Edward Frenkel’s new book Love and Math is now out. It’s a must-read for those who share the interests of this blogger, so go get a copy now.
The “Love” of the title is much more about love of mathematics than love of another person, as Frenkel provides a detailed story of what it is like to fall in love with mathematics, then pursue this deeply, ending up doing mathematics at the highest level. Along the way, there are lots of different things going on in the book, all of them quite interesting.
A large part of the book is basically a memoir, recounting Frenkel’s eventful career, which began in a small city in the former Soviet Union. He explains how he fell in love with mathematics, his struggles with the grotesque anti-Semitism of the Soviet system of that time (this chapter of the story was published earlier, available here), his experiences with Gelfand and others, and how he came to the US and ended up beginning a successful academic career in the West at Harvard. I remember fairly well the upheaval in the mathematics research community of that era, as the collapse of the Soviet system brought a flood of brilliant mathematicians from Russia to the West. It’s fascinating to read Frenkel’s account of what that all looked like from the other side.
Russia at the time had a vibrant mathematical culture, but one isolated from and quite different than that of the West. Many of its most talented members had rather marginal positions in official academia, and their community was driven much more by a passion for the subject than any sort of careerism. Frenkel comes out of this background with that passion intact, and it shines throughout his book. In some other ways though, he’s more American and less Russian than just about anyone I know. Part of the Russian mathematical culture has sometimes included a certain cynicism and vision of great mathematics as an esoteric subject best closed to outsiders, with little interest in communication with the non-initiated. I confess to a personal sympathy with the cynicism part (as any reader of this blog has probably figured out) but no sympathy for obscurantism about mathematics research.
Frenkel’s sunny optimism and cheerful enthusiasm for his subject and life in general is very American, and in his writing he often gets through to melt the cynical part of this reader. What’s really wonderful though is his dedication to the cause of the opposite of obscurantism, that of doing the hard work of trying to explain mathematical insights to as wide an audience as possible. His book is packed with mathematics and physics, full of enlightening explanations of difficult topics at all different levels of mathematical sophistication.
Perhaps the most remarkable part of the book though is the way it makes a serious attempt to tackle the problem of explaining one of the deepest sets of ideas in mathematics, those which go under the name of the “Langlands program”. These ideas have fascinated me for years, and much of what I have learned about them has come from reading some of Frenkel’s great expository articles on the subject. To anyone who wants to learn more about this subject, the best advice for how to proceed is to read the overview in “Love and Math” (which you likely won’t fully understand, but which will give you a general picture and glimpses of what is really going on), and then try reading some of his more technical surveys (e.g. here, here and here).
The Langlands story is a complex one, but it starts with a very deep and beautiful idea that brings together different parts of mathematics: one way to think about number theory is to think of rational numbers as rational functions on a space, the space of primes. One then ends up seeing all sorts of parallels between the study of Riemann surfaces and number theory. Frenkel explains this in detail, including André Weil’s description of a “Rosetta stone”, a translation between aspects of number theory, aspects of Riemann surface theory, and yet a third intermediate parallel theory, that of algebraic curves over a finite field.
He goes on to explain the subject of “geometric Langlands theory”, the transposition of the Langlands program from the number theory to the Riemann surface case, creating a whole new area of mathematics, one with deep connections to quantum field theory. The book includes extensive discussion of discoveries by Witten and others linking duality in four-dimensional quantum field theory to the fundamental mysterious Langlands duality in the geometric Langlands case. Frenkel has been in the middle of these developments and is the ideal person to tell this story.
The connection between these ideas and two-dimensional quantum field theory seems to me to be a subject for which we have so far only seen the tip of an iceberg, with much more to come in the future. One part of this that I don’t think Frenkel discusses is early work by Witten (before geometric Langlands was formulated) giving explicit analogies between 2d qft and reciprocity laws in number theory. For more about this, see Witten’s 1988 Quantum field theory, Grassmanians and algebraic curves, or a more recent paper by Takhtajan. Working on writing up the material about the harmonic oscillator and representation theory from my last year’s course has gotten me interested again in the number-theoretical version of that particular story. Unfortunately I don’t know a really readable reference, hope some day to write something myself once I have a better understanding of the subject.
So, I heartily recommend this book to all with an interest in mathematics or its relation to physics. If the “Love” of the title has you hoping for a tale of romance between two people, you’re going to be disappointed, but you will find something much more unusual, a memoir of the romance of mathematics and its relation to the physical world.
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