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	Mathematics: Frontiers and Perspectives Mathematics Unlimited: 2001 and Beyond
	Reviewed by David P. Roberts

Frontiers and 2001 are both inspired by David Hilbert's famous address to the International Congress of Mathematicians in 1900 in Paris. Hilbert asked, "What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?" These books in one sense answer Hilbert's question, as they form a huge montage of where mathematics is now. But in another sense, they raise Hilbert's question again: where is mathematics going?

Here are rough statistical profiles of the two books.

Many readers will be drawn to the general interest material. In Frontiers, I was sympathetic to Gowers' elegant essay, which asks from theory-builders more respect for problem-solvers. I felt informed and amused by Koblitz's 2001 paper on cryptography, which concludes by describing how the professional culture of cryptographers differs from that of other mathematicians. In both books, I detected a general reaction against Hilbert's "axiom of solvability of every problem." Mumford's Frontiers paper even concludes with "The intellectual world as a whole will come to view logic as a beautiful elegant idealization but to view statistics as the standard way in which we reason and think." Readers in search of controversy will find plenty.

The bulk of Frontiers and the biggest chunk of 2001 concern pure mathematics. Articles by Jones, Margulis, Sarnak, Smale, and Stanley in Frontiers follow Hilbert's model and center on lists of unsolved problems. Many of the other surveys also highlight unsolved problems or point to promising areas. I am sure many beginning graduate students will closely inspect some of these articles to guide them with their career choices. But even the casual reader can profitably troll for mathematical tidbits. In 2001, Bailey and Borwein present beautiful examples, communicable to undergraduates, of very believable conjectures which turn out to be false for interesting reasons. Baez and Dolan discuss how one can generalize taking the cardinality of a finite set; the number e is naturally a generalized cardinality, but it seems that pi is not. Kontsevich and Zagier reformulate a lot of fancy arithmetic geometry in terms of definite integrals called periods; this time pi is a period but it is conjectured that e is not. In Frontiers, I was especially intrigued by Arnold's elaborate schema in which the triple tetrahedron-octahedron-icosahedron corresponds to the triple reals-complexes-quaternions.

A much-celebrated feature of our time is that theoretical physics and pure mathematics are again making intimate contact at a deep level. Both books treat this topic very well. Papers by Witten and Penrose in Frontiers take somewhat opposing viewpoints. In 2001, Marathe discusses knots and Morrison discusses mirror symmetry, two areas where insights from physics have been crucial to advances in pure mathematics. In Frontiers, Vafa predicts that in the next century "quantum theory will be completely reformulated and that number theory will play a key role in this reformulation." One could argue that the connection between multiple zeta functions and Feynmann integrals presented by Bailey and Borwein in 2001 is already a step in this direction.

We are of course in the midst of a computer revolution. Outside of an excellent article by Lax, Frontiers is mostly silent about this. In 2001, there are articles on quantum computing, computational complexity, field visualization, algorithms, and the mathematics of the internet, among other things. There are also a number of articles representing the fact that nowadays machine computations often play an essential role in the discovery of new results in traditional pure mathematics. The space allocated to computer-connected material is altogether true to the reality that we are entering a brave new world. Many readers will be interested in an article of Langtangen and Tveito, "How should we prepare the students of science and technology for a life in the computer age?" The authors write in a tone which alternates from a scold to a call to arms; they argue for a computer- and modeling-based curriculum. But every issue has two sides: Antman calls for a reemphasis on core math and science courses, as modeling courses are "a grossly inadequate substitute."

The computer revolution is vastly increasing the usefulness of mathematics to society at large. If there are any doubters left, they should consult 2001. There are articles on fluid dynamics, climate, financial markets, materials science, control theory, neuroscience, the cardiovascular system, liver surgery planning, molecular evolution and the entertainment industry. The articles are absolutely convincing that mathematics through computers has a central role to play, even in the less traditional fields. In the future, we may expect fewer students to ask "what's all this for?", at least if we are properly prepared with better answers!

Hilbert's turn-of-the-century address has a special place in the mathematics literature. It is impossible not to detect an admirable ambition behind both of the books under review. Each book wants to be to the year 2000 what Hilbert's address was to the year 1900. The organizations behind the books pulled out all the stops to achieve this end. The makers of Frontiers trumpet that 15 of the contributors are Fields medalists; they would have been justified in adding that the remaining contributors are comparable in stature. Springer probably didn't even make a profit on 2001. Not only is 2001 a truly huge book with a low price, the paper is top-quality and there are many gorgeous color plates. The special care put into 2001 is further indicated by the inclusion of twenty-four pages consisting of short biographies and color portraits of the contributors.

The specialness of Hilbert's place in the literature is two-fold. First, Hilbert's 23 problems were viewed by many as a canonical list of problems, all truly deserving of special attention. It's generally viewed that the Clay Mathematical Institute's seven Millennial Prize Problems serve this function for the year 2000. But second, Hilbert captured and to a great extent defined the mathematical spirit of his times. We live in very different times now. Frontiers is an excellent book, but it is 2001 which has a better claim to capturing and defining the mathematical spirit of our times.

Publication Data: Mathematics: Frontiers and Perspectives, edited by V. Arnold, M. Atiyah, P. Lax, and B. Mazur. American Mathematical Society, 2000. Softcover, 459pp, $39.00. ISBN 0-8218-2697-2. There is also a hardcover edition, ISBN 0-8218-2070-2.

Mathematics Unlimited--2001 and Beyond, edited by Björn Engquist and Wilfried Schmid. Springer, 2000. Hardcover, 1237pp, $44.95. ISBN 3-540-66913-2. There is also a two-volume collector's edition, ISBN 3-540-67099-8

The written version of Hilbert's original paper is available at several places. One is at the beginning of a two-volume set, celebrating the 75th anniversary of Hilbert's address, Mathematical developments arising from Hilbert problems, edited by Felix Browder. (Proceedings of Symposia in Pure Mathematics, Volume XXVIII-Parts 1 and 2 AMS, 1976. Softcover, 628pp, $34.00. ISBN 0821814281.) A second place is in a recent more historical book, The Hilbert Challenge, by Jeremy J. Gray. (Oxford University Press, 2000. 240pp, 34.95. Hardcover, ISBN 0198506511.) A third place is at a web page maintained by David Joyce.

The Clay Mathematical Institute's has both general interest and detailed mathematical descriptions of its seven millennial prize problems.

David Roberts is an assistant professor of mathematics at University of Minnesota, Morris.

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