by Keith Devlin, Granta Books. Hardback ISBN 1862076863, £20.00 ($29.95).
On 24 May 2000 in a lecture hall at the College de France in Paris, Michael Atiyah from the UK and John Tate from the US announced that a $1 million prize would be granted to those who first solved one of the seven most difficult open problems in mathematics. These are known as the “Millennium Problems”, and the whole idea was an initiative of the Clay Mathematical Institute (CMI), which was established one year earlier by magnate Landon Clay. The list of problems was selected by a committee of top mathematicians, including – along with Atiyah and Tate – Arthur Jaffe, the current director of the CMI, Alain Connes, Andrew Wiles and Edward Witten, the only physicist, who is also a Fields medallist.
One-hundred years before, also in Paris, David Hilbert had given the famous address laying out an agenda for the mathematics of the 20th century. He proposed a total of 23 problems; a few turned out to be simpler than anticipated, or were too imprecise to admit a definite answer, but most were genuine, difficult and important problems that brought instant fame and glory (if not necessarily wealth) to those who solved them.
Some of the differences between the two sets of problems should be mentioned. While Hilbert’s set provided a guideline for mathematics research, the Millennium Problems provide a description of the current frontier of knowledge in the subject. The other important difference is that among the CMI set, two are inspired by deep physics problems: fluid dynamics and the structure of gauge theories.
In this book, Keith Devlin, a well known mathematician who writes excellent books and articles for a lay audience, takes up the daunting task of explaining these problems as best as can be done to an audience assumed to have no mathematical sophistication beyond the high-school curriculum – and all this in only about 200 pages! Although such an ambitious aim is nigh on surreal, the results are quite satisfactory. The book is able to communicate to a large extent the depth and importance of the problems, and to give a glimpse of the deep elation whoever solves them will feel. For anyone involved in research, the $1 million prize is almost beside the point.
The author chooses to present the problems from the “simplest” to the most arcane. The first is the Riemann hypothesis. Deeply related to the distribution of prime numbers, this is the only one of the problems proposed by Hilbert that has not been settled. Technically, one needs to find the location of the zeros of a certain function (Riemann’s zeta function). If proven true, there are literally hundreds of important results in number theory that would follow, and it is quite likely that a poll among mathematicians would identify this as the most important open problem in mathematics.
The second problem has a physics flavour to it. All the basic interactions in the Standard Model of particle physics are gauge interactions. If we consider the idealized situation of a pure gauge theory, we would like to have a mathematically sound proof that the quantum theory exhibits confinement, and that the mass of the first excited state is definitely positive (there is a mass gap). Most physicists would agree with these properties, however, a real proof may provide completely new methods in quantum field theory that may bring a revolution similar to the invention of calculus by Leibnitz and Newton.
The third problem is related to computational complexity where one can ask, among those functions or propositions that can be computed, which are “easy” and which are “hard”. Problems that can be solved in polynomial time with respect to the length of the input (in bits) are assigned to complexity class P, while the class NP contains problems that are considered hard because, so far, any algorithm used to solve them requires exponential time. Among the latter, one of the most famous is the travelling-salesman problem. Nobody knows whether the classes P and NP are equivalent. This is a central problem in computational theory and its resolution may have far-reaching technological consequences.
The fourth problem again has a physics flavour, and is related to the Navier-Stokes equation describing the fluid flow of an incompressible fluid with viscosity. It is difficult to exaggerate the importance of this equation in the design of aeroplanes and ships. Although there are plenty of approximation and numerical methods, as in the case of gauge theories, we are still lacking a deep mathematical methodology that will allow us to understand in detail the space of solutions for given initial data. To appreciate the difficulty of the problem, a solution would imply a detailed understanding of the phenomenon of turbulence – no small accomplishment.
The fifth problem is at the root of modern topology: the Poincaré conjecture. Roughly speaking, topology is the study of spaces that are stable under arbitrary continuous deformations. Thus a tetrahedron or cube can be continuously deformed to a sphere, while it is impossible to do the same with the surface of a doughnut. It is a very interesting and legitimate question to ask for the complete set of topological invariants in a given dimension. In the case of a two-dimensional sphere it is clear that we cannot lasso it. This property is known as simple-connectedness and it characterizes the sphere topologically. Poincaré asked whether a similar property would completely characterize a three-dimensional sphere. During the 20th century we have achieved a topological classification for spaces whose dimension is different from three, curiously the dimension where Poincaré’s conjecture was formulated. Progress in the past two decades seems to indicate that it may be settled in the positive in a few years time.
The last two problems are truly arcane. They go under the names of the Birch and Swinnerton-Dyer, and the Hodge conjectures. To exhibit the difficulty in explaining the first, it should be noted that definite support for it came from the settling of Fermat’s last theorem by Wiles and the subsequent progress in the so-called Taniyama-Shimura conjectures. The problem is deeply rooted in the study of rational solutions to special types of equations (elliptic curves), which in turn are contained in the apparently unfathomable world of Diophantine equations. The Hodge conjecture is, according to Devlin, the one most difficult to formulate for a lay audience. Its resolution would provide deep insights into analysis, topology and geometry, but its mere formulation requires advanced knowledge in all three subjects.
Devlin comes out with flying colours in his effort to make these fascinating problems accessible to a wide audience. It is inevitable that there are some dark corners and a few inaccuracies in such a challenging task. However, for anyone interested in the frontiers of mathematics and scientific knowledge, this book provides enthralling reading.
