@BalarkaSen: Yes, I'm going to outline a proof in the smooth case. One issue is that it's very technical, so it's hard to know what details to include and what to skip; another is that it's going to need me to make a bunch of pictures, so it's going to take a while
i just googled and saw that it requires Morse theory. i've always wanted to know that stuff. maybe you can give a short intro to the prereqs first and then do the proof, @Mike
@Balarka: It's used for two technical (but obviously important) parts of the proof. It's just that knowing why these things are true doesn't really help you with understanding what's going on at all.
I'm not too familiar with the various people in the field
@BenLim: I guess UCLA probably wouldn't be a good fit for your tastes; the only person who really does (classical? idk; I mean "not homotopy theory" or "not K-theory") algebraic geometry is Totaro
@MikeMiller Yea. I'm quite surprised that so many of the prospies at Stanford want to do number theory. The latter is not my forte, I have very minimal experience in it.
@MikeMiller Actually, I don't mind algebraic K-theory say. If I had gone to UCLA, probably Rouquier was my first choice. I'm more algebraic minded, and ppl have actually told me that de Jong would be the best fit. Unfortunately, columbia waitlisted me....
well, i was invited to a modular forms school on goa some few months earlier, and i was told that soundarajan was going to come there. but i finally decided against going there.
i missed yesterday's lecture, since i'm home sick right now :( the speaker defined sites, stacks, and sheaves on them, and tried to get everyone comfortable
but yea it's like a formal way to say the two things are different.
I want to say this. There's a functor from $M_{1,1}(Spec Q) \to M_{1,1}(Spec Q \to Spec Q(\sqrt{3}))$, and that $M_{1,1}$ is a stack is precisely that this is an equivalence of categories. The one on the right is the "category of descent data".
@BenLim I attend grad school at Tata Institute of Fundamental Research.
@robjohn It would be great if you could change the mathjax links to "https" instead of "http" since firefox has stopped loading http content on https pages by default.
if you have a matrix, and you swap the ith row with the jth row and the ith column with the jth column, does the resulting matrix have the same characteristic polynomial?
but again, permutating A-xI twice and having an unchanged determinant doesn't imply that permutating A will be the same. That is if p is two permutations applied to a matrix, that det(p(A)-xI) is the same as det(p(A-xI))
the head of my math department once submitted a few-page paper which basically amounted to the fact that taking transposes commutes with squaring a matrix
Hello. I have a question about terminology. I have a definition about a certain "ordering" of sets (well it's not just any ordering, it also imposes some other restrictions but it doesn't really matter). I proved that each of the concerned sets has an "ordering" satisfying the definiton. Can I say "[this proves] the definition is well-founded"? Or is there a better term than well-founded?
@Andrew123321 was there any reason to suspect the definition to be ill or ambiguous in the first place?
I suppose if the definition implicitly claims to be applicable to all of the "concerned sets," then you do need to justify such an ordering is possible on all of the concerned sets
an abstract algebra teacher I helped TA with emphasized "well-defined" too much in my opinion. the only time it need be discussed is in the context of defining maps to or out of collections of equivalence classes. (actually I remember thinking this true of one other context too but can't remember what it was.)
@Hippalectryon These days I felt the need to attend some far far more advanced mathematics than the one I attend now. After publishing my book I'll definitely try to do that.
Presently all gets reduced to having the proper ideas (in the ends it's all about ideas). Just writing the proofs is not exciting anymore.
@Hippalectryon I was also expecting you to recommend me a profound study in the area of elliptic functions. Actually this is what I plan to do in the near future in terms of math. :-)