Over at the site of the AMS, there is an interesting overview article by J. Milnor on the ideas behind the Poincare hypothesis and Perelman's proof. You don't have to be an expert in low dimensional topology to read this...
Milnor's article
Alan Hatcher's "Algebraic Topology" is, besides being freely available on his homepage, one of the best & most elegant textbooks I've ever come across.
He also has some other books on more advanced topics in algebraic topology, in various stages of completion, but I haven't read those yet.
(BTW the Schwartzchild solution doesn't really have a singularity. The singularity is an artifact of the coordinate system used, just like the singularity of latitude and longitude of the earth -- and we do believe in the north and south poles here, right? Kruskal exhibited coordinate systems in which there is no singularity.)
There's still a singularity for r=0. Kruskal coordinates get rid of the coordinate breakdown at r=2m.
I feel much more like doing research. I'm a mathematics student, and if you get a non-research job with maths, you usually end up as a programmer or a statistician. Whereas, as you do research you are being payed a decent amount of money to read all the books you can get your hands on, investigate the topics you like, etc. I think that's rewarding in itself.
The amount of jobs is limited. You're a physics student, but you won't find an industry job where you have to study eg. the quark-gluon interaction for moderate energies:)
BTW, you can reward yourself, seeing that you study physics, just discover some nifty device like a handheld quantum computer and you'll be rich _and_ a good scientist in no time:))
Slashdot Mirror
Over at the site of the AMS, there is an interesting overview article by J. Milnor on the ideas behind the Poincare hypothesis and Perelman's proof. You don't have to be an expert in low dimensional topology to read this...
Milnor's article
He also has some other books on more advanced topics in algebraic topology, in various stages of completion, but I haven't read those yet.
There's still a singularity for r=0. Kruskal coordinates get rid of the coordinate breakdown at r=2m.
I feel much more like doing research. I'm a mathematics student, and if you get a non-research job with maths, you usually end up as a programmer or a statistician. Whereas, as you do research you are being payed a decent amount of money to read all the books you can get your hands on, investigate the topics you like, etc. I think that's rewarding in itself. The amount of jobs is limited. You're a physics student, but you won't find an industry job where you have to study eg. the quark-gluon interaction for moderate energies :)
BTW, you can reward yourself, seeing that you study physics, just discover some nifty device like a handheld quantum computer and you'll be rich _and_ a good scientist in no time :))