*The Mathematical Century: The 30 Greatest Problems of the Last 100 Years*, Piergiorgio Odifreddi, 2004. 224+xvi pp. $27.95 cloth. ISBN 0-691-09294-X. Princeton University Press, 41 William Street, Princeton, NJ 08540. (800) 777-4726, http://www.pup.princeton.edu/

If you are like me, you would like to include the history of mathematics of the twentieth century in your classes. Teachers in the sciences frequently bring modern ideas into the classroom to elicit the "wow factor" from their students. However, perhaps like me you know that the mathematics of the twentieth century is just too hard and too difficult to get across in the classroom. If you are indeed like me, this book is for you.

This book is not about 30 problems, per se. Rather, it is about 30 subjects or groups of theories. For each problem Odifreddi touches on the history of the problem, the people involved in the solution, and the often diverse ideas that lead to the solution. In the process, we are treated to a history of Hilbert’s problems, the Fields Medal, and the Wolf Prize, all of which Odifreddi takes as a kind of criteria for "greatness." Each topic in a traditional undergraduate curriculum, both pure and applied, is represented. An undergraduate instructor is bound to find something that can be used as motivation in the classroom. This book could help inspire some students, but at the very least it will help answer the age-old questions "what is this good for?" and "why would anyone think of this?" A high school student may not understand the mathematics in the book, but a high school teacher will, and can make use of the problems and solutions as a way to motivate students.

Odifreddi has a wonderful way of explaining complicated theorems, and he includes a taste of the mathematics necessary to prove these theorems. What Odifreddi points out in this book is that finding and then studying the underlying structure of a problem or problems is how mathematics is done. Mathematicians know that, but we cannot always communicate that to our students or to the public. Odifreddi communicates this very well, from sections on the foundations of mathematics (from sets, to structures, to categories and then to functors) to applied mathematics (Hilbert spaces and General Relativity) and full circle to a section on formal languages.

Readers of *Convergence* and supporters of the ideas it espouses will enjoy this book, and make use of it in the classroom

Gary Stoudt, Professor of Mathematics, Indiana University of Pennsylvania

See also the MAA Review by P. N. Ruane.