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Selected Expository Works of Shing-Tung Yau with Commentary, Volume I

Lizhen Ji, Peter Li, Kefend Liu, and Richard Schoen, editors
International Press
Publication Date: 
Number of Pages: 
Advanced Lectures in Mathematics
[Reviewed by
Michael Berg
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This selecta of articles by S. T. Yau, one of the premier geometers of our age, is a treasure trove not just for geometers but for all who have an interest in how geometry fits in modern mathematics in a greater sense. The articles in this two-volume collection are presented as expository, even though they are generally pitched at a high level and contain serious mathematics, including a lot of theorems and proofs. Thus, these articles are not accessible to the average non-specialist and a non-geometer coming to these papers had better evince a keen interest in the field, more than merely tangential, and he should come to the game with some genuine experience in place.

To give an indication of what we’re dealing with, here is an almost random set of six titles: from volume I, “Estimates of eigenvalues of a compact Riemannian manifold,” “A review of complex differential geometry,” and “Geometric aspects of the moduli space of Riemann surfaces,” and from volume II, “A survey of Calabi-Yau manifolds,” “String theory and the geometry of the universe’s hidden dimensions,” and “Mirror symmetry and localizations.” Many of these articles, including some of the preceding six, have co-authors, and all of them come equipped with a preceding commentary section by Yau himself. This feature ads weight to the characterization that these offerings are expository: the reader is treated to Yau’s own insights and evaluations of the papers in question, their greater purpose and role, their place in the scheme of things, and, en passant, the larger geometric picture — obviously the latter has to be understood in the broad modern sense, with analysis and topology on the scene, and also physics of the hypermodern sort. Yau’s commentary is rich with personal asides and, for lack of a better word, social commentary, and this gives a clear idea of Yau’s role as a major statesman of geometry in the broadest sense. For instance, we learn of Yau’s contact with Stephen Hawking, and get a glimpse of the marvelous interplay between geometry and physics in the hands of two of its greatest modern proponents.

Each volume sports an explanation of the set of books’ raison d’etre in the form of an introduction by one of the editors, Lizhen Ji, as well as a photo gallery and Yau’s daunting curriculum vitae. After this it’s off to the races.

Obviously it’s invaluable to have such a detailed and ramified presentation of so much of modern geometry, in its broadest sense, by a grandmaster, replete with his own meditations of his subject and his work, what he has done and where he proposes to go, and indeed what has happened in differential geometry over the last so many decades and what the future holds. The only other work I can think of that possesses a corresponding quality of a masterful insider’s view of a large swath of mathematics is the Grothendieck-Serre correspondence, and there the objective is implicit: in the present case Yau is explicitly involved in evaluation and prognostication. It’s a unique work, and a very important one. 

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