Review of Galois Knot Theory
I was given a copy of Galois Knot Theory by J. Maruyama for Christmas last year. It is one of the greatest presents I have ever received, and is a book that truly highlights everything that is beautiful about mathematics.
The book describes itself as “intended for researchers and advanced graduate students” as well as “strongly recommended as a high school textbook”. Unbelievably, it manages to offer an awful lot to both these groups of people with its unique mix of rigour and readability.
Although admittedly a little slow to start, after around twenty pages you’ll be engrossed by the most important definitions, informative lemmas, and decisive theorems. Highlights include the crucial definition 2.2.8 and theorem 3.1.15: you’ll be kicking yourself for not coming up with them yourself. But the true highlight of the book is the result that underlies the entire topic, theorem 3.3.9.
If I had to pick one result to use to explain the beauty in mathematics to a non-mathematician, this result would be it. Its proof is simply delightful:
