The heat equation is magic. It is a staple of any introductory course in partial differential equations, for instance the ubiquitous course on “methods of applied mathematics” taught in possibly every mathematics department in the world — or nearly so. The heat equation serves as a canvas on which to depict such powerful techniques as separation of variables, integral transforms (especially Laplace’s), and Fourier series, and it presents us with the Gaussian density, a.k.a. the Gaussian kernel, a.k.a. the heat kernel, whose utility straddles a host of mathematical areas including hard analysis (and PDE, of course), mathematical physics (obviously), and even the theory of numbers. Indeed, Gaussian kernels are essentially the summands of theta series (which, *via *superposition, solve the heat equation subject to certain boundary conditions) and we are immediately face to face with half integral weight modular forms. Once theta functions are on the scene, so are L-functions, really, the prototypical case arising in Riemann’s second derivation of his functional equation for the zeta function in *Über die Anzahl der Primzahlen unter einer gegebenen Grösse*, the most important single paper in analytic number theory (and the only paper Riemann ever wrote in that field!). This having been said, let’s add to the mix that Schrödinger’s wave equation is in the form of the heat equation and that the most prevalent distribution of all, the Gaussian one, is of course given in terms of the Gaussian kernel, i.e. the heat kernel,

So it should come as no surprise that with geometric analysis currently being all the rage, the heat kernel is also featured prominently in this context. And we accordingly find on the back cover of *Heat Kernel and Analysis on Manifolds* the following description: “The heat kernel has long been an essential tool in both classical and modern mathematics but has become especially important in geometric analysis as a result of major innovations beginning in the 1970s [particularly with Yau’s work] … This book is a comprehensive introduction to heat kernel techniques in the setting of Riemannian manifolds, which inevitably involves analysis of the Laplace-Beltrami operator and the associated heat equation.” Beautiful.

And the book under review is indeed a wonderful source for this material. The author’s approach is both broad and deep, and extremely thorough and rigorous. In Grigor’yan’s own words: “The purpose of this book is to provide an accessible for graduate students introduction to the geometric geometric analysis of the Laplace operator and the heat equation, which would bridge the gap between the foundations of the subject and the current research … [focused] on the following[:]

[l]ocal geometric background … [s]pectral-theoretic properties … Markovian properties and maximum principles … [s]moothness properties … [and g]lobal geometric aspects.”

Accordingly, starting with classical material (the Green formula occurs on page 4), Grigor’yan deals out a rich load of hard analysis in short order (weak derivatives and Sobolev spaces occur on page 34, for example) and then proceeds to perhaps the heart of the matter: we encounter regularity theory in Chapter 6, spectral analysis in Chapter 10, advanced material on Green’s functions and Green’s operators (with functional analysis now entering into the game in a bigger way) in Chapter 13, then hard analysis is added in Chapter 14 (inequalities galore: Faber-Krahn, Nash), and finally pointwise Gaussian estimates in Chapters 15 and 16.

Grigor’yan’s prose is clear and tight, and the book also contains over 400 exercises.

*Heat Kernel and Analysis on Manifolds *is poised to be an important book in the field and a valuable pedagogical contribution. It is bound to be an important player on the scene for years to come.

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