I have to do an exam on Differential Geometry and my teacher wants that I prepare a choosen topic, outside lectures program, that I will talk about at the oral part of the exam. I am interested in group theory, so I would want to prepare a topic that uses it, something between differential grometry and group theory or that use group-theoretic tools and arguments. Can you give me some ideas please?
Be careful: this is a master degree exam, so don't propose me something that would be good for a phd thesis ^^ ! Thank you!
- My backround in differential geometry:
manifolds, bundles, tensor bundles, foundations of lie groups and lie algebras, cohomology (only until definition of cohomology groups), connections until levi-civita theorem, parallel transport and riemannian manifold.
- My backround in group theory:
Sylow thoerems, Hall theorems, Schur-Zassenaus, regularity on groups (nilpotents, solvables, supersolvables, polycyclics etc), kinds of product of groups (direct, cartesian, semidirect, wreath, free product, amalgamed etc), free groups and presentations, hnn extensions, permutation groups (k-transitivity, primitivity etc), foundation of transfer and some applications and foundations of finitely groups.
p.s. topics in differential geometry that use very beatiful algebra arguments are welcome!
I agree with Mike that you should ask a faculty member about this, since we have no concrete idea about how deep your lecture program goes, nor what is of interest to your professor.
That being said, there is an answer lying perfectly in the intersection of your two areas, and that is Klein Geometry. Klein started the so called Erlangen Program in the 1870's, whose agenda was to unify the several different notions of geometry (euclidian, affine, projective, hyperbolic, elliptic, spherical, conformal, etc.) using Lie groups to distinguish geometries by their underlying symmetries. The idea is that given a Lie-group $G$ acting transitively on a manifold $X$, we have $X \cong G/H$ with $H$ being the stabilizer subgroup of $G$ of a single point in $X$. We can thus define a homogenous space to be a space of the form $G/H$ with $G$ being a Lie-group and $H$ a closed subgroup. Euclidian space, affine space, projective space, etc. are all examples of homogeneous spaces, and it might be a useful exercise trying to figure out which Lie-groups give those. The more modern approach is then to study manifolds that locally look like a given homogenous space, this is done with Cartan Connections. Riemannian geometry is what you get when taking Euclidian geometry as your homogenous model.
The other possibility I'm aware of is, which is fairly modern and in essence the reverse of the Erlangen program is Geometric Group theory which studies groups by finding topological/geometric spaces on which a group acts. Since I don't know much about it in detail, I can't really tell you about it, but a single google search reveals several survey papers and lecture notes.