Topology
School has been very busy lately– I had a test on Monday, and two on Tuesday, and a lab due today– but I’m seeing the light at the end of the tunnel: summer. If I knew for certain that the offer to do math research this summer with a prof here at UH still stands, then I’d be set.
Despite the best efforts of my professors to deprive me of an increments of free time with duration of more than 5 minutes, I’ve been able to continue trying to gain the background necessary to understand differential forms in their natural setting of manifolds. I’m still reading Vector Analysis by Kanich, but at a snail’s pace, and skipping a few ideas; my entire goal with that book is to just get the gist of the stuff before differential forms so I can study that section, and then stop. I’m not ashamed to admit that book is way over my head– I wonder if it has to do with the fact that it was written for students studying under a German curriculum? A friend who’s also trying to learn differential forms pointed out to me a Cal. I book used in Germany that covers everything we’re doing in Real Analysis, an American senior level course (at UH, the hardest, hands down)!
More productively, I’m reading ‘Introduction to Topological Manifolds’ by Lee, and am finding it smooth going. Weird, considering that Vector Analysis is in the Springer undergraduate series, and Lee’s book is in the graduate series. Lee seems to expect less of his readers, and spends more time on the basics. I now have a firm grasp of what a manifold is: a second countable, Hausdorff, locally Euclidean topological space, and even more importantly, why it is defined that way. Even the exercises are interesting. Unfortunately, Lee’s book doesn’t cover differential forms (maybe those can’t be defined on topological manifolds without extra properties?), but it is interesting in itself, and will probably make it a lot easier to read about differential forms in the future. An unexpected side-effect of my reading is a new found interest in topology itself, which before I couldn’t really get into, except to see that the definition of a topology is pretty cool. With the examples Lee gave, I finally saw the connection between deformation of surfaces— you know, how people explain topology by talking about how it shows certain things are equivalent in special ways, like a donut and a coffee cup, as if that really explains anything in a useful way—, topology, and that weird term, homeomorphism.
Add to that, I just got “How the Universe Got Its Spots” by Janna Levin yesterday, who seems to be a cosmologist concerned with the topology of the universe, as well as an engaging writer, and it seems that I may be distracted from my primary goal of differential forms, and go off on a tangent into topology. Oh well, that always seems to happen.
April 28th, 2005 at 5:26 pm
It is pretty well accepted in the physics community that our universe is flat (neither open or closed) due to the spectacularly precise measurments by WMAP in 2001. Outside our universe’s horizon (i.e the light cone), it may well be the universe has zome crazy topology with our universe only being locally Euclidean. However such theories are not falsifiable. It is also probably true that our universe is only flat macroscopically. It is theorized in most theories of quantum gravity that the topology of spacetime itself ungoes flucuations on Planck-length scales (cf. Google: Quantum Foam).
PS. Tiger is here! An excellent and thorough review: http://arstechnica.com/reviews/os/macosx-10.4.ars/
April 28th, 2005 at 8:00 pm
I like Lee’s book too. I think it’s a good book to start learning topology with manifolds as main examples. Although if I remember correctly, he skips a few important stuff on general topology, one of which is the Tychonoff theorem for countable product topology. You need some sort of smoothness in order to talk about differention. So differential forms are in Lee’s other book. Introduction to Smooth Manifolds. (Haven’t read that one.)