Mochizuki

He has good web design <http://www.kurims.kyoto-u.ac.jp/~motizuki/top-english.html>, just like this ABC conjecture page <http://www.math.unicaen.fr/~nitaj/abc.html>.


The work he did is too hard for anyone to understand right now. It's a very interesting situation. I read some new information about the status of people trying to extract value from his mathematical work and proofs:


Other blogs

Other blogs that I came across which look interesting:


There was something about "bhargavology" which I noticed because I heard about it yesterday relating to that weird PhD thesis: <https://theconsciousmathematician.files.wordpress.com/2015/02/bhargavology-learning-seminar.pdf>


There was a lot of cool stuff on that, here's a nice bit about analytic number theory: <https://theconsciousmathematician.wordpress.com/2014/06/24/some-gems-from-analytic-number-theory/>


Integrality

I had a skim over some notes that I found <http://people.fas.harvard.edu/~amathew/chintegrality.pdf> because I'm trying to understand algebraic characterizations of integrality. The really difficult part of the proof in DVRs is making good use of integrality, and to do this we need an algebraic format for it.

There seems to be two. One is is that $\alpha$ is integral iff there exists a faithful $R[\alpha]$-module that is finitely generated as an $R$-module. To prove this involves the determinant. According to wikipedia <https://en.wikipedia.org/wiki/Integral_element> Nakayamas lemma is a consequence of this (of the proof, not the integrality conditions).


Random curious thing

Someone linked this video <https://www.youtube.com/watch?v=zeRXVL6qPk4> which apparently shows a much faster way to prove with less galois theory - I had a look and it doesn't seem to use Galois theory at all! It's "topological" proof that involves monodromy groups.


Keepin on

I have been spending a long time honing in on and thinking about integrality. I was asking on a math site for help about it but it's hard to get help because I didn't have any specific question - just needed to talk to another person about it, you know?

It's always hard to do that but a really smart cryptographer was able to tell me that (1) integral closure really is important, which was very good for me to hear as it vindicates my efforts that might seem pointless since I could easily have just copied one of the many proofs from a book into my notes and got done with it and (2) they recommended me Dino Lorenzi's book - Invitation to Arithmetic Geometry. Chapter 1 is about integral closure and he said "making it chapter 1 means respect".


So the "algebraic" way to see integrality is very closely related to the Cayley-Hamilton theorem or maybe a slight generalization of it that proved Nakayama's lemma. I'll put some work into understand that result better and hopefully the DVR facts I need will follow.