NOT Your High-School Geometry...

Book Review...

I briefly mentioned Shing-Tung Yau's new book, "The Shape of Inner Space" in previous posts (a book about the geometry underlying string theory), but have now read a review copy sent along by the publisher, and can say a little more about it.

In the last couple of decades many books about cosmology have been published (by physicists or science journalists) for a general audience. And many of those volumes have been quite good and accessible to non-professionals, even when the more technical material is not very comprehensible to the average-Joe. Yau's book is somewhat different, on several counts.

"A compact Kahler manifold with a vanishing first Chern class will admit a metric that is Ricci flat."

If the above sentence leaves you in a fog (or worse), I'm not sure this will be a book for you. Yau's book is filled with such language (every time I thought I was entering a more layman-friendly section, it would be short-lived, before I was once again in way over my head). If you are not very familiar with string theory, M-theory, Calabi-Yau manifolds, black holes, branes, and the like, you may want to pass on this volume; having a casual interest in physics/cosmology won't get you through it.
The book involves a lot of heavy-duty mathematics (moreso than other cosmology-type books for the masses; indeed some of it reads more like a textbook than a trade book). I actually enjoy the challenge of reading certain science material that is beyond my comprehension, but I suspect I'm an anomaly, and most folks don't have the patience for reading through material they simply don't understand. In short, I think there is probably a very limited audience for this offering, though that particular audience may very much relish it (...and I stand in awe of them!).

A little background: For any who don't know, Yau is a mathematician (not physicist) and Fields Medalist, with a focus on very advanced/abstract geometry, who's theories (especially proof of 'Calabi-Yau manifolds') came to underlie the mathematics of string theory. For those who don't follow such things, string theory is much more controversial now then when it was first introduced and there seemed to be an almost faddish bandwagon in its direction. Yau's book appears at a time when interest in string theory may even be waning, or at least taking a lot of heat. Yau is quite cognizant of the difficult road ahead for string theory, and how dominant views could change; at one point he writes:

"I personally think Calabi-Yau manifolds are the most elegant formulation [of the underlying geometry of the universe], as well as the most beautiful manifolds constructed so far among all the string vacua. But if the science leads us to some other kind of geometry, I'll willingly follow....
"Despite my affection for Calabi-Yau manifolds --- a fondness that has not diminished over the past thirty-some years --- I'm trying to maintain an open mind on the subject, keeping to the spirit of Mark Gross's earlier remark: 'We just want to know the answer.' If it turns out that non-Kahler manifolds are ultimately of greater value to string theory than Calabi-Yau manifolds, I'm OK with that. For these less-studied manifolds hold peculiar charms of their own. And I expect that upon further digging, I'll come to appreciate them even more."

Indeed one of the charms of this book is that while so many popular cosmology books beat the drum of the author's particular hardened point-of-view, Yau, as a mathematician, recognizes that he is somewhat apart from these physicist wrestling matches, and can step back, still offering his own personal leanings, while remaining more freely open to new conceptualizations than some other debaters seem to be.
The two chapters I most enjoyed (comprehended) came toward the end of the volume, "Truth, Beauty, and Mathematics," and "The End of Geometry?," where he waxes somewhat philosophical, even poetic, about the nature of mathematics/geometry and its interplay with physics, and also speculates about a future entailing what he terms 'quantum geometry'... but by then I was pretty tuckered out from the 280 pages that preceded those chapters!

If you are considering purchasing this book I would recommend that you read Peter Woit's review, AND the comments that follow it, here:

http://www.math.columbia.edu/~woit/wordpress/?p=3165

(You might also want to read the Wikipedia entry for "Calabi-Yau manifolds" to get a sense of whether or not you can follow this material.)

And I would also recommend that anyone choosing this volume initially read the 12-or-so pages of glossary at the back of the book just to familiarize yourself with many of the more heavily used terms ahead-of-time (unfortunately, many lesser-used, but difficult, terms are not included in the glossary).

One last ironic note: this book is published by "Basic Books"... one thing it is NOT though, is "basic!"
I don't doubt that it is an excellent exposition of its subject, but it is a challenging read to-be-sure. (In fairness to Basic let me say that I'm also currently reading another of their prior cosmology offerings, Frank Wilczek's "The Lightness of Being," from 2008, and enjoying it considerably more than the Yau volume.)

There is a lot more on Calabi-Yau manifolds around the internet, as well as many more reviews of this particular book available.