Fearless Symmetry:
Exposing the Hidden Patterns of Numbers

Avner Ash + Robert Gross

Princeton University Press 2006
A book review by Danny Yee © 2010 https://dannyreviews.com/
Fearless Symmetry "began as an expository article for professional mathematicians about the proof of Fermat's Last Theorem" but is, as its back-cover blurb claims, "written in a friendly style for a general audience". This seems like a contradiction, and indeed Fearless Symmetry is somewhat unbalanced.

The approach spans the largely unoccupied space between popular books and actual mathematics. Ash and Gross present key definitions and results, work through detailed illustrative examples, and attempt to provide intuition for key concepts. They largely omit the details of proofs, but stay focused on the mathematics, with pretty much no digressions to biography or history. What broader ruminations there are are attempts to give non-mathematicians a feel for concepts such as "representation" and "conjecture".

Fearless Symmetry proceeds in three parts of increasing difficulty. Part one covers some "algebraic preliminaries". Chapters on representations, groups, permutations, modular arithmetic, and complex numbers assume almost no background and will be refresher material for readers with any kind of training in formal mathematics. They provide just enough foundation for what is to come, rather than attempting any systematic presentation. Material on varieties and quadratic reciprocity may be less familiar but still shouldn't pose any difficulty.

This is the foundation for part two, which is an introduction to Galois Theory. This introduces the absolute Galois group of Q, linear representations of Galois groups and their characters, and Frobenius elements. As background to this, there's some basic material on matrices and matrix multiplication and a rapid introduction to elliptic curves.

The difficulty level ramps up sharply in part three. This gives some detailed examples of how Galois representations and reciprocity laws can be used, looking at the roots of unity and various kinds of elliptic curves. It then presents some more algebraic number theory and uses that to get a new perspective on quadratic reciprocity, before very sketchily introducing ├ętale cohomology and modular forms and attempting to place generalized reciprocity laws in a broader context. Finally the series of results leading up to Wiles' proof of Fermat's Last Theorem are outlined.

It's hard to know where Fearless Symmetry fits. Newcomers to formal mathematics may enjoy part one but will probably find getting through part two a challenge. Mathematicians, on the other hand, will probably wish part three had been expanded at the expense of the rest. Fearless Symmetry probably works best for those with some background in mathematics. Falling in this category myself — I last looked at Galois Theory twenty years ago — I found the foundational material trivial but interesting in places, the core material on Galois Theory a nice mix of the half-familiar and the novel, and the last half of part three only barely comprehensible.

In any event, it's nice to see a book which tries to make actual mathematics accessible to non-mathematicians, and which is prepared to push readers out of their comfort zone. I recommend Fearless Symmetry to anyone who enjoys mathematics, has some background in abstract algebra, is curious about Fermat's Last Theorem, and doesn't mind being confronted with some material that they probably aren't going to understand.

July 2010

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Related reviews:
- Avner Ash, Robert Gross - Elliptic Tales: Curves, Counting, and Number Theory
- books about mathematics
- books published by Princeton University Press
%T Fearless Symmetry
%S Exposing the Hidden Patterns of Numbers
%A Ash, Avner
%A Gross, Robert
%I Princeton University Press
%D 2006
%O paperback, bibliography, index
%G ISBN-13 9780691138718
%P 272pp