the cokernel is the codomain mod the image, equivalently (when the image is closed) the orthogonal complement of the image, equivalently the kernel of the adjoint
@PVAL-inactive My advisor once had me read the thesis of a student of a particularly famous Floer homologist. He told me that he quit math because he was inable to write any of ideas down even though he had plenty of good ideas. I think only 70% of the reason he told me about this is because the thesis would be useful for my work.
@MikeMiller: I suppose it's not, yeah. I realize it's a pain to keep typesetting the margin or something for the whole day even when you've got a whole theorem you want to write down.
@TedShifrin I got some more topic suggestions from my supervisor! :) If you're interested, I linked them around here.
I also talked to him about the "isolated" position I've been experiencing---and he had some nice news to deal with that, too: He might start holding (bi?)weekly meetings for his students (he calls it "research tutorials")
@TedShifrin There's a famous theorem that if you have two connections on a $G$-bundle such that the holonomies around every loop are conjugate, the connections are gauge equivalent. I'm working out how to do this again, which will probably take me a couple hours (to do that and the enhancement I want). Do you know a reference I can look at where this is proved?
I think I am falling into the pattern of writing down theorem/lemma/prop-> proof-> completely meaningless remark to fill space before next theorem/lemma/prop/ and repeat.
@TedShifrin I did do (and liked!) the ones you recommended before :) I didn't find that particular chapter very exciting, except the (statement of) Theorema Egregium and Hilbert's theorem.
I think understanding that constant principal curvature happens only for tubes is cool (albeit intuitively plausible). Two constant principal curvatures?
What can you say if you have lines on a surface making constant angle at each point? What about the corresponding question for geodesics (next section)?
Hello all! I've got a really quick, dumb question that I guess is fairly perennial about standard terminology -- if, in some well behaved vector space, a function maps numbers to numbers, operators map functions to functions, and functionals map functions to numbers, what is the name given to an object that maps operators to operators?
@TedShifrin Eh... $H^1(X,\mathcal O)=H^{0,1}(X)$ which is half dimension of $H^1(X,\Bbb C)$, as is $H^1(X,\Bbb Z)\otimes_{\Bbb Z}\Bbb R$ as a real vector space, since $H^1(X,\Bbb C)=\text{that}\otimes_{\Bbb R} \Bbb C$?
@Balarka: I don't understand that attitude. It's the result of a bunch of formulas. If you want more intuitive understanding, you'll see it with moving frames/forms.
@TedShifrin If I have a linear vector field of dimension $2$, a periodic orbit means that there exist T>0 such that $\phi_T(x)=x$ (where $\phi$) is the flow, right?
@TedShifrin Okay. I have almost never done symbol-pushing without understanding the intuition before (and inevitably get tripped up in such calculations), and I want to get this chapter done before the moving frames section in the next one.
@Balarka: I think it's intuitive that (at least with a reasonable parametrization) Codazzi exactly tells you about rates of change of the principal curvature along the lines of curvature.
@Danu: I guess I was wanting you to say well-defined mod a translation of the torus.
