*Fearless Symmetry*. 2006, with Robert Gross2012, with Robert Gross*Elliptic Tales*.*Summing It Up*. 2016, with Robert Gross

Written in a friendly style for a general audience,
*Fearless Symmetry* is the first popular math book
to discuss symmetric patterns of numbers
and the ingenious techniques mathematicians use to uncover them.
The book starts with basic properties of integers and permutations
and ends with current research in number theory.
Along the way, it takes delightful historical and philosophical digressions
on French mathematician Évariste Galois
and well-known problems such as Fermat’s Last Theorem,
the Pythagorean Triples, and the ever-elusive Riemann Hypothesis.
Required reading for all math buffs,
*Fearless Symmetry* will appeal to anyone curious
about popular mathematics and its myriad contributions to everyday life.

This book explains the Birch–Swinnerton-Dyer Conjecture in the mathematical field of elliptic curves. We get 14 chapters of background before the conjecture is stated in chapter 15; by that point we have learned a wide range of interesting mathematics, and are in a position, if not to fully understand, at least to appreciate the Conjecture and its importance.

The route to this point covers a lot of ground. Each new piece of mathematics introduced is (relatively!) straightforward, but by the end, there is just so much machinery in play, that it all becomes a little overwhelming. That feeling is good for understanding just how deep this Conjecture is.

I was reminded of things I learned long ago, and learned lots of new interesting pieces of mathematics: an algebraic definition of the projective plane, how points on a curve can have a group structure, group generators, analytic continuation, series expansions, and much much more. What is great about this book is the way each new piece is slotted into the picture with a route map of where each chapter is going, explanations of how the pieces fit, and explanations of the importance and meaning of the concepts.

There are exercises along the way, of a form that deepens understanding, there for the serious reader. I was more of a visitor, looking at the interesting details, frankly skimming a few of them, but not putting in the work needed to live there. But I had a good time as a tourist.