In Love... With Math (Frenkel's new book)

Mini-review of Edward Frenkel's "Love and Math":

   It took me two-and-a-half days to read Martin Gardner's autobiography and jot down enough notes to write a lengthy review of it. I purchased Edward Frenkel's new, rich volume, "Love and Math: The Heart of Hidden Reality," a few days later and took over three weeks to finish it. I won't do a detailed review simply because there's so much of it I don't grasp well enough! That might sound like a negative…but in this case it ISN'T; in fact, it's a glowing positive… I love the book, from its Vincent van Gogh cover to its endnotes and "glossary of terms" (even though there is much in-between I don't yet comprehend)!

Let me explain: I once commented to a friend that most people seem to enjoy attending talks/lectures where they understand (or agree with) everything that is being said… I find those b-b-boring! Rather, I enjoy going to talks/lectures that are way over my head (or offer viewpoints I'd never considered), and being challenged to pull out of the air whatever bits of new learning I can draw from them… a talk that is 80% incomprehensible to me, but that I learn new ideas from, and stretch my mind from the other 10-20%, is a great, almost exhilarating, talk to me. THAT'S the sort of feeling I get reading Frenkel's new volume.

I've seen references on math and physics blogs for awhile now to the "Langlands Program," but without getting much sense of what it is. Frenkel's book covers a lot of ground, but with a primary purpose of elucidating to a general audience the Langlands Program (his specialty), and why it is so important.
The book is an odd mix of personal history and introduction to real and advanced mathematical ideas. The early chapters are interesting and important foundationally, but the volume really takes off on its mathematical excursion with chapter 9 on Andre' Weil's "Rosetta Stone" of the connection between number theory and geometry through the three "parallel tracks" of "number theory," "curves over finite fields," and "Riemann surfaces." From there on, lots of discussion of manifolds, sheaves, Kac-Moody algebra, Lie groups, gauge theory, SO(3) groups.... latter chapters becoming increasingly difficult if you haven't fully fathomed earlier chapters…. heavy-duty stuff for the average person who finds this book lying next to Tom Clancy or Barbara Kingsolver on a table at their local bookstore!!

Here's an example of the sort of content you can expect along the way:

"As we established in the previous chapter (see the diagram on p. 161), in the version of the Langlands relation that plays out in this column, the cast of characters has 'automorphic sheaves' in the role of automorphic functions (or automorphic representations) associated to a Lie group G. It turns out that these automorphic sheaves 'live' in a certain space attached to a Riemann surface X and the group G, called the moduli space of G-bundles on X. It's not important to us at the moment what it is."

Got that!? Me neither, but I have fun trying. Frenkel ardently tries to walk the reader through many ideas and how they tie together, and to do so at a level that a lay person can follow. His own fervent passion for his subject exudes off almost every page.

Frenkel's life-story, which is embedded in the volume, is itself fascinating, from growing up and being educated in the Soviet Union to his current position at UC Berkeley, but the mathematical portions are clearly not a fast-read… much of the book is a slow-savor-contemplate-re-read endeavor… BUT, worth it! I plan to re-read parts of this volume several times. The final chapter covers Frenkel's award-winning short film "Rites of Love and Math" quite a different topic, but at least a more reader-comprehensible narrative to end the book with. Prior to that he summarizes the volume this way:

"The Langlands Program has been the focus of this book. I think it provides a good panoramic view of modern mathematics: its deep conceptual structure, groundbreaking insights, tantalizing conjectures, profound theorems, and unexpected connections between different fields. It also illustrates the intricate links between math and physics and the mutually enriching dialogue between these two subjects. Thus, the Langlands Program exemplifies the four qualities of mathematical theories that we discussed in Chapter 2: universality, objectivity, endurance, and relevance to the physical world."

Last year I commented that one of the many things I liked about Paul Lockhart's book Measurement was that the author made no pretense that math is easy… he warned readers at the start to be prepared to really slow down and think while proceeding through the book… that parts would be tough-going. Too many popular math books hype themselves as volumes that will finally make you enjoy or connect with math… when it just ain't so. I still believe that Steven Strogatz's book, "The Joy of X" is one of the ONLY books that actually achieves such a goal of wide accessibility to non-mathy readers.
Frenkel likewise falls short of this noble goal, BUT what I love about it is his utter sincerity and hard work in wishing/striving to reach the masses. One can tell by the way the sentences and paragraphs are crafted that he (or a good editor ;-) is truly attempting to make plain to a general audience very, verrrry deep, rich, often labyrinthian mathematical ideas. The old Dr. Seuss adage says, "Don't be sad that it's over, be happy that it happened!" Well, I would say of Frenkel's volume, 'Don't be unhappy that he fails to make everything crystal clear, be thrilled that he's made such an arduous effort!'

And here are some other Web reviews of the book:

http://tinyurl.com/muungoj

http://www.math.columbia.edu/~woit/wordpress/?p=6266