Today I Learned: On De-Klein-ing Lie's - Part 0

Update: Part 1 Part 2.

Note: for AMSI peeps who are bored and need some homework, here're the slides. =)

Last Thursday, a friend of mine - Nick Sheridan gave a seminar at my maths department on homological mirror symmetry. It was a pretty sweet talk - I really ought to go through his slides again at some point and write up the big picture. It was especially sweet because it's the first time that I've helped to organise one of these "proper" departmental seminars.

To secure the usual AGT (algebra-topology-geometry) seminar room for the talk, I had to obtain the booking from the AMSI 2013 Summer School organisers. So I wandered somewhat sheepishly into Jan de Gier's office and before I know it I'm walking out of his office with the requisite room booking in one hand, and an invitation to speak at a special seminar.

This post is an expanded summary (yes, it's oxymoronic, whatevs man!) of what I talked about.

Talk title: On De-Klein-ing Lie's

Talk title justification:

A small discourse between two mathematicians: Sophus Lie and Felix Klein.
L: Hey Felix, hey man, hey, tell me my beard's awesome.
K: I de-Klein.
L: Aww man, come on...just Lie a little.

Yep, puns and pictures are all I've got to go on.

The point of the talk was to walk through something that I'd picked up fairly recently on one of my daring excursions to the university library: a tiny volume entitled "Lectures on Mathematics ~ Klein". It's basically a collection of a whole bunch of lectures that Felix Klein gave at the Congress of Mathematics held in Chicago during August 1893. There's some pretty sweet stuff here, albeit written in slightly archaic language. My talk is motivated by/very very very loosely based on Lecture II: Sophus Lie.

Actually, this post is getting longer than I expected. Maybe I'll just give an overview of what I said.

Part 1: Projectivization

I'll be honest - I probably should have said projective completion instead of projectivization. But hey, I'm not an algebraic geometer, and projectivization sounds a bit better to me than "projective completion" or the "\(\mathrm{Proj}\)" construction.

The idea is to take \(\mathbb{R}^n\) or an algebraic variety (solution set to a polynomial) in \(\mathbb{R}^n\) and to "add stuff at infinity" so that we wind up with a compact space. I sort of emphasised the notion of "adding stuff at infinity" quite a bit during my talk, although "compactification" is probably slightly more accurate.

Part 2: The Erlangen Program.

This wasn't really necessary, but I thought it a cute introduction to the Erlangen program - which was at the time an attempt to unify all of these new non-Euclidean geometries that people were discovering. I basically just pointed out that \(\mathrm{GL}_{n+1}(\mathbb{R})\) acts on the projective space \(\mathbb{RP}^n\) transitively, and that Klein's Erlangen program is all about groups acting on spaces:
- the space tells you the "elements" of your geometry;
- the group tells you the "symmetries" of your geometry.

Part 3: Plucker's Line Geometry and Lie's Sphere Geometry.

This was the actual meat of the talk (it was a very fluffy and recreational talk). I gave the defining equations in \(\mathbb{RP}^5\) for the space of lines and the space of spheres in \(\mathbb{R}^3\) (*). As well as nice characterisations of when two points in the space of lines represented intersecting lines and when two points in the space of spheres represented tangentially meeting spheres. I concluded with a nice picture of how hyperboloids of 1-sheet and Dupin's cyclides are respectively corresponding objects in Plucker's line geometry and Lie's sphere geometry.

I then shamelessly advertised this blog and went home. =P

I'll try to flesh out part 3 a bit more next time, there's some pretty nice mathematics there - albeit a lot of it is just algebraic manipulation. Maybe I'll think about things more geometrically in the future.

(*) I lied a little here too, for expositional purposes, I should really be thinking about lines/spheres in \(\mathbb{RP}^3\). It's just a bit annoying/distracting having to clarify what that means.