Books : reviews

Avner Ash, Robert Gross.
Fearless Symmetry: exposing the hidden patterns of numbers.
Princeton University Press. 2006

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.

Avner Ash, Robert Gross.
Elliptic Tales: curves, counting, and number theory.
Princeton University Press. 2012

rating : 3 : worth reading
review : 6 December 2020

Elliptic Tales describes the latest developments in number theory by looking at one of the most exciting unsolved problems in contemporary mathematics—the Birch and Swinnerton-Dyer Conjecture. In this book, Avner Ash and Robert Gross guide leaders through the mathematics they need to understand this captivating problem. The key in the conjecture lies in elliptic curves, which may appear simple, but arise from some very deep—and often very mystifying—mathematical ideas. Using only basic algebra and calculus while presenting numerous eye-opening examples, Ash and Gross make these ideas accessible to general readers, and, in the process, venture to the very frontiers of modern mathematics.

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.

Avner Ash, Robert Gross.
Summing It Up: from one plus one to modern number theory.
Princeton University Press. 2016

Summing It Up uses addition as a springboard to present a compelling and accessible look at numbers and number theory, and how we apply beautiful numerical properties to answer math problems. Mathematicians Avner Ash and Robert Gross explore addition’s most basic characteristics as well as the addition of squares and other powers before moving onward to infinite series, modular forms, and issues at the forefront of current mathematical research. Employing college algebra, the first part of the book examines such questions as, can all positive numbers be written as a sum of four perfect squares? The second section of the book incorporates calculus and examines infinite series, and the third section ties together the first two parts of the book through a discussion of modular forms. Appropriate for numbers novices as well as college math majors, Summing It Up delves into mathematics that will enlighten anyone fascinated by numbers.