The Millennium Problems:

The Seven Greatest Unsolved Mathematical Puzzles of Our Time

(A brief review)

Popular math and science books have to walk a very fine line: on the one hand, if the material gets too complicated or abstract, you’ll lose readers. The corollary to this is fairly straighforward: editors and publishers tend to be wary of the complicated and the abstract, because they’re afraid of losing readers. (Stephen Hawking was apparently informed that each equation he included in A Brief History of Time would halve the sales. He therefore limited the number of equations to one, E = mc2.)

On the other hand, in the quest to make everything as manageable and benign as possible, writers can oversimplify to the point that they misrepresent the actual material. This is both irritating (and/or confusing!) for a reader who does have some background in the subject, and – in my opinion – does a disservice to the then (misinformed) public and (misrepresented) scientific community.

I have immense respect, therefore, for those authors who manage to produce engaging, accurate, comprehensible, and accessible accounts of their research – especially considering that most of the scientists and mathematicians who undertake to write these popular books are not trained writers.*

The latest entry on my reading list does a superb job of carrying off this balancing act, and quite honestly, it’s a book that I’d love to go back in time and give to my fourteen-year-old self, or that I’d recommend as a gift for any budding young student with a good grasp of high school math.

The book is The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time, by Keith Devlin, who is currently a visiting professor at Princeton, and who normally teaches at Stanford, where he is the Executive Director of the Human Sciences and Technologies Advanced Research Institute.

The premise of the book is quite simple: the Millennium Problems are, as the subtitle suggests, seven of the greatest mathematical problems for the 21st century, which have baffled all of the world’s leading mathematicians for up to a couple of hundred years. In 2000, the Clay Mathematics Institute, after conferring with leading experts around the world, announced a competition: they selected a list of seven problems, that are not only extraordinarily difficult, but whose solutions will have extraordinary implications in advancing the frontiers of mathematics, physics, computer science, or engineering. Anyone who can solve one of these problems will be awarded a prize of $1 million. (Now almost fourteen years after the contest was announced, exactly one of those problems – the Poincaré Conjecture – has been solved.)

It should be clear right now that none of these problems are remotely easy. To give you an idea of the challenge Devlin took on in agreeing to write a book that would explain all these problems to a general audience, let me quote the technical formulation of the Hodge Conjecture, one of the seven problems:

Every harmonic differential form (of a certain type) on a non-singular projective algebraic variety is a rational combination of cohomology classes of algebraic cycles.2

I suspect even those with undergraduate or graduate degrees with mathematics might have difficulty understanding that one! Some of the other problems – the Riemann hypothesis or P vs. NP, for instance – are a little less complicated at first glance, but none of them could ever be termed “easy”. Nonetheless, Devlin does an admirable job of introducing the history of each problem, sketching in some detail the implications that a solution would have for modern science and technology, and translating what is often highly technical mathematics into intelligible English prose.

The book is also an accessible (!) tour of some of the most fascinating branches of modern mathematics that students are rarely exposed to unless they are studying to become mathematicians themselves – like algebraic topology (third-year class at most universities), complex analysis (also a third-year class), number theory (ditto), and partial differential equations (fourth year/graduate level). Do you have to be a mathematician to read this book? Absolutely not – Devlin assumes only a reasonable knowledge of high school level mathematics; some background in calculus would be helpful, but is not – in my best judgment – essential. That being said, if the idea of any equation harder than E = mc2 has you running for the hills, this is not your book: the level of mathematics presented is a good deal higher than in most popular books I’ve encountered. (So do be prepared to do some thinking – these problems have been giving the professionals headaches for years!)

But if you’d like an introduction to the frontiers of mathematical research today, or you happen to know a student who likes math and is wondering what one might, well, do with that interest, I’d say that this would be a great place to start. It’s also sufficiently accurate and well-written that (as a reader who does have a somewhat more advanced background) I found it a really worthwhile and engaging read.

(Now I need to go find a textbook on cohomology classes …)

1Some other names that come to mind, off the top of my head, of scientists who are also excellent writers for a general audience: Leonard Susskind, Stephen Hawking, and Richard Feynman (physics); Jeffrey Rosenthal (statistics); Ed Frenkel (mathematics – Langlands program).

2 Devlin, p. 214

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