on 6 November 2013
This is a wonderful book. It is a voyage through mathematics, seen
and illustrated as an ethereal and universal body of knowledge (though
one which is capable of making the most precise and concrete predictions
about the world we live in) yet one which is at the same time very
much a human endeavour. Not in the sense that mathematical truths
are subjective, but in the sense that the circumstances that bring about
their discovery are very much steeped in human and historic circumstances.
Would so many brilliant Russian jews have devoted themselves to
Mathematics if discrimination had not closed so many other paths
for them? Would we in the West have known about the mathematics
they discovered if the Eastern block had not collapsed? These are,
in my mind fascinating questions raised by the book.
And yet the book is a lot more: it offers a glimpse of one of the most
advanced areas of present day mathematics (the Langlands program).
Will you close the book with a complete understanding of what that
program is? Probably not, but then again its ultimate implications are
still being worked out, and the book is not a scientific tract. It offers
a panoramic view of a magnificent landscape, a Rosetta stone of
Mathematics, and as such conveys what it is that mathematicians
do and what living mathematics is.
As a side comment, I am amused by the statement of one of the
reviewers that Frenkel's book "parades a constant reference to
the plight of Jewish students in Russia during the Stalin epoch".
The book begins in the mid eighties, more than thirty years after
Stalin's death... Perhaps the reviewer might give the book, or better
a history book a closer read? And even it Frenkel's book were an
"anti Stalin tract" (it isn't at all), would there be something wrong
in criticizing Stalin?
on 9 November 2013
This book combines a variety of unusual features which make it absolutely unique.
A distinctive feature of this book is that its author is one of the best mathematicians of our time. He combines his deep mathematical work with applying his time and effort to explain art of mathematics in different forms, ranging from movies to books.
He is a most gifted presenter who is able to explain mathematical concepts in a nice and understandable way. He does this very emotionally and this book exerts a deep impact.
The book contains a very sincere description of the author's path in mathematics, which is very dramatic. The narrative reads like a text of a movie.
The author convincingly writes about many ways how love is important for mathematical work and how mathematical discovery is similar in its nature to various forms of art.
Do not hesitate to order and read this book, you will be so much rewarded. Most likely, you will discover how much different is mathematics from what you thought it was. And then, give it to your friends and relatives to read.
on 15 January 2014
I read a lot of books about popular maths and this is one of the best I've ever read. Its a very human book - the author weaves a lot of autobiographical material into his account and some of this is very moving - particularly the sad story of how institutional anti-semitism prevented him from gaining entry as an undergraduate into Moscow University.
One of the main scientific themes of the book is an account of the "geometric Langlands programme" which is one of the most exciting areas of modern mathematics. This is tough stuff, but readers should stick with it because it is so fascinating, and you won't find a better non-technical account than Frenkel's. Broadly speaking, the original Langlands programme, which was proposed by Robert langlands in the 1960s, is a web of conjectures relating the theory of numbers to properties of certain curves. The recent solution of Fermat's last theorem by Andrew Wiles exploited exactly this sort of connection. Now the geometric Langlands programme takes this one stage further and says that there is a third piece of the puzzle which links numbers, curves and high-dimensional geometry. An amazing new twist to the story is that string theory, which has been much publicised recently as a "theory of everything" in fundamental physics, is also weaved into this web of relationships. So even if string theory turns out to be a disappointment in physics, it still has a major role to play in pure mathematics! Wouldn't it be wonderful if string theory could have something new to say about the properties of prime numbers? Well from what I've read in this book, that may not be such a far-fetched idea. Frenkel is an expert on this stuff. He writes about it with authority and passion, and has the rare gift of being able to write clearly for non-experts. This is a really inspiring read.
on 28 December 2013
This is a wonderful read. Frenkel's intelligent enthusiasm makes this a genuine page turner. He has a wonderful ability to distil and explain some fairly esoteric mathematics in a way that is accessible and illuminating.
I really highly recommend this book. Essentially a love letter to mathematics, that cannot but fail to win you over.
Really just buy it! You will not be disappointed.
on 16 December 2013
Like one or two other books I've read, I had to read this one twice to really enjoy it, as the first time through I did find it hard going, with 'sheaves' and the like, and not really sure where the book was going. I had almost decided to pass it on to Oxfam as soon as I finished!
But I hadn't really taken in the intro, where it was made clear that it would relate to advances in Quantum theory, which I have read a bit about, and once I got through to that late in the tale, I realised that the book was quite a masterly, ordered, presentation of how his maths experiences worked through to the culmination of explaining how the phsysics developed by Gell-Mann, etc., grew so naturally from the maths of previous generations (as was the case with Einstein).
So, I gave it a second go, and found the book then most interesting on several levels:
- his love of maths and physics, and why, and the fact that he made a film to try and popularise the subject;
- the problems of being a Russian citizen, together with the interesting comment that 1984 Russia was eerily similar to George Orwell's book of the same name (written in 1948 of course);
- his broad interests outside of maths;
- the interesting way (second time through!), the various chapters worked clearly and steadily towards a peak in his career, and logically towards the quantum physics work;
- some new maths concepts he introduced me to, including the way the Langlands Program (a principal maths project he was working on) was illuminating deep connections in quite separate areas of maths;
- comments on how 'pure' mathematics relates to the physical world, and whether it exists 'out there' or relates to our consciousness; a discussion I have come across several times, and always find intriguing, in spite of, or perhaps because of, the lack of any definitive answer!;
- the personal interest (not a major topic, but still quite touching) of and in his family, and his pride when they attended a seminar he was presenting at.
So, in the end definitely worth the time and effort.
on 4 January 2014
For me this is a book which doesn't really live up to it's billing. The dust jacket claims that it will dazzle you with its lucid explanation of the elegant beauty of mathematical principles and inspire in the reader the author's own passion for the subject. If you are expecting the illuminating narrative of a Simon Singh or a Brian Cox you will be disappointed. Inspiring wonder in others requires a writer to do more than repeatedly say "isn't this amazing". This is especially so since the subject matter covered - braid groupings, Galois Groups, Rieman functions etc is actually pretty dry and not really of much interest to the non-technical reader.
Whilst the autobiographical passages of the book are interesting and readable (if somewhat too self-congratulatory), this is not really a book for the mathematical novice at all. If you don't know your polynomials from your cubic equations you are likely to end up either switching off or skipping significant amounts of text
This is, comfortably, the best popular math book I’ve ever had the fortune to lay my eyes on.
I have a couple degrees in the subject. One in applied math, that I studied in college, and one in pure math that I got twelve years later. Indeed, I coincided with the author at Harvard, and his description of the Math department in the Science Center, the ping pong table and that hidden gem of a library brought back memories of my first semester in college, which was largely spent poring over impossible math assignments. (It also reminded me of the rather poor personal hygiene of some of the guys there, which goes unmentioned in the book!) This is now my favorite popular math book.
I don’t know math anymore, but I regardless LOVED LOVED LOVED reading this. The man has an unbelievable gift for explaining stuff and flattering you into thinking you understand it too. I can probably now blag about SO(3) and SU(2) and SU(3) and fibres and sheaves and fundamental groups with the best of them and I feel like I own the material (which of course I don’t.)
For three days of my life I did math with Grothendieck and Gelfand, Drinfeld and Kac, basically, I sat there at the table with them. It was a massive high!
A couple months ago, to resolve a dispute with an almost equally spent former mathematician, I wasted hours scouring the Internet to remind myself what the real meaning of the cross product is, to no avail. Frenkel slips the answer en passant on page 121: the tangent space to the unity element of a Lie group is a simpler construct called a Lie algebra. The Lie algebra of an n-dimensional Lie group is a vector space of dimension n, basically. Well, the operation under which 3D space is such an algebra is the cross product. Cool, eh?
And so on. You close your eyes and you follow this boy, who grew up as a little dreamer in a city two hours out of Moscow and just wanted to know how the universe works. And you meet his teachers and take part in his search as he tries to solve the problem that consumed Einstein in the last 20 years of his life, that of coming up with a single theory that can reach beyond the “standard model” and also cover gravitation. You start with his first ever big triumph (something to do with the “knot groups,” which are explained beautifully) and you move on from there to the most intuitive ever explanation of Galois’ insight regarding the solution of polynomials (basically drawing an analogy between irrational numbers and imaginary numbers I wish somebody had told me about 30 years ago) and eventually you make it to the “Langlands problem” which I must admit went 101% over my head.
But I had enough there to cheer the author on, all the way.
The bit about the “equation of love” was beyond weird, of course, but I’m alright with it. Nobody forced me to read it. It was a bit as if somebody cooks you the best meal in history and then offers you the option of flamingo feather pie as desert: you don’t have to eat it. It did not undo the amazing first 228 pages, or that phenomenal set of notes in the back.
Also, the guy loves math. It’s so evident, it’s infectious. And he gives you a very long warning that you cannot judge math from the abject crap that passes for math in the world’s high school curricula. He hated that too, despite the fact that he could do it. Math is the amazing construct that, whether we like it or not, surrounds us, underpins the physical world and is waiting for us to go, discover it and get the answers we seek.
Chris Isham aside (who does not know me from Adam, but I once took his class), this is the most a man has ever inspired me to read math, and he did it from a book. What can I say?
I’m not sure I’ve had more fun reading a book, frankly. Did not really learn anything, it would be beyond presumptuous to say so, but It flattered me so hard, I got a proper high.
Thank you, Bernard, for suggesting this!
on 23 January 2015
Frenkel’s book deserves a much wider audience this side of the Atlantic than it currently has.
I read Love and Maths alongside Hacking’s “Why is there a Philosophy of Maths at all”, and at least in the first few days, I persevered with the dual task until I eventually focused on Frenkel almost exclusively. I’ve now read the book twice and must admit to having been moved from being merely entertained to being seriously impressed.
The title and the first few largely biographical chapters are, in truth, slightly misleading. Once Frenkel gets going, however, he impresses as a serious writer with no quarter given for the less than serious reader. But the “cost”, as can happen so often, is not clarity of accessibility. This last point is interesting. The way that Frenkel ensures no compromise is by providing, in the main body of the book, a fairly low lying terrain. Think of this as a strenuous but ultimately achievable trek up Kilamanjaro. Yes, you have to be able to breath the thin air, but there are no serious 5:5 stretches. You don’t need your climbing rope and crampons or ice pick. However, laced, literally page by page, are footnotes that are more like the diversion to K2. And what an amazing diversion. You can (if you want) be seriously addressed with credible mathematical discourses, and yet stay the course if you feel threatened. Its a smart way of delivering a superb read.
Leaving aside this very clever mechanism, the heart of the book brings together a beautifully crafted exposition of the importance of the Langlands programme with a topical weft and weave of contemporary maths.
A few months after I read the book a second time I had the chance to meet Frenkel very briefly. He is a disarming and charming man with a steely eyed determination to convey his feelings. The book is the same. Don’t be fooled by the title, and don’t give up in the foothills. The peaks are what count. You will be infected by the love of the subject that so many of us wish we could share with loads more people. I just hope the book gets more of an audience in Britain.
Frenkel also wrote the obit for Grothendieck in the NY Times. Well worth the read for those who dont know a mathematician who may well deserve the title of the 20th century's greatest (and yes, I include Godel, Hilbert and Weyl et al in that comparison).
on 15 April 2015
The author is a professional mathematician and describes what it means to do mathematical research. There are some real gems in this book such as:
Which functions (Euler phi function) are used to encrypt credit card numbers.
Which is larger 2/3 or 3/5 – Most people know that 2 bottles of vodka for 3 people is better than 3 bottles for 5 people.
How we can obtain the Fibonacci sequence from its generating function.
What does a finite field mean.
There is also a good explanation of the Langlands Program.
Very few typos (the only two I picked up were both on page 85 - should be divisible not visible and penultimate paragraph should start with ‘As we’.)
It is really pleasing to see that the author does not shy away from the mathematics in his writing.
However this is not a book for the layman because the mathematics is totally inaccessible to general audience. This is not a piece of writing for a popular audience as I mistakenly believed it was after reading a review before purchasing. To fully appreciate this book you should at least be an undergraduate in mathematics or physics as it is tough going in places.
Throughout the book the author has highlighted his personal struggles of being Jewish with the regime of the Soviet Union. This is really interesting as I was unaware of how the regime thought of all Jews as opponents, criminals, foes (these are my words). When depicting this the author mentions a number of locations in Russia which he should have illustrated with maps as most of us in the West will not be able to visualize the locations.
Additionally I have a few minor quibbles:
There are a number of terms omitted from the glossary such as invariant, winding number.
Should have explained the term monodromy through illustrations.
The author claims ‘It is customary to exclude 1 from this list” (of primes). I always thought of 1 as neither prime nor composite, just a unit.
I conclude that this is not a book for the layman.
Kuldeep Singh 15th April 2015
on 18 October 2015
This was disappointing. It's a combination of an autobiography, which is quite interesting, especially his early mistreatment during Communist rule, and an awful attempt at presenting the mathematics he specialises in. Unless you already understand the field in which the author specialises, you will not gain anything from this, and presumably if you do understand it, there will be no need to read this book. There's no structure, no order, no flow. Sometimes concepts are referenced before being explained or even introduced. Long passages are impenetrable to anyone who isn't an expert. I really don't know who this is for. I had hoped it would convey some of the joy of mastering a complex subject such as mathematics and revealing its inner beauty, but while this may be its intention, it doesn't succeed.