In May 2000, the Clay Mathematics Institute elevated seven long-standing open problems in mathematics to the status of "Millennium Prize Problems," endowing each with a million-dollar prize. The seven particular problems were chosen in part because of their difficulty, but even more so because of their central importance to modern mathematics. The problems and the corresponding general areas of mathematics are as follows.
||The Riemann Hypothesis
||Yang-Mills Existence and Mass Gap
||The P versus NP problem
||Navier-Stokes Existence and Smoothness
||The Poincaré Conjecture
||The Birch and Swinnerton-Dyer Conjecture
||The Hodge Conjecture
The Navier-Stokes equations were first written down in the early 1820's, Riemann made his hypothesis in an 1859 paper, and the Poincaré conjecture dates from 1904. The remaining problems arose in the period 1950-1971.
In The Millennium Problems, Keith Devlin aims to communicate the essence of these seven problems to a broad readership. It is, of course, a very ambitious goal. The preface makes it clear what Devlin's ground rules are. First he assumes only "a good high school knowledge of mathematics." Second, he is writing "not for those who want to tackle one of the problems, but for readers — mathematician and non-mathematician alike — who are curious about the current state at the frontiers of humankind's oldest body of scientific knowledge." He is clear that the readership drives the level of the book, so that precise statements of the problems will not always be given. Rather the goal is "to provide the background to each problem, to describe how it arose, explain what makes it particularly difficult, and give... some sense of why mathematicians regard it as important."
After the short preface, the book has an interesting Chapter 0, and then one chapter for each problem in the above order. These seven chapters are constructed similarly. Most have a long historical component, generally including biographical information about the person or persons after whom the conjecture is named. Each has substantial background mathematical information, with topics ranging from complex numbers in Chapter 1 and group theory in Chapter 2 to congruences in Chapter 6 and algebraic varieties in Chapter 7. Applications are emphasized when possible. A nice theme in Chapters 2 and 4 is that mathematicians are behind physicists and engineers and just trying to catch up. Each chapter concludes with a discussion of the millennium problem itself.
Chapter 5 illustrates how Devlin ties the various units of a chapter into a coherent narrative. It begins with four pages about the life and work of Henri Poincaré. It moves on to introduce "rubber sheet geometry" in terms of how subway maps and refrigerator wiring diagrams are not geometrically faithful to the physical objects they represent, but nonetheless clearly capture all relevant information. This unit is important as it will make readers feel that topology is natural, rather than weird. Chapter 5 next introduces the concepts of vertices, edges, faces and finally Euler characteristic in terms of the Königsberg bridge problem. It introduces non-orientable surfaces and makes the introduction of an ambient four-space seem natural, since it is necessary for an embedding of the Klein bottle. It topologically classifies closed surfaces first crudely in terms of their orientability, and then completely in terms of networks drawn upon them and the Euler characteristic of these networks. It gives a very attractive example of two seemingly linked rings that in fact can be pulled apart. This example shows the reader that not everything is geometrically obvious, and thus underscores the utility of algebraic invariants that can rigorously confirm that two objects are topologically different. It discusses how the ordinary two-sphere is characterized among all closed surfaces by having the property that any loop on it can be shrunk continuously to a point. Finally, by way of this two-dimensional analogy, it discusses the actual three-dimensional Poincaré conjecture.
The strain imposed by the challenge of communicating all seven millennium problems to a broad readership naturally shows at times. In the Navier-Stokes chapter, for example, the background mathematical information presented is calculus and specifically differentiation. Readers are instructed that "dy/dx" is to be read "dee-wye by dee-ex." Some seven pages later, the Navier-Stokes equations themselves are presented. They are four coupled non-linear partial differential equations in four independent variables. The exposition is gentle, but readers new to calculus will only understand at a superficial level. The strain is felt somewhat more in Chapter 6 and particularly so in Chapter 7. But these various strains are unavoidable, and I think in general Devlin has done a very good job giving general readers a feel for the seven millennium problems.
The Millennium Problems concentrates on the past and present of the problems, but it's also natural to wonder about their future! Can we expect to see some prizes handed out within our lifetimes? Devlin raises this question at the end of the various chapters, but always in a noncommittal way. His mention of the "twenty-fifth century" in the preface may incline some readers to be pessimistic. My personal feeling is that there are good reasons for optimism. I'll take this opportunity to put down my guess that the torrid pace of mathematical progress in the 21st century will include the solution of at least two of the millennium problems before 2020 and at least five before the end of the century. When solutions to the millennium problems do come, it would be nice if the general public recognized them for the monumental achievements that they will be. Books such as Keith Devlin's The Millennium Problems will help a great deal.
Part of the interesting Chapter 0 appears in slightly reworked form in the November 2002 installment of Devlins's angle, The inaccessibility of modern mathematics.
More information on the millennium problems, including official statements of the problems, is available at the Clay Mathematics Institute. Devlin in his preface refers also to a forthcoming official CMI book, to which he will be a contributor.
There are several shorter introductions to the seven millennium problems at a somewhat more advanced level than Devlin's book. One is by Barry Cipra in a volume reviewed by MAA Online. While Devlin's book is generally free from errors, and Cipra's book has been especially well edited, both make the same error with respect to the now proved higher-dimensional Poincaré conjecture. In dimensions four and greater, a simply-connected compact manifold can still exhibit great topological complexity. Only after one assumes that homology groups are also trivial in degrees one through the dimension minus one is such a manifold guaranteed to be homeomorphic to a sphere.
David Roberts is an assistant professor of mathematics at the University of Minnesota, Morris.