, its spectral analysis and applications. It is a recurring theme, tying the book together. We learn about the Laplacian in the commutative setting (Section 2.4), on the platonic solids (Section 3.2), on the n-sphere (Section 4.3), on symmetric space (Section 5.7), and on hyperbolic surfaces (Section 7.6).
It must be obvious to readers of this review by now that Gurarie’s book contains a lot of beautiful mathematics. In it is material that can be used for a year-long graduate course, assuming the instructor will fill in some of the technical details and provide some of the missing proofs. A motivated person can read the book alone, though a complete novice in the field might find it a bit hard to follow at times. Overall it is a solid text that does a whole lot. It goes a long way to bringing its reader up to speed with the concepts and constructions of classical and modern harmonic analysis.
Gizem Karaali is Assistant Professor of Mathematics at Pomona College.