“Fearless Symmetry” by Avner Ash, Robert Gross

At a book store in a shopping center by the coast of California I found this gem of a book. I skimmed through the content list, and bought it without much more thinking. In retrospect, it is safe to say that it was worth the $23.95 plus Californian tax.

As the title suggests, the book is much about symmetry  – but it is also slightly misleading. The book is really about number theory and the theory that led to the solution of Fermat’s  Last Theorem.

The book’s main mission is to explore the absolute Galois group $$G=G(\mathbb Q^{alg}/\mathbb Q)$$ through representations, that is, morphisms from $$G$$ to more known groups, such as matrix groups and finite fields. As such, the book is more about representation theory than symmetry. But it doesn’t stop there! A main theme in the book is how representation theory is behind generalized reciprocity laws in number theory and how reciprocity laws are used in advanced mathematics (an example of a reciprocity law is $$(p/q)=(-1/q)(q/p)$$ where $$(p/q)$$ is the Legendre symbol. That is, knowing if $$p$$ is square mod $$q$$ tells us if $$q$$ is square mod $$p$$ and conversely).

The book is written in a leisurely language and contains no difficult proofs and avoids technical definitions – without losing substance. Number theory is presented as a rich subject with lots of tools and abstractions.

The presentation was very inspirational, and this next semester will be like Christmas for me.