bemonocled wrote:How did the functional analyst in your department scare you off?
gromov wrote:I would say that subject test score is fine - it's not that bad. Of course, raising it some could be nice, but yours isn't bad enough to raise concern.
That said for the very hardest to get into schools, a much higher test score can sometimes help.
I would be more concerned with your fundamentals (a lot of people applying have very high math GPAs). Your advanced classes will keep you in the running, but you should apply widely (perhaps consider some of the choices of previous poster).
You sound like the type of applicant where letters of recommendation could just about squeeze you in.
Basically, your GPA in advanced math coupled with test scores makes it clear you probably won't flunk out of the program's basic requirements, nor are you scared of trying advanced material. After that, it's often letters.
EDIT: The reason your professor's extra cautiousness about your subject score might be warranted, however, is that it might compensate a little for concern about your GPA.
Anyway, I asked him about going into NCG, and he said that it isn't a fertile field at the moment, which is a serious red flag in my mind.(More important of course is that it is fertile in 3-4 years,)
bemonocled wrote:I don't know exactly what he means by that. The non-commutative geometers I have talked to (who admittedly may be biased) seem to think that it is a great field for a young scientist to go into because there are so many unanswered basic questions. For example, there is not even a good way to define functions between spectral triples yet (the non-commutative generalization of Riemannian manifolds) - though there are some candidates.
Maybe he means that non-commutative geometry still needs to prove its worth to mathematicians/physicists in general? I would agree with that to an extent, though I've been told that a large reason for this is because it is such a sophisticated subject that people don't want to put the effort into learning it until it becomes really necessary to advance (especially among physicists). For example, the only rigorous model of the integer quantum hall effect that takes into account all of the relevant physics comes from non-commutative geometry, but it does not really tell you anything new about the quantum hall effect other than models that do not use NCG are obscuring the full picture. And with stuff like applications of NCG to the Standard Model, the masses are wrong. Connes' proposal on how to prove the Riemann hypothesis via NCG has decades of work left to be done, etc.
If you go to my older posts, I definitely overestimated how well I thought I'd do on the GRE, and I'm gonna have to retake it(thats what my professor is suggesting, he says a 74% might hold me back).
Hmmm... that's interesting. Maybe he meant its too difficult for most people to start out in? I'm not really sure what the meaning of it was. I'll probably learn about it regardless. Right now my major (hypothetical research) interest is in dualities between topological/geometric objects and algebraic objects, of which NCG is arguably based on. Of course algebraic geometry brings in its own dualities, and I believe you see a few in differential geometry/topology(all I know of are Serre Swan theorem and spectral triples really).
bemonocled wrote:I am really interested in the same kind of stuff. If you haven't before, you should give the paper A Mad Day's Work library.msri.org/books/sga/from_grothendieck.pdf a read. It gives a birds-eye view of a bunch of different perspectives on generalizations of geometry and I'm sure that you would enjoy it.
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