It doesn't take a genius to deduce that Petersen and I have different teaching styles and views of geometry. But my style displeases plenty of people, too.
But Cartan thought of torsion as coming from the infinitesimal displacement of the basepoint of vectors (affine, as opposed to linear) when you parallel translate.
Teaching has also gotten more fun now that my classes are willing to ask questions... this week I've been able to use the first half or so of discussion for answering questions
It takes a knack to encourage student comfort and interaction, but once you learn to do it, you'll enjoy teaching way more than the average bear, @Mike. Of course, most mathematicians don't give a s***.
@TedShifrin For line bundles we have an inverse line bundle, i.e., another for which $E \otimes E^* = \varepsilon$, where $\varepsilon$ is the trivial line bundle. This is essentially because 1-matrices commute, so we can just define $g^\ast_{\alpha \beta} = g^{-1}_{\alpha \beta}$, and this new set of things still verify the cocyle condition. For a general vector bundle $E$, is there a "stable inverse" - another vector bundle $E^*$ (I'd like of the same rank)
such that $E \otimes E^\ast = \varepsilon^n$ for some $n$?
What would Chern class formulas tell you, @Mike? I suspect it's wrong. Sadly, one of our grad students stole my Hirzebruch (forever) ... but even though I know how to do this, I don't want to sit down and write it out.
I'm also looking at some stuff Hatcher wrote about vector bundles and characteristic classes... I might be one of two people who isn't that fond of Milnor's style
@user939259 You have $y = 2442556$ for $x = 10.844712025993891$, so the distance cannot be more than $7.8447...$. Moving $x$ a little towards $3$ will decrease the distance a bit, but very soon will increase it again, so the distance will be close to that.
I need to learn some characteristic classes and Morse theory, @Ted, so I checked out those two books. I have only looked at characteristic classes so far and I really dislike it. Do you have a nicer source?
I don't know Hatcher's stuff on that, @Mike. I learned most of it (and have taught it) from the diff geo/curvature perspective. Even though it's formal, I like Hirzebruch's classic.
Is it ok to edit and extend an OP such that the question becomes more general , when a bounty has already been placed and that answer has already been given ?
@TedShifrin This is all differentiable, despite the title? At least at present, I'm looking for the algebraic topological/differential geometrical POV... if there's an algebraic geometrical one, it'll have to come later :)
I am trying to prove that convex sets in R^n can be separated by open balls, R^n being equipped with the usual topology. Should I think algebraically or geometrically?
@DanielFischer it is easy to show that $\frac{\log(n)}{n-1}$ is decreasing for $n\ge2$. Thus, $$\sum_{k=2}^\infty\frac{\log(n)}{n!}\le\log(2)\left(\frac1{2!}+\frac2{3!} +\frac{\log(4)}{3}\sum_{n=4}^\infty\frac{n-1}{n!}\right)$$
OK. $A$ and $B$ be compact convex sets in $\Bbb R^n$. $d(x, A)$ is a nonnegative real for any point $x \in A$ as $A$ is closed. Take the segment $\ell = \inf_{x \in B} d(x, A)$ and take the segment in $\ell$ of length $\ell/2 + \varepsilon$ with $0 < \varepsilon < \ell/2$.
The open sphere tangent to the line perpendicular to $\ell$ at $\ell/2 + \varepsilon$ (and tangent to that very point) of radius $r > \ell/2 + \varepsilon + d(A)$ contains $A$ entirely. Doing the same with $B$ gives an open sphere which contains $B$ entirely.
I can't figure out how convexity comes to play, though. Apparently it does.
@TedShifrin You can flag any post of the user for mod attention, pointing out deliberate impersonation of you [violation of SE terms of service). I think they will rename.
@BalarkaSen Ummm. If I understood it correctly, it contains the right idea and you can make a proof from it if you take the time to actually properly define everything and write down the details.
@MartinSleziak, the reason why I was asking was that I wanted to know whether you assign homework, and if so, how much time one would spend on it weekly and how much it would account for when calculating the final grade. Do you have any further input on that? I realize it differs from country to country, but how is it at your university?
