intuitively I'd say no, because I need an open ball around every point of my open set, but I know not to trust my intuition so I'll think about it for a moment
I'll do reverse-order since everybody seems to be more excited about the last option :P
So that was doing something based on a paper by Thurston, namely this one on "slithering" (something combining foliations and fibrations). It's low-dimensional topology type-stuff
The second option was something related a little bit to complex geometry---he said it was essentially about exploring the consequences of some formula for the so-called $\chi_y$-genus, using a general version of Hirzebruch-Riemann-Roch. Based on this paper
The first one was looking at this paper, and perhaps trying to prove (at least a special case of) the main theorem by more "elementary" methods, in particular avoiding something like a Atiyah-Singer index type thing
ok, I now think I can choose the same $x$. I don't need to find a ball around $y$, since I'm proving it for arbitrary $x$ and $\epsilon$ I'm also proving it for a ball centered in $y$ and small enough that $B_d(y,\epsilon ')\subseteq B_d(x,\epsilon)$, so the ball with respect to $d'$ contained in $B_d(y,\epsilon ')$ is also contained in $B_d(x,\epsilon)$
Now, given $\epsilon>0$, can you find $\epsilon'$ so that $B_{d'}(x,\epsilon')\subset B_d(x,\epsilon)$ and $\epsilon'$ so that $B_d(x,\epsilon')\subset B_{d'}(x,\epsilon)$?
@TedShifrin In response to this, I mentioned also to Kotschick that I'm not really anywhere near familiar with things I need to deal with e.g. Atiyah-Singer, and he didn't seem to mind much. He basically said I wouldn't need to understand the full proof to still be able to get the statement and use it
And the first section of chapter 3 seemed "trivial" in comparison to what I'm doing now---I read through it and didn't see many hard statements except maybe the Kaehler identities
@TedShifrin Eh, yeah---I didn't really think in terms of explicit expressions for the vector fields involved but after I finished the proof I looked around more on the internet and found an MO thread that discusses exactly what you said
@TedShifrin I'm wondering how physicists interpret them!
I actually wrote a MSE thing on the Euler sequence, too. .... The Euler vector field moves you along the lines through the origin (the rulings of the cones I keep mentioning).
...since it's "well-known" that you need Calabi-Yau (and in particular Kaehler) to get supersymmetry in the effective 4d theory coming from superstring theory in 10d
there has to be some interpretation
@TedShifrin I think I saw that too, though I didn't spend as much time looking at that thread.
I think I do, that was the first exercise in which I had to prove that $2$ topologies are the same, but I have a few more similar to this one, I think I should be able to do them on my own, tomorrow though! For now thanks a lot for your help @Ted and good night!
Hey @TedShifrin, Kotschick said today that a pullback bundle is something like a "[insert word I forgot] product" because you can define as $\{(x,v)\in M\times V\mid f(x)=\pi(v)\}$ where $f:M\to N$ and $V$ is a bundle over $N$ with projection $\pi$
You'll learn it eventually to learn about various associated bundles (like building different vector bundles from a principal bundle and a representation of the group)
@TedShifrin If you have a functor that distinguishes non-isomorphic objects, then the result of applying uniquely determines your object, i.e. it's a "perfect" invariant.
