Viewpoint

by Joseph Lipman

Recently the Clay Mathematics Institute issued an announcement calling for the solution of seven of the most famous outstanding problems in mathematics. (See <www.ams.org/claymath>) The challenge was widely publicized in the popular media because of the million-dollar prize being offered for the solution of each problem. PUrview asked Professor Joseph Lipman to talk about the significance of the Institute¹s announcement and how it might relate to mathematics at Purdue.

The significance of the Clay problems must be considered in the context of mathematics as a whole. Mathematics is, among other things, an essential part of technology. Information processing, coding, data compression and transmission, cryptography‹all depend on centuries-old number theory, originally pursued without any idea that someday it would affect the life of everyone who uses a CD, an ATM, or the internet. Computers themselves were invented by mathematicians, who could hardly have imagined all the consequences. Designing planes and rockets would be impossible without the theory of fluid dynamics originated over two hundred years ago and being developed more than ever today (as is number theory.) Some properties of DNA depend on the topology of the knots into which it folds. Modern physics is based, miraculously, on totally mathematical descriptions of phenomena on inconceivably large and small scales (relativity, quantum field theory, ...), as though math were a sixth sense for apprehending the world around us.

These are some of the areas touched on by the problems. And in all these instances, the necessary math was in place prior to the applications, having been discovered through curiosity and fascination with purely intellectual constructs. The point is that pure mathematics is fundamentally important and needs to keep on developing as an organic body, in all its interrelated aspects, because no one can say which particular areas will have spectacular uses fifty or a hundred years from now.

Mathematics develops through exploration and opening up of fertile regions of the mind. Discovering such territories in the first place is a highly-regarded form of mathematical accomplishment. Usually these discoveries are associated with internally driven individuals struggling at the frontiers of knowledge with challenging problems‹of which the seven Clay problems are top-of-the-line examples. Mathematicians are drawn to such problems because they point to profound connections between fundamental concepts, offering the potential of breakthrough discovery, and because so much work has already been done on each of them that a solution will have to be the intellectual equivalent of reaching the summit of Mount Everest, without extra oxygen. Working on any one of them requires deep knowledge of diverse mathematical relations developed over decades, if not centuries, by a worldwide community of researchers. Significant progress toward a solution will almost certainly involve finding unexpected mathematical relations and creating new, fruitful methods. (One could think of the analogy with evolution as a process of emergence of new species arising in response to challenges posed by the environment.)

The million dollar prizes are, because of their magnitude, being talked about in the news media and so may help to raise the visibility and importance of mathematics in the perception of the general public. They have, of course, grabbed the attention of mathematicians and may stimulate some additional work on the problems: mathematicians are not immune to the seductions of fame and fortune. However, insofar as large prizes draw attention to this aspect‹rather than the aesthetics of elegant patterns and logical interconnections, the satisfactions of assembling lower-level structures into higher-level systems, the awesome grandeur of an intellectual edifice built up over centuries, the thrill of discovery, the lure of limitless possibilities‹they may also tend to trivialize what research in pure mathematics is really about. The most influential list of problems in the past century, formulated by David Hilbert at the International Congress of Mathematicians in 1900, carried no monetary reward. The most powerful motivation for the best research has been, and will remain, that offered by supreme challenges in and of themselves.

What does math research done at Purdue have to do with these problems? Well, if and when one of them is solved, it will be like Neil Armstrong walking on the moon. He couldn't have done it without the joint efforts of thousands of people, each contributing to the overall project (not to mention all the science developed over the centuries that went into it). You won't hear about 99% of these people, but their efforts were all necessary. Of course there was a well-defined goal there. When it comes to the Clay problems, the mathematical community will not be so narrowly focused. But still, any solution will build on the efforts of hundreds who have devoted their professional lives to mathematical research, most of which will not be directly noticeable, but will still form part of the necessary infrastructure.

Is it fair to reward only the person who finally creates a solution, when his or her success may just be the peak of a pyramid built by the achievements‹sometimes crucial‹of many, many others? Perhaps not. On the other hand, since so many exceptionally able people have already worked on each of the Clay problems, it appears that few on this planet could have the vision, single-minded perseverance, and sheer mathematical genius to put together a solution. (Those qualities were exhibited, for example, by Andrew Wiles of Princeton University in solving the 350-year old Fermat problem, as reported since 1994 in the press, in a TV documentary, etc.) Talent of such extreme rarity surely deserves high recognition.

Closing on a personal note, I don¹t expect my own research (on some algebraic structures which appear in numerous mathematical situations, and which exhibit intriguing relations) to play much of a role in the solution of any of these wonderful problems; and if it did, chances are it wouldn¹t be visible. But to any extent that it might contribute, I would feel gratified and delighted.