Review of Galois Knot Theory

A thrilling review of this truly enlightening book

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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.

Theorem 3.3.9. Assume
\begin{align*}
M^{(V)}(|y_{\zeta,\mathcal{C}}|
&\to\int\int_1^{-1}\tan^{-1}(\overline{d})d\mathcal{J}_m\cup\cdots+\exp^{-1}\left(\sqrt2^{-9}\right)
\\
&\geq \frac{\overline{\frac1x}}{\tan^{-1}(0\omega^{(\Sigma)}}.
\end{align*}
Then $\|\hat{\Lambda}\|\lt y$

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:

Proof. This is trivial.

Reading Galois Knot Theory, you will find yourself constantly surprised that you haven’t heard of J. Maruyama: she’s the author of 60 books and 82 articles, and you’ll find yourself wanting to track down and study each one. If they’re anywhere near as eloquent as this book, you’ve been missing out for a long time.

… Ok, time to come clean. There’s another reason you haven’t heard of J. Maruyama. She’s made up.

In fact, the whole book is made up.

The ruse quietly revealed in small text at the bottom of page ii:

The contents of this book were randomly generated by mathgen, a context-free grammar to produce random mathematical writing.

That’s why the result above appears to be complete nonsense: it is complete nonsense!

mathgen works by starting with a basic template for a paper containing blanks. These blanks are replaced with words, phrases or equations, which in turn may contain blanks. By repeatedly filling the blanks, the software eventually arrives at a final paper. On their website, you can even generate a paper with yourself as the author: I’m not sure my supervisor is going to be very impressed with my latest paper

mathgen was written by Nate Eldredge, and is partly based on SCIgen, by Jeremy Stribling, Max Krohn, and Dan Aguayo. It is entirely open source and available on GitHub, so why not adapt it to use more phrases from your area of research and exponentially increase your paper output (while simultaneously exponentially decreasing their quality).

Matthew Scroggs is a PhD student at UCL working on finite and boundary element methods. His website, mscroggs.co.uk, is full of maths and now features a video of him completing a level of Pac-Man optimally.
Twitter  @mscroggs    Website  mscroggs.co.uk    + More articles by Matthew

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