Book: The Millennium Problems
Author: Keith Devlin
Price: Granta, ?20
Reading "The Millennium Problems: the Seven Greatest Unsolved Mathematical Puzzles of Our Time" could be your ticket to a prize of $7m - $1m for each of the seven problems - to say nothing of the adulation you will receive from mathematicians around the world and the likelihood of a Fields medal, the equivalent in mathematical circles to a Nobel prize.
There is, however, a catch. Most of the problems in Keith Devlin's book are so hard that not only will you have no idea how to begin solving them unless you have a PhD in mathematics already, you will be fully stretched even to reach the base camp of understanding the problem. It is noticeable that the endorsements on the book's cover are all from professional mathematicians. Devlin himself admits in the early pages: "How can the non-mathematical reader understand these words, when they in turn don't link to everyday experience?... it's not that the human mind requires time to come to terms with new levels of abstraction - that's always been the case - rather, the degree and pace of abstraction may finally have reached a stage where only the expert can keep up."
So is it worth struggling through a book where your grasp of the meaning behind the words is likely at times to be shaky? Well, yes. At times I felt frustration with Devlin's treatment of his subject, not least because of his habit of referring forward to explanations that will come later. But by the end of the book I did feel that a window had been opened for me on to the frontiers of modern mathematics. As human knowledge becomes more specialised, it is important that we at least know the lie of other minds' lands.
The seven millennium problems, in the order Devlin presents them, are these: the Riemann hypothesis; Yang-Mills theory and the mass gap hypothesis; the P versus NP problem; the Navier-Stokes equations; the Poincar? conjecture; the Birch and Swinnerton-Dyer conjecture; and the Hodge conjecture. Since Devlin takes a whole book to explain what these problems are, only the briefest description can be attempted here. Note that five of the seven are described as a "hypothesis" or "conjecture," so the essence of the problem is to prove what someone has suggested and many believe to be true, but no one has yet been able to prove. In this sense they are similar in principle to, although different in nature from, the renowned Fermat's last theorem (this was called a theorem in deference to Fermat, who claimed to have proved it, but for the rest of us it remained a conjecture until proved by Andrew Wiles in 1994).
The two millennium problems closest in kind to Fermat's last theorem are the Riemann hypothesis and the Birch and Swinnerton-Dyer conjecture. The Riemann hypothesis, frankly described by Devlin as an "obscure-looking" question, dates from 1859 and concerns how the prime numbers are distributed among the familiar positive integers (1,2,3,4,5) particularly when the numbers get too large to work this out by hand, or even computer. For what it is worth, the Riemann hypothesis can be stated as the assertion that "all of the zeros of the Riemann zeta function which are not negative even integers (-2,-4,-6) have the form H + bi where b is some real number and i is the imaginary square root of -1." Now you may be familiar with i from schooldays and you may know that "real" is used here in a rather special mathematical sense, but you are unlikely to know what the Riemann zeta function is, and I am not about to tell you. Devlin does, and I think I understand what he means. Even so, it is hard to get a sense from the statement of the problem of why this can possibly matter and why, as Devlin puts it, "Ask any professional mathematician to name the most important unsolved problem of mathematics and the answer is virtually certain to be the Riemann hypothesis."
Is there any reason to crack the problem of the Riemann hypothesis beyond the fact that "it is there" and has continued to taunt mathematicians for a century and a half? The answer is yes. For a start, knowing more about the pattern of prime numbers matters to anyone who shops over the internet, because the public-key encryption methods that are used to make such transactions secure depend on the difficulty in factoring very large numbers into primes. It is not that proving the Riemann conjecture would immediately reveal your credit card details to the world, but that the techniques used to prove it, if ever this happens, might involve some mathematical breakthrough in factoring techniques that would render public-key encryption much less secure.
It is also intriguing that, as often happens, results in one domain of mathematics turn out to have surprising relevance in another. There is a tantalising parallel between the spacing of the zeros of the Riemann zeta function and the spacing between the energy levels of what theoretical physicists call a "quantum chaotic system." It is these unexpected cross-links that add so greatly to the intellectual beauty of mathematics.
Another of the millennium problems, the Navier-Stokes equations, relates to fluid dynamics. This is one of the simpler problems to state, since, effectively, all that one has to do is solve a given equation governing the mathematics of fluid flow. Mathematicians are reasonably sure that the equation, which is written in the terse but information-dense notation of "vector calculus," describes how fluids behave. Nature is thus a master at solving the Navier-Stokes equation in real time. Computers, too, can simulate special cases. But can a mathematician come up with a general solution? If we lived in a flat, two-dimensional world, the problem would no longer exist: the two-dimensional solution of the Navier-Stokes equation is old hat; unfortunately it provides no clue as to what happens in three dimensions, which is what really matters.
As a biologist, I work far from the world of professional mathematics, but I studied maths as an undergraduate at Cambridge. During this time my moment of greatest satisfaction came from the realisation that a theorem from an abstract branch of pure mathematics could be mapped into the world of fluid dynamics to solve a problem by a means that was faster and more elegant than that recognised by the official examiners. This pleased me immensely, even though I realised that a career in mathematics was not for me.
