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Mathematics Unlimited: 2001 and beyond edited by Björn Engquist
and Wilfried Schmid, Springer-Verlag, £27, ISBN 3540669132

AT THE Congress of Mathematicians held in Paris in 1900, David Hilbert gave a
lecture on mathematical problems. He covered 10 topics, but the written lecture
contained another 13 problems. His lecture influenced the development of
mathematics throughout the 20th century.

Hilbert did not necessarily intend to set an agenda, nor did he consider his
list to be comprehensive. He described his paper as “merely a sample of
problems”. What he really wanted his audience to grasp was that mathematics was
alive and kicking.

So what’s the state of mathematics a century after Hilbert’s lecture?
Editors Björn Engquist and Wilfried Schmid tackle this question in
Mathematics Unlimited. They’ve assembled more than 60 articles on topics from
quantum computing to the Diophantine equations, for example.

The range of applications covered, including economics and even the film
industry, reflects a trend of the last half of the 20th century: mathematicians
found that problems from other disciplines gave rise to new mathematical
ideas.

The book has its limitations, but if you are a working mathematician it is
well worth owning a copy of this book. It illustrates that mathematics is still
far from exhausted, with enough problems for the next millennium. It reminds us
all what a wonderful and diverse subject it is.