I come to you today with a recommendation for the book Fearless Symmetry by Avner Ash and Robert Gross. I started it this summer and finally had a chance to finish it over the Christmas break. I didn’t understand the last half-dozen chapters, but my dad did warn me that would happen. I wouldn’t even attempt reading it unless you’ve already been exposed to some undergraduate mathematics. But if you have, or if it’s been a while and you need a refresher, I highly recommend the book.
In the book, Ash and Gross attempt to explain some of the math underlying Wiles’ proof of Fermat’s Last Theorem. So you can understand why the math gets a bit hard at the end.
Along the way, you’ll get a very conversational, well-written, fun-loving introduction to the Absolute Galois Group of the Algebraic numbers. This is a group that is so complicated and messy and theoretical that we can only explicitly write down two elements of the group. In order to talk about it, we need representations, which the authors also introduce in a gentle way. In particular, we need linear representations.
Elliptic curves become very important too. I have studied elliptic curves in two of my classes before, but I really liked the way they introduced them here: We know everything about linear equations (highest exponent 1), and everything about conics (highest exponent 2 on x and y), but suddenly things become very interesting when we allow just ONE of the exponents (on x) to jump to 3. These are elliptic curves. Amazingly, you can define an arithmetic on the points of an elliptic curve that yield both a GROUP and an algebraic VARIETY. Incredible. Of course, the authors introduce what a variety is too.
After reading this, I also gained a much bigger view of abstract algebra–a course I’ve taken, but I found myself guilty of seeing the trees but not the forest. I loved the way Ash and Gross introduce the group SO3 and relate it to A4 with the rotations of a sphere inside a shell. Very nice visualization!
I could go on, but just know that there are lots of little mathematical gems scattered throughout this book. It’s a refreshing jaunt through higher-level mathematics that will demystify some of the smart-sounding words you’ve been afraid to ask about :-).
Go check it out!