The topology seminar has come and gone...How did it go, I hear you ask? I've really no idea...
Ursula told us how to go about knitting Klein bottles and Mobius strips, and directed us to this website for further details, and to see one person's valiant attempt to knit the projective plane. Her Klein bottle was quite splendid: I am tempted to learn how to knit so that I can make one of my own. (You know I'll try wearing it as a hat.) I'd always pictured the projective plane as looking a little bit swirlier, though...I think of it as being some kind of funky seashell. There's really no mathematical reason why I should; I just do.
I'm not sure how well my talk went over; it may have been nonsense. I took a lot longer than I'd expected, even without giving the details. And my subject matter was, to say the least, eclectic. I spoke on the de Rham cohomology of compact Lie groups; my goal was to relate them to objects defined only in terms of the Lie algebras. To everyone's shock and disgust, I started off doing Riemannian geometry, with metrics, Hodge stars, harmonic forms...Used a lot of Lie derivatives...I finally related that to topology and de Rham cohomology with the Hodge Decomposition Theorem, which (as a corollary) tells one that the kth de Rham cohomology group of a compact Riemannian manifold is very isomorphic to the space of harmonic k-forms; in the bi-invariant metric I put on my group G, harmonic forms turned out to be precisely bi-invariant forms, as one would hope. I used that to translate de Rham cohomology into the cohomology of the Lie algebra, and, upon translation, found that the Hodge decomposition gave a completely natural and completely representation-theoretic description of both Lie derivatives and the cohomology spaces: the cohomology spaces turned out to be the submodules of invariants in the cochain spaces.
So de Rham cohomology on a compact Lie group is really a special case of the representation theory of Lie algebras. I find this splendid.
Geometry, topology, representation theory...I even fielded a question on measures from someone in the audience. I got some weird, weird looks from the topologists, let me tell you...Analysis is rubbish.
I hope they realise that I really am one of them. They should not be afraid. I'm just...interdisciplinary, that's all.
Another seminar tonight...This time on fluid dynamics, in particular the dynamics of really cheap gin and tonics as they pass through my esophagus. The Rosebud has a happy hour running from 4 'til midnight daily: $2 well drinks. The price is right. This is especially splendid.
Posted by aloysius at May 11, 2004 05:21 PM | TrackBack |