lie algebra


049/100 days of productivity

It was this week’s time to explain linear algebra to people, so I spent some time talking about the Jordan normal form.

Then I tried to finish my geometry homework, which resulted in the frustrating experience of being stuck at the same (probably rather simple) exercise for the last few hours.

I kind of hope that I will figure it out tomorrow.


Revised the first half of topology today and remembered how much I like it. Its one of the few courses where I can still sit down and scribble pictures and stare in to space and use plenty of intuition and eventually arrive at the correct reasoning; pretty much what got me into maths in the first place. It really is a beautiful and rich subject!

Going to start the algebraic topology part tomorrow, much harder but also really interesting. The thrust of algebraic topology as I understand it is to translate topological problems into problems in group theory. These are generally much easier to understand and a great deal of group theory is very well understood.

And then the ideas of Lie Algebras help us translate harder group theory problems into linear algebra which is easier again! I love maths, why don’t I listen in lectures?

“Here’s the thing: I lose sleep because of you. Every week.

Before I tell you why, you should understand the truth about school. You see, the main event of school is not academic learning. It never has been. It never will be. And, if you find someone who is passionate in claiming that it is about academics, that person is lying to himself or herself and may genuinely believe that lie. Yes, algebra, essay writing, Spanish, the judicial process — all are important and worth knowing. But they are not the MAIN event.

The main event is learning how to deal with the harshness of life when it gets difficult — how to overcome problems as simple as a forgotten locker combination, to obnoxious peers, to gossip, to people doubting you, to asking for help in the face of self-doubt, to pushing yourself to concentrate when a million other thoughts and temptations are fingertips away.

It is your resilience in conquering the main event — adversity — that truly prepares you for life after school. Because, mark my words, school is not the most challenging time you will have in life. You will face far greater challenges than these. Sure, you will have times more amazing than you can imagine, but you will also confront incomparable tragedy, frustration, and fear in the years to come.

But, you shouldn’t be worried about the fact that you will face great adversities. You should be worried because you’re setting yourself up to fail at overcoming them. Here’s the real reason I lose hours of sleep worrying about you: You are failing the main event of school. You are quitting. You may not think you are quitting, but you are because quitting wears many masks.

For some, you quit by throwing the day away and not even trying to write a sentence or a fraction because you think it doesn’t matter or you can’t or there’s no point. But it does. What you write is not the main event. The fact that you do take charge of your own fear and doubt in order to write when you are challenged — THAT is the main event.”


From “What Students Need to Hear” by teacher Chase Mielke

Read the whole essay. You’ll be glad you did. 

Generalisation is not the point of mathematics. To be honest, it’s usually rather dry. The challenge is to generalise in a rich and revealing direction.

Terry Gannon, Moonshine Beyond the Monster §3.3

(generalisations of the affine algebras: Kac-Moody algebras, Borcherds’ algebras, toroidal algebras)