Mathematical Perspectives on Theoretical Physics: A Journey from Black Holes to Superstrings

This is a huge book (over 800 pages) on a theme that could easily take up many, many more pages. As Prakash conveys in the book's preface, her goal is to present to the reader a self-contained means whereby to get to "know the basics of string and superstring theories, just as [we know] calculus, linear algebra, geometry and analysis." This evidently prodigiously diligent and energetic reader will then be provided with "the elements of all the prerequisites" for attacking the aforementioned string and superstring theories equipped with "an overview of other great theories that have preceded it" as well as "a motivational thread to reach the end goal." The question is, of course, whether Prakash succeeds in her ambitious undertaking.

But this question is an unfair one: the titanic size of the subject she is dealing with, coupled with its youth (and early evolutionary stage) and sophistication, render it intrinsically far less accessible than the branches of mathematics Prakash singled out for comparison. Preliminaries and prerequisites to calculus, linear algebra, geometry and analysis are by no means as broad nor as deep as what one requires for string theory and can be taken for granted; the same cannot be said for the material with which Prakash concerns herself. Her stated attempt to include all the indicated mathematical prerequisites for these avant garde topics in current physics between the covers of a single book reminds this reviewer of trying to force the classification of finite groups, soup to nuts, into one volume. And Grothendieck's magnum opus comes to mind, too: his Seminaire de Geometrie Algebrique notes come to over 6000 pages and are characterized as only the beginning (in a sense, at least). Is it not a priori impossible to do justice to subjects this huge (and novel) in works which succeed as textbooks or self-study books? Do such herculean efforts not almost always end up as reference-books? I think so. And this is not a problem: these sources are hugely valuable even though a cover-to-cover reading is too daunting.

I would consider Prakash' book in that light. Its chapters are many, long, and dense, but the individual sections are short and well-constructed. The exercises (Yes, there are exercises!) are generally speaking accessible and hints are provided. So I do recommend the book. But be forwarned, even when browsed in quanta (i.e. in discrete packets) it is not for the timid.

Michael Berg (mberg@lmu.edu) is professor of mathematics at Loyola Marymount University in Los Angeles, CA. His research interests are algebraic number theory and non-archimedian Fourier analysis.