Love and Math

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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21 Responses to Love and Math

  1. His last name is Frenkel, not Frankel (you misspell it 3 times in the post).

  2. Paulo Guerra says:

    “Frenkel’s sunny optimism and cheerful enthusiasm for his subject and life in general is very American…” (!!!)

    No comments.

  3. Peter Woit says:

    Paulo Guerra,

    OK, I probably should have said “used to be very American”, things have changed somewhat in this country in recent decades…

  4. Michael Welford says:

    Amazon lets you peek at the end material for this book!!! It’s not so frighteningly advanced as you might suppose from Peters talk about ‘geometric Langlands’ and ‘Riemann curves’.

    If you get straight A’s in calculus while barely opening your textbook, then this book looks like a good book to measure yourself against. See how many chapters you can get through before you finally bog down. Of course if you describe yourself as, for instance, a Frenkel-chapter-12 you’d better be prepared to back it up, because the math nerds will be gunning for you.

    I’ll be waiting for a paperback edition.

  5. Anonymous says:

    Hard to digest… I thought that geometric Langlands is one of the most esoteric subjects closed to outsiders?

  6. Peter Woit says:

    Anonymous,
    Yes, geometric Langlands is one of the more difficult areas of mathematics to understand, because it brings together some very deep and not completely understood ideas. Frenkel’s more technical expository papers are by among the most accessible things to read, but there is no royal road, this subject is very demanding. That he’s trying to explain it at a popular book level is a very remarkable feat.

  7. Anonymous says:

    Many fields of mathematics have deep ideas and difficult problems. The reasons why some areas are considered more deep and difficult are purely psychological and cultural. Some of these areas close themselves to outsiders to perpetuate the myth of greatness. Just like string theorists, mathematicians working in these areas enjoy getting all the admiration. This includes prestigious awards, positions in top schools and publications in the best journals for making even the slightest progress on some famous problem. Brilliant mathematicians working in these areas are no more brilliant than top mathematicians in many other areas who unfortunately are often judged by a different standard.

  8. Armin Nikkhah Shirazi says:

    Peter,

    Based on Paulo Guerra’s name (which does not sound typically American) I surmise that his “no comments” comment was a subtle criticism of the implication of your claim that “sunny optimism and cheerful enthusiasm” exclusively characterize American culture, not this was once the case and may not be so any longer.

    Thank you for the book review, though.

  9. Ponder Anew says:

    “…The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena. That it may completely fulfil this high mission, may the new century bring it gifted masters and many zealous and enthusiastic disciples!” –David Hilbert 8.8.1900.

    I came to mathematics late in life but I have certainly developed a love affair with it; it’s seeing the problem all the way through, the clarity of vision that oftentimes seems to take eons to obtain, which is so addictive – similar to those bursts of creative insight which lead to novelty in so many fields! This is the first time I’ve encountered this “geometric Langlands” entity and it sounds quite fascinating; I wonder, is it the solution to the enigma regarding the unity of mathematics and the intimate connection between mathematics and physics? Of course, to date, I don’t have the knowledge base required to tackle such a book but perhaps one day the Love will lead me there . . .

  10. Peter Woit says:

    Armin,

    A bad idea for me to be trafficking in national stereotypes at all, of course they are of very limited significance and dubious value, but, in any case, plenty of other national stereotypes include even more optimism than that of Americans. In terms of stereotypes, from IP address you’re right, but these days “Paulo Guerra” is a pretty American name…

  11. Peter Woit says:

    Ponder Anew,

    Frenkel is known for referring to the Langlands story as a “Grand Unified Theory of mathematics”, with some justice. The connections to physics make the unification theme even more impressive.

  12. Mathematician says:

    Thanks for the review. I’ve ordered the book.

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  14. anon says:

    Grand unified theory of nothing. The last time this Frankel quote came up on this blog, David Ben-Zvi wrote (July 24, 2008): ‘Of course Frenkel’s comment on the “grand unified theory” is tongue in cheek, but in any case he was referring to the
    entire Langlands program, not just to its geometric aspects.’

  15. Chris Austin says:

    If Frenkel thinks he can “popularize” maths by making a movie that puts forward the worse-than-silly propositions that there exists a “formula for love”, and that after discovering this “formula”, he has no alternative but to kill himself, why should anyone think there is anything more to his “deep” maths than the emperor’s new clothes?

  16. uair01 says:

    Michael Welford says:
    Of course if you describe yourself as, for instance, a Frenkel-chapter-12 you’d better be prepared to back it up, because the math nerds will be gunning for you.

    I wonder where this book lies between:
    1) The “Road to reality” where I got to chapter 10 and from there I had to read “in between the equations” (just as Mr. Penrose suggests).
    2) This interesting number theory book (Fearless symmetry) which I read in its entirety.

    I still wonder how many people buy those books. They’re too difficult for the common public but are (I assume) too trivial for the specialists. But it’s very sympathetic of the authors to present “unadulterated” science.

    One author (forgot name) once wrote that: “every extra formula halves the possible audience”.

    1) http://www.amazon.com/The-Road-Reality-Complete-Universe/dp/0679776311/ref=sr_1_1?ie=UTF8&qid=1379850599&sr=8-1&keywords=road+to+reality
    2) http://www.amazon.com/Fearless-Symmetry-Exposing-Patterns-ebook/dp/B005K46YVU/ref=sr_1_1?ie=UTF8&qid=1379850665&sr=8-1&keywords=fearless+symmetry

  17. Peter Woit says:

    Chris Austin,
    One more reason for reading the book is that there’s a chapter about the film and what he was trying to do with it. One can like, dislike, not care, about his film (for options other than “not care”, you might want to read what he has to say), but I don’t see how this has anything to do with the issues about mathematics. I don’t see any reason why doing great mathematical work should correlate at all with whether one’s art-film project is wonderful or misguided.

    uair01,
    The book is not like Penrose’s, which was a graduate level textbook disguised in the cover of a popular book. It’s much closer to “Fearless Symmetry”, which covers related mathematics, at a similar level. The difference is that Frenkel’s book is not written at all at a specific level aimed at an audience with a certain background. Much of the book is a memoir that could be enjoyed by anyone with zero knowledge of math and zero interest in learning more. Of the rest, a lot is explanations of mathematical ideas at a popular level, with no equations. In parts though, he’s getting into ideas that it’s just not possible for people without a lot of background to follow. Those will be understandable by people with some sophisticated math training, but for the average person will just be a glimpse at something they’re not going to really understand (but such a glimpse might give them some insight into what math at this level is like).

  18. Shantanu says:

    Peter, OT.
    Nobel prize annoucement in 2 weeks. Any guesses?
    Will it be awarded for Higgs discovery?

  19. Marco Masi says:

    I’m no longer sure that falling in love with math is healthy for everyone. It is fine for mathematicians, but for physicists it can be dangerous. I once used to be it too until I realized that it distracted me from understanding better the foundations of physics and the deeper meaning of some of its principles. In a certain sense I believe this is one of the causes of today’s failure in theoretical physics to go beyond the SM.

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