Exam went well yesterday. They did call me once, but that was really a non-issue. Just a student wondering whether I really did mean to write $\sigma^{-1}$ when I asked them to find a $\gamma$ such that $\gamma\sigma\gamma^{-1} = \sigma^{-1}$. And yes, I really did mean that, since otherwise the exercise would be trivial.
In the same paper there is mention that $d(\alpha(X))_x = \alpha_x\circ A_x$ for a (contact) 1-form $\alpha$, a contact vector field $X$, and its linearisation at $x$, $L_x$.
two of the students handed in 25 pages each. I hope this is because they used a new page for each subquestion, but I didn't open the envelopes to check yet
@Semiclassical As I said, hopefully they just wasted a bunch of paper. I mean, I don't see how you could meaningfully write 25 pages in just 4 hours by hand
Interesting proof of Poincare's lemma: Say $\omega$ is a closed $k$-form on a ball $B$ centered at 0 in $\Bbb R^n$. Consider the flow $f_t : \Bbb R^n \to \Bbb R^n$ such that $f_t(x) = e^{-t} x$, which preserves the ball. This gives rise to a vector field $X$ on $\Bbb R^n$.
Define $F(\eta) = \int_0^\infty f_t^* \eta dt$. Then $F(\mathcal{L}_X \omega) = \omega$. By Cartan's magic formula, $\omega = F(d \iota_X \omega) + F(\iota_X d\omega)$. The second term vanishes as $\omega$ is closed; $F$ and $d$ commutes by differentiation under the integral sign. So $\omega = d(F(\iota_X \omega))$.
@AlessandroCodenotti algebra. One question on CRT, one on permutations, one on rings (and unique factorization), and one on Sylow theory with a bit of representation theory
Hello, is it true: $f:X\to R$ is not bouded from bellow means that there exists $u_n\in X$ such that $||u_n||\to+\infty$ implies that $f(u_n)\to-\infty$ ?
i mean i think both of the writers you mentioned write better things but they're also not writing in english, shakespeare is eminently english and that's an important thing
@Eric I like to say that Borges's Labyrinths, Dostoyevsky's "Notes from the Underground" and Eliot's The Waste Land are respectively my versions of Quran, Bible and Upanishad
