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Keith Devlin

The Millennium problems

To celebrate the coming of the new millennium, the Clay Mathematics Institute offered a $1,000,000 for the solution of each of seven mathematical problems. (If you fancy having a go at one of them, well the Poincaré conjecture has already been solved, but you might be interested in my thoughts on the P vs NP problem) In The Millennium Problems Keith Devlin gives an introduction to each of these problems. The book is written in a non-technical style without too much mathematics, and so is suitable for any reader who wants to get an idea of the nature of these seven problems.

My main criticism of the book is that the task Devlin has taken on is too much to fit into one book. Each chapter has a lot of introductory material, which doesn't leave much space for the description of the problem itself.. Most of the problems really need a book to themselves, such as those that have been written on the Riemann Hypothesis. For the last two chapters Devlin doesn't attempt to explain the maths leading up to the problem, he just tries to give the reader an overall feel for the problem, which in some ways is a more satisfactory way of using the space available.

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Paperback 256 pages
ISBN: 0465017304
Salesrank: 67078
Weight:0.59 lbs

Published: 2003 Basic Books

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Paperback 256 pages
ISBN: 1862077355
Salesrank: 231714
Weight:0.09 lbs

Published: 2005 Granta Books

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Paperback 256 pages
ISBN: 0465017304
Salesrank: 194298
Weight:0.59 lbs

Published: 2003 Basic Books

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Product Description
In 2000, the Clay Foundation announced a historic competition: whoever could solve any of seven extraordinarily difficult mathematical problems, and have the solution acknowledged as correct by the experts, would receive $1 million in prize money. There was some precedent for doing this: In 1900 the mathematician David Hilbert proposed twenty-three problems that set much of the agenda for mathematics in the twentieth century. The Millennium Problems--chosen by a committee of the leading mathematicians in the world--are likely to acquire similar stature, and their solution (or lack of it) is likely to play a strong role in determining the course of mathematics in the twenty-first century. Keith Devlin, renowned expositor of mathematics and one of the authors of the Clay Institute's official description of the problems, here provides the definitive account for the mathematically interested reader.

Product Description

In 2000, the Clay Foundation announced a historic competition: whoever could solve any of seven extraordinarily difficult mathematical problems, and have the solution acknowledged as correct by the experts, would receive $1 million in prize money. There was some precedent for doing this: In 1900 the mathematician David Hilbert proposed twenty-three problems that set much of the agenda for mathematics in the twentieth century. The Millennium Problems--chosen by a committee of the leading mathematicians in the world--are likely to acquire similar stature, and their solution (or lack of it) is likely to play a strong role in determining the course of mathematics in the twenty-first century. Keith Devlin, renowned expositor of mathematics and one of the authors of the Clay Institute's official description of the problems, here provides the definitive account for the mathematically interested reader.

Good introduction, but very superficial ***
This is an honest attempt to introduce a wide range of mathematical problems to a general readership; problem is: only readers with some background in mathematics are likely to be drawn to a book like this, and for them this book is simply too superficial. I read about half of it, selecting the problems I was interested in: I learned nothing I didn't already know -- and I don't know much at all. In my opinion the pitch of the book is wrong: readers with a background in mathematics will find it shallow and useless; readers with no background are more likely to want to read a Newyorker- or Newsweek-type article, rather than a whole book. So I think the book is honest and well intended, but flawed from the start.

Good introduction, but very superficial ***

This is an honest attempt to introduce a wide range of mathematical problems to a general readership; problem is: only readers with some background in mathematics are likely to be drawn to a book like this, and for them this book is simply too superficial. I read about half of it, selecting the problems I was interested in: I learned nothing I didn't already know -- and I don't know much at all.

In my opinion the pitch of the book is wrong: readers with a background in mathematics will find it shallow and useless; readers with no background are more likely to want to read a Newyorker- or Newsweek-type article, rather than a whole book. So I think the book is honest and well intended, but flawed from the start.

A General Introduction to the Official Problem Book *****
The goal of Keith Devlin's "The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time" is "to provide the background to each problem, to describe how it arose, [to] explain what makes it particularly difficult, and [to] give you some sense of why mathematicians regard it as important." "In May 2000 ... the Clay Mathematical Institute (CMI) announced that seven $1 million prizes were being offered for the solutions to each of [the] seven unsolved problems of mathematics..." Devlin's book is a "general introductions to ... the official book on the problems..." "... readers ... wishes to ... solve one of the Clay Problems should read the definitive description ... in the CMI book." "The official CMI book consists primarily of detailed and accurate descriptions of the seven problems..." Keith Devlin was asked "to provide short introductory accounts of the problems to make the book more accessible to mathematicians...journalists...readers..." "To read my [Keith Devlin's] book, all you need...is...high school knowledge of mathematics...You will also need sufficient interest in the topic." The book has eight chapters. Chapter zero is the general introduction to the problems. Chapter one is about the Riemann Hypothesis. Riemann suggests that for Riemann's Zeta function to be zero, the roots have the form ½ + bi for some real number b. Chapter two is about Yang-Mills Theory and the Mass Gap Hypothesis. The Yang-Mills equations describe all of the forces of nature (electromagnetic force, the weak nuclear force, and the strong nuclear force) other than gravity. The hypothesis provides "an explanation of why electrons have mass." The problem asks for "missing mathematical development of the theory, starting from axioms." The third chapter is about computer (The P Versus NP Problem). "Computer scientists divide computational tasks into two main categories: Tasks of type P can be tackled effectively on a computer; tasks of type E could take millions of years to complete. Unfortunately, most of the big computational tasks that arise in industry and commerce fall into a third category, NP, which seems to be intermediate between P and E. But is it? Could NP be just a disguised version of P? ... no one has been able to prove whether or not NP and P are the same." Chapter four is about the Navier-Stokes Equations. The equations describe "the motion of fluids and gases--such as water around the hull of a boat or air over an aircraft wing." They are partial differential equations (PDE). "To date, no one has clue how to find a formula that solves these particular equations." Chapter five is about the Poincare Conjecture. "If you stretch a rubber band around the surface of an apple, you can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface...if you imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut...when you ask the same shrinking band idea distinguishes between four-dimensional analogues of apples and doughnuts...no one has been able to provide an answer." Chapter six is about the Birch and Swinnerton-Dyer Conjecture. The conjecture suggests that "there are infinitely many rational points on E [the elliptic curve] if and only if L(E,1)=0." Birch and Swinnerton-Dyer "creates" an counting device L(E,1) for rational points. Chapter seven is about the Hodge Conjecture. "The basic idea was to ask to what extent you can approximate the shape of a given object by gluing together simple geometric building blocks of increasing dimension ... The Hodge conjecture asserts that for one important class of objects (called projective algebraic varieties), the piece called Hodge cycles are, nevertheless, combinations of geometric pieces (called algebraic cycles)."

A General Introduction to the Official Problem Book *****

The goal of Keith Devlin's "The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time" is "to provide the background to each problem, to describe how it arose, [to] explain what makes it particularly difficult, and [to] give you some sense of why mathematicians regard it as important." "In May 2000 ... the Clay Mathematical Institute (CMI) announced that seven $1 million prizes were being offered for the solutions to each of [the] seven unsolved problems of mathematics..."

Devlin's book is a "general introductions to ... the official book on the problems..." "... readers ... wishes to ... solve one of the Clay Problems should read the definitive description ... in the CMI book." "The official CMI book consists primarily of detailed and accurate descriptions of the seven problems..." Keith Devlin was asked "to provide short introductory accounts of the problems to make the book more accessible to mathematicians...journalists...readers..." "To read my [Keith Devlin's] book, all you need...is...high school knowledge of mathematics...You will also need sufficient interest in the topic."

The book has eight chapters. Chapter zero is the general introduction to the problems. Chapter one is about the Riemann Hypothesis. Riemann suggests that for Riemann's Zeta function to be zero, the roots have the form ½ + bi for some real number b. Chapter two is about Yang-Mills Theory and the Mass Gap Hypothesis. The Yang-Mills equations describe all of the forces of nature (electromagnetic force, the weak nuclear force, and the strong nuclear force) other than gravity. The hypothesis provides "an explanation of why electrons have mass." The problem asks for "missing mathematical development of the theory, starting from axioms." The third chapter is about computer (The P Versus NP Problem). "Computer scientists divide computational tasks into two main categories: Tasks of type P can be tackled effectively on a computer; tasks of type E could take millions of years to complete. Unfortunately, most of the big computational tasks that arise in industry and commerce fall into a third category, NP, which seems to be intermediate between P and E. But is it? Could NP be just a disguised version of P? ... no one has been able to prove whether or not NP and P are the same." Chapter four is about the Navier-Stokes Equations. The equations describe "the motion of fluids and gases--such as water around the hull of a boat or air over an aircraft wing." They are partial differential equations (PDE). "To date, no one has clue how to find a formula that solves these particular equations." Chapter five is about the Poincare Conjecture. "If you stretch a rubber band around the surface of an apple, you can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface...if you imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut...when you ask the same shrinking band idea distinguishes between four-dimensional analogues of apples and doughnuts...no one has been able to provide an answer." Chapter six is about the Birch and Swinnerton-Dyer Conjecture. The conjecture suggests that "there are infinitely many rational points on E [the elliptic curve] if and only if L(E,1)=0." Birch and Swinnerton-Dyer "creates" an counting device L(E,1) for rational points. Chapter seven is about the Hodge Conjecture. "The basic idea was to ask to what extent you can approximate the shape of a given object by gluing together simple geometric building blocks of increasing dimension ... The Hodge conjecture asserts that for one important class of objects (called projective algebraic varieties), the piece called Hodge cycles are, nevertheless, combinations of geometric pieces (called algebraic cycles)."

Seven of the greatest mathematical problems *****
If you want to know about seven of the most difficult unsolved math problems for which the Clay Mathematics Institute offers 1 million dolars a piece to whoever can solve them, this is the right book. Actually, we might talk about six unsolved problems since Perelman apparently solved the famous Poincaré conjecture. A quite readable account for someone who has some training in math.

Seven of the greatest mathematical problems *****

If you want to know about seven of the most difficult unsolved math problems for which the Clay Mathematics Institute offers 1 million dolars a piece to whoever can solve them, this is the right book. Actually, we might talk about six unsolved problems since Perelman apparently solved the famous Poincaré conjecture.

A quite readable account for someone who has some training in math.

The best that could be done in a linear medium ****
This compromise between the desire to be comprehensible to a wide audience, and to describe aspects of highly abstract mathematics, works better than one might expect, but it is still a compromise. (I skipped years of school and took my Math degree at London University too immature to be successful.) In my opinion, the only way this (book)/project could have been successful would have been for the Clay Institute to have commissioned a website with home pages for each of the problems, and a large web of explanatory pages for the various mathematical concepts involved. There was one place I thought the author (no doubt overwhelmed with the purely mathematical difficulties of the task he had set himself) missed an opportunity to be clear. His Navier-Stokes equations describe a perfectly incompressible fluid. Clearly this is a mathematical abstraction - the speed of sound in such a fluid would be greater than the speed of light, indeed infinite. I think the whole thing would have been clearer if he had noted that the real question to be answered is "are the equations for an incompressible fluid a useful approximation to the behaviour of a real fluid, or does the attempt to approximate inevitably lead to nonsense?". Attempts to simulate multi-body gravitational interactions on a PC screen, for example, seem easy to program, but simple programs that calculate forces at an instant and then step positions forward a finite time, inevitably lead to all the particles eventually shooting off the screen, simply because two particles very close together at the instant have mometary huge forces on them, and the approximation that the force is constant over a step is then nonsensical. So far, the Navier-Stokes equations seem to fail in the same way. The question is, can this be fixed? At least that's my understanding, but it does not come through in this book.

The best that could be done in a linear medium ****

This compromise between the desire to be comprehensible to a wide audience, and to describe aspects of highly abstract mathematics, works better than one might expect, but it is still a compromise. (I skipped years of school and took my Math degree at London University too immature to be successful.)

In my opinion, the only way this (book)/project could have been successful would have been for the Clay Institute to have commissioned a website with home pages for each of the problems, and a large web of explanatory pages for the various mathematical concepts involved.

There was one place I thought the author (no doubt overwhelmed with the purely mathematical difficulties of the task he had set himself) missed an opportunity to be clear. His Navier-Stokes equations describe a perfectly incompressible fluid. Clearly this is a mathematical abstraction - the speed of sound in such a fluid would be greater than the speed of light, indeed infinite. I think the whole thing would have been clearer if he had noted that the real question to be answered is "are the equations for an incompressible fluid a useful approximation to the behaviour of a real fluid, or does the attempt to approximate inevitably lead to nonsense?". Attempts to simulate multi-body gravitational interactions on a PC screen, for example, seem easy to program, but simple programs that calculate forces at an instant and then step positions forward a finite time, inevitably lead to all the particles eventually shooting off the screen, simply because two particles very close together at the instant have mometary huge forces on them, and the approximation that the force is constant over a step is then nonsensical. So far, the Navier-Stokes equations seem to fail in the same way. The question is, can this be fixed? At least that's my understanding, but it does not come through in this book.

Good reading for non-experts ****
It is probably impossible to satisfy everyone when writing a book about modern mathematics: no matter how good the book, some readers are bound to find it too primitive, while others will be hopelessly lost. The author seems to have tried to find the middle ground, perhaps a little on the "simple" side. A professional mathematician would probably find this book far too elementary; as a chemist, I found it educational. In places, it goes on and on about elementary concepts instead of progressing quickly to something more advanced. But overall, it was a good and stimulating reading that provided a glimpse of contemporary mathematics. Recommended if you are a non-mathematician with an interest in mathematics.

Good reading for non-experts ****

It is probably impossible to satisfy everyone when writing a book about modern mathematics: no matter how good the book, some readers are bound to find it too primitive, while others will be hopelessly lost. The author seems to have tried to find the middle ground, perhaps a little on the "simple" side. A professional mathematician would probably find this book far too elementary; as a chemist, I found it educational. In places, it goes on and on about elementary concepts instead of progressing quickly to something more advanced. But overall, it was a good and stimulating reading that provided a glimpse of contemporary mathematics. Recommended if you are a non-mathematician with an interest in mathematics.

Not easy but worth the effort ****
In 2000 the Clay Institute proposed the seven current mathematical problems that they hoped would guide mathematical research in our current era. The millenium problems are a respectful nod to a similar set of mathematical problems compiled by David Hilbert in 1900; one of the original Hilbert problems, the Riemann Hypothesis, has found its way into the Clay Millenium problems list. The millenium problems are unimaginably (to most of us) abstract and intractable, and to even attempt to explain them to the lay-person is an impossible task. Nevertheless Devlin has made a brave and worthy attempt. Each of the problems: The Riemann Hypothesis, The Yang-Mill Theory and Mass Gap Hypothesis, The P v NP problem, The Navier-Stokes Equation, The Poincare Conjecture, The Birch and Swinnerton-Dyer Conjecture, and the Hodge Conjecture, has its own chapter. Each chapter gives some historical background to the problem, a mathematical overview, the possible implications of its proof (or disproof) and as lucid an explanation as is possible of the problem itself. The cover reviews state that you'll come away feeling much the wiser after reading this book and whilst this is true, wisdom in this instance is a double-edged sword, the insight gained from Devlin's explanations bring home the realisation of just how difficult and obscure modern day mathematical research is. A very good book, although taxing in parts; if you're at all interested in mathematics, and unless you're here by accident I assume you must be, then I would recommend reading it.

Not easy but worth the effort ****

In 2000 the Clay Institute proposed the seven current mathematical problems that they hoped would guide mathematical research in our current era. The millenium problems are a respectful nod to a similar set of mathematical problems compiled by David Hilbert in 1900; one of the original Hilbert problems, the Riemann Hypothesis, has found its way into the Clay Millenium problems list.

The millenium problems are unimaginably (to most of us) abstract and intractable, and to even attempt to explain them to the lay-person is an impossible task. Nevertheless Devlin has made a brave and worthy attempt. Each of the problems: The Riemann Hypothesis, The Yang-Mill Theory and Mass Gap Hypothesis, The P v NP problem, The Navier-Stokes Equation, The Poincare Conjecture, The Birch and Swinnerton-Dyer Conjecture, and the Hodge Conjecture, has its own chapter. Each chapter gives some historical background to the problem, a mathematical overview, the possible implications of its proof (or disproof) and as lucid an explanation as is possible of the problem itself.

The cover reviews state that you'll come away feeling much the wiser after reading this book and whilst this is true, wisdom in this instance is a double-edged sword, the insight gained from Devlin's explanations bring home the realisation of just how difficult and obscure modern day mathematical research is.

A very good book, although taxing in parts; if you're at all interested in mathematics, and unless you're here by accident I assume you must be, then I would recommend reading it.

Pedantic ***
This book is more pedantic than I thought it would be. Being a smallish book and a smaller audience, it is understandable that the mathematical details are trimmed down (almost excised, you might say). Still, there is too much history and not enough details. Too often the author says that it may be above the reader's level. Overall, I was disappointed, but it was not a waste.

Pedantic ***

This book is more pedantic than I thought it would be. Being a smallish book and a smaller audience, it is understandable that the mathematical details are trimmed down (almost excised, you might say). Still, there is too much history and not enough details. Too often the author says that it may be above the reader's level. Overall, I was disappointed, but it was not a waste.

Oddly uneven exposition.... **
Someone may want to point out to Mr Devlin that the kind of person who picks up this book and who is willing to plough through it probably has a high level of native intelligence (even if she does not have an extensive math background) and the repeated and protracted "apologies" for the high level of abstraction of the mathematics which these problems require is both annoying and patronizing. I would guess that a bright 15 or 16 year old would have made that assumption before opening the front cover. It's simply unnecessary. And amounts to filler after a while. Now to the math. A reader having an extensive math background (say college major or above) will find little real math of interest here. Be prepared to face a page and a half explanation of the amazing growth in magnitude of factorials. There is astoundingly some algebra done wrong here (see page 192). The research in some of the chapters appears to have been done hastily and the exposition is not clear (P vs NP Problem for example). There are factual errors, from the minor (the year of Isaac Newton's birth) to Mr Devlin having Daniel Bernoulli and Euler working in the 19th century (p. 132; it was the eighteenth). This gives the feel of a book which has been written hastily and one wonders if it was reviewed by another mathematician before publication. In fairness, the description of most of the problems is deftly handled and will draw the interest of most readers. Please give the climbing the mountain to view the math landscape analogy a rest. The book is ridiculously overpriced at US $16.00 and Canadian $25.00.

Oddly uneven exposition.... **

Someone may want to point out to Mr Devlin that the kind of person who picks up this book and who is willing to plough through it probably has a high level of native intelligence (even if she does not have an extensive math background) and the repeated and protracted "apologies" for the high level of abstraction of the mathematics which these problems require is both annoying and patronizing. I would guess that a bright 15 or 16 year old would have made that assumption before opening the front cover. It's simply unnecessary. And amounts to filler after a while.
Now to the math. A reader having an extensive math background (say college major or above) will find little real math of interest here. Be prepared to face a page and a half explanation of the amazing growth in magnitude of factorials.
There is astoundingly some algebra done wrong here (see page 192).
The research in some of the chapters appears to have been done hastily and the exposition is not clear (P vs NP Problem for example).
There are factual errors, from the minor (the year of Isaac Newton's birth) to Mr Devlin having Daniel Bernoulli and Euler working in the 19th century (p. 132; it was the eighteenth).
This gives the feel of a book which has been written hastily and one wonders if it was reviewed by another mathematician before publication.
In fairness, the description of most of the problems is deftly handled and will draw the interest of most readers.
Please give the climbing the mountain to view the math landscape analogy a rest.
The book is ridiculously overpriced at US $16.00 and Canadian $25.00.

It's Not An "Idiot's Guide to the Millenium Problems" ****
Let's be frank: most people have a better chance of winning $1M at the state lottery than by proving any of these "millennium problems". Keith Devlin does a good job of explaining why. A little reverse psychology here and there ("...if you find the going too hard, then the wise strategy might be to give up.") just makes us want to push on toward the more difficult problems. The going isn't too hard thanks to Devlin's expository ability, but alas, I think this will be true only for aficionados of mathematics and physics. In his columns for the Mathematical Association of America, Keith has always had in mind a varied audience of readers. But how can he hope to communicate to the non-mathematician when so much meaning resides in the equations that appear throughout the book? Still, his pedagogy prevents this from becoming "The Idiot's Guide to the Millennium Problems". (I suppose it'll appear real soon.) Devlin hints at a disturbing idea. Will cutting edge problems become so abstruse some day that it will take the best minds all the fruitful years of their lives just to arrive at a position of comprehension? What then, mathematical AI? There are some silly mistakes, perhaps caused by a looming deadline. One involves a mix-up between the relativistic precession of Mercury's orbit and the relativistic bending of light rays. A logical error appears in a footnote on pg.54, where the word "a" should replace "no". Another one appears in the caption of Fig. 5.5, where "Example" should replace "Proof". Would it be too much to ask that copy editors who are assigned technical books have a dim awareness of mathematical argumentation?

It's Not An "Idiot's Guide to the Millenium Problems" ****

Let's be frank: most people have a better chance of winning $1M at the state lottery than by proving any of these "millennium problems". Keith Devlin does a good job of explaining why. A little reverse psychology here and there ("...if you find the going too hard, then the wise strategy might be to give up.") just makes us want to push on toward the more difficult problems.

The going isn't too hard thanks to Devlin's expository ability, but alas, I think this will be true only for aficionados of mathematics and physics. In his columns for the Mathematical Association of America, Keith has always had in mind a varied audience of readers. But how can he hope to communicate to the non-mathematician when so much meaning resides in the equations that appear throughout the book? Still, his pedagogy prevents this from becoming "The Idiot's Guide to the Millennium Problems". (I suppose it'll appear real soon.)

Devlin hints at a disturbing idea. Will cutting edge problems become so abstruse some day that it will take the best minds all the fruitful years of their lives just to arrive at a position of comprehension? What then, mathematical AI?

There are some silly mistakes, perhaps caused by a looming deadline. One involves a mix-up between the relativistic precession of Mercury's orbit and the relativistic bending of light rays. A logical error appears in a footnote on pg.54, where the word "a" should replace "no". Another one appears in the caption of Fig. 5.5, where "Example" should replace "Proof". Would it be too much to ask that copy editors who are assigned technical books have a dim awareness of mathematical argumentation?

Inspiring. And, it's from Keith Devlin. *****
Great book. Although pushing the limits of "accessible", those who understand it will be intrigued by Devlin's discussion of the Millenium Problems, a set of puzzling mathematical problems that, if solved, could redefine and revolutionize mathematical science. If you thought cryptography was interesting, wait until you see this. Also recommended: The Math Gene (also by Keith Devlin)

Inspiring. And, it's from Keith Devlin. *****

Great book. Although pushing the limits of "accessible", those who understand it will be intrigued by Devlin's discussion of the Millenium Problems, a set of puzzling mathematical problems that, if solved, could redefine and revolutionize mathematical science. If you thought cryptography was interesting, wait until you see this.

Also recommended: The Math Gene (also by Keith Devlin)

Another sloppy physics history ****
Reviewer Ted Sung pointed out a sloppy remark of physics history made by the book. There is another serious mistake appeared on Page 91: The Yang-Mills gauge theory has never been awarded a Nobel Prize. C. N. Yang did get the Prize in 1957, but it was conferred to him and his another collaborator T. D. Lee for a different contribution to physics (postulating that parity is not conserved under weak interactions). Unfortunately the author did not perform a simple research on history but rather believed in his speculation. My four-star rating does not count in this unforgivable carelessness. Devlin has done a good job in popularizing these great mathematical problems. Perhaps someone could deliver even a finer exposition. However, it would definitely demand the readers another level of mathematical sophistication.

Another sloppy physics history ****

Reviewer Ted Sung pointed out a sloppy remark of physics history made by the book. There is another serious mistake appeared on Page 91: The Yang-Mills gauge theory has never been awarded a Nobel Prize. C. N. Yang did get the Prize in 1957, but it was conferred to him and his another collaborator T. D. Lee for a different contribution to physics (postulating that parity is not conserved under weak interactions). Unfortunately the author did not perform a simple research on history but rather believed in his speculation. My four-star rating does not count in this unforgivable carelessness. Devlin has done a good job in popularizing these great mathematical problems. Perhaps someone could deliver even a finer exposition. However, it would definitely demand the readers another level of mathematical sophistication.