This marks the first of an eventual multitude of posts on my new blog, the alien number!

What is the alien number?

It’s a page where I hope to track and categorise my progress in understanding arithmetic algebraic geometry, number theory and hopefully plenty of other exciting and strange areas in modern mathematics.

I came up with the name for this blog after reading Jordan Ellenburg’s excellent post about Mochizuki’s apparent proof of the ABC conjecture. He writes:

“Looking at [the proof], you feel a bit like you might be reading a paper from the future, or from outer space.”

And having looked at some of the papers myself, it did feel exactly like that!

Mochizuki’s “proof” of the ABC conjecture brings together my two favourite areas of mathematics – abstract category theory (in the guise of a suped-up, extraterrestrial mutation of arithmetic geometry) and concrete number theory. I love the idea that such general nonsense can (or needs to?) be used to study the deep structure of the integers $\mathbb{Z}$, arguably the most familiar object in maths. More generally, I am interested to understand how to use tools from algebraic geometry and category theory to solve other problems, both in and out of number theory.

I have absolutely no idea what Mochizuki’s proof is about; it’s an eventual goal of mine to be able to read it and be able to decide for myself whether or not it is valid. But it certainly looks like an interesting enough potential new area in mathematics to devote a good deal of time studying the basics to ultimately lead me there.

This is what this blog is about: it is my own “lab book”, starting from the very basics of scheme theory and eventually building up to more advanced topics in arithmetic geometry, with diversions to various other things that I like the look of on the way. Explaining this content concisely on here will help me learn it, and perhaps you might find it interesting to read too.

A word of warning: this is going to take a very long time! In particular, it’ll be years before I can properly look at Mochizuki’s proof properly. That isn’t what this is about. The journey to that point will be the most interesting thing, and hopefully along the way we’ll uncover many other weird mathematical gems.