Hmm, I mentioned that Atiyah paper on non-multiplicative signature. Still love that. Chern-Lashof's papers were great. I liked a lot of Lawson's work, and he writes well.
A number of Griffiths's papers. Particularly Variations on a Theorem of Abel.
Ok, I'm now wondering whether every $\sigma$-algebra has a minimal generating family of sets, it feels like this should be true, but I know my intuition is always wrong so...
@MikeMiller: I did go to a seashore a couple times, but mostly spent time on the hotel whenever my parents go on a trip. I made a decision a year ago that I won't go to family-trips anymore.
@Alessandro: I'm puzzled that they say it works for countably generated. What if we use the usual $\sigma$-algebra generated by open intervals in $\Bbb R$?
I was thinking about that too, that $\sigma$-algebra should countably generated since every open interval can be written as countable union of open intervals with rational endpoints so it should coincide with the $\sigma$-algebra generated by those
I found the paper referenced in the MO question and it seems readable, but half past midnight seems to be more appropriate of a time to go to sleep than starting reading it, I'll see tomorrow if I can understand what's going on
What do you think, does the definition of twisted homology $H_n(X;V) = H_n(C_*(\widetilde{X}) \otimes_{\mathbb{Z}[\pi_1(X)]} V)$ for a left $\mathbb{Z}[\pi_1(X)]$-module $V$ admit some kind of long exact sequence for a pair $(X,A)$?
This reminds me of one of my all-time favorite questions (which @MikeM will be glad to know was first posed to me by Jerry Kazdan, of Kazdan-Warner). Give a function with a unique critical point, which is a local minimum but not a global minimum.
you can also do it by imitating the definition of the ovals of cassini: $$f(x,y)=((x-1)^2+y^2)^2((x+\frac12)^2+(y-\frac{\sqrt{3}}2)^2((x+\frac12)^2+(y+\frac{\sqrt{3}}{2})^2$$
@MikeMiller I was thinking one could define $H_n(X,A;V) = H_n(C_*(X,p^{-1}(A)) \otimes_{\mathbb{Z}[\pi_1(X,x_0)]} V)$ where $p \colon \widetilde{X} \to X$ is the universal covering.
Yes, May, Davis-Kirk, Whitehead and Hatcher all use bundles as coefficients system when it comes to saying that twisted homology defines a homology theory with a long exact sequence
The thing is, I found a paper where, for a special kind of subspace A, a long exact sequence is constructed and wanted to generalize it / understand it.
I sure understand that, thanks for thinking about it :) I now have come to the point that I think this is what is needed.
(just for the record, above I made a mistake, $C_*(A)$ needs to be a $\mathbb{Z}[\pi_1(X,x_0)]$-module, not $C_*(A;V) = C_*(A) \otimes_{\mathbb{Z}[\pi_1(X,x_0)]} V$)
now that you mention it, I had to work through numerical analysis things for an exam which I found pretty boring, but did it anyway.
of course, you always find something interesting, even if its just unrelated stuff like thinking about if there is anything categorial or something in it :)
@Ted: Got it. My curves lies on a sphere, so $\gamma \cdot \gamma' = 0$. Differentiate: $\|\gamma'\|^2 + \gamma \cdot \gamma'' = 0$. Assume $\gamma$ is arclength parametrized to get $\gamma'' \cdot \gamma = -1$.
That is, $k \mathbf{N} \cdot \gamma/\|\gamma\| = -1/a$, i.e. $k \mathbf{N} \cdot \mathbf{n} = -1/a$. Left side is exactly $-\Bbb{II}(\gamma', \gamma')$, so by Meusnier $\kappa \cos(\theta) = 1/a$. Since $\theta$ (the lattitude) goes from $0$ to $\pi$, it's positive and less than $1$, hence $\kappa \geq 1/a$.
@Balarka: You can get it directly from Meusnier (without differentiating your $\gamma$) if you remember that you know what all normal curvatures are on the sphere.
So you know how a the graph of a cube is just the graph of two squares connected at common vertices? Is there a term for doing that in general to a graph?
