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Dec
29
comment Contracting a contractible set in $\mathbb R^2$
$S^1\subset\mathbb R^2$ is compact and connected but not contractible.
Dec
29
comment Concrete non trivial computation of Morse homology
[Let's wait for answers but] I've always thought it's mainly useful for (co)homology of loop space and such [and not in finite-dimensional situation]...
Dec
27
comment Applications of universal coefficient theorem
possible duplicate of Why homology with coefficients?
Dec
27
comment Conditions for a Topological space to be a Spectrum
@UserSomeNumber Why, of course everything can (and should) be considered up to (coherent) homotopy equivalence
Dec
26
comment Relationship between topological and Quillen's K-theory
(Somewhat) related: Topological vs. Algebraic K-Theory
Dec
26
comment Conditions for a Topological space to be a Spectrum
@UserSomeNumber One space can have different structures of infinite loop space. So ($\Omega^\infty$-)spectrum is not just a space that can be infinitely delooped, it's a space together with some choice of such deloopings.
Dec
25
comment Intuition of Chern-Weil theory
Good question! Basically, Chern–Weil formulas come from (Lie algebra) cohomology of $g$ and its close relative $(S^\ast g[2])^{g}$ as an algebraic model for the universal bundle $EG\to BG$. (I hope to write an answer later.)
Dec
25
comment “Why” is $[\mathbb{C}:\mathbb{R}] < \infty$?
@omar That's more or less the reply I was typing :-) (Related question @ MO: mathoverflow.net/q/8756 )
Dec
25
comment “Why” is $[\mathbb{C}:\mathbb{R}] < \infty$?
For one thing, any polynomial of odd degree has a real root...
Dec
24
comment Solving 5th degree or higher equations
One way to prove this, btw, is Lagrange inversion formula.
Dec
24
comment Does a four-variable analog of the Hall-Witt identity exist?
Related: negative answer @ MO
Dec
24
comment Does a four-variable analog of the Hall-Witt identity exist?
See also lamington.wordpress.com/2011/11/20/the-hall-witt-identity
Dec
23
comment The close form expression of a Pfaffian
$\operatorname{Pf}(1/(x_j-x_i))$ can be computed using Cauchy determinant identity, perhaps
Dec
22
comment Lie algebra of Derivations as a functor?
$\operatorname{Der}(A,M)$ ($M$-valued derivations) is a functor in $M$. As for $\operatorname{Der}(A,A)$, it's in a sense an algebraic version of tangent bundle...
Dec
20
comment Triple Euler sum result $\sum_{k\geq 1}\frac{H_k^{(2)}H_k }{k^2}=\zeta(2)\zeta(3)+\zeta(5)$
Related: math.stackexchange.com/q/606070
Dec
20
comment What comes after $\cos(\tfrac{2\pi}{7})^{1/3}+\cos(\tfrac{4\pi}{7})^{1/3}+\cos(\tfrac{6\pi}{7})^{1/3}$?
(Re: What Ramanujan formula) see (3.8) and (3.18) in maa.org/sites/default/files/pdf/upload_library/22/Ford/…
Dec
20
comment What comes after $\cos(\tfrac{2\pi}{7})^{1/3}+\cos(\tfrac{4\pi}{7})^{1/3}+\cos(\tfrac{6\pi}{7})^{1/3}$?
All cosines in (2) are roots of a (rational) cubic equation. So one can use Ramanujan's formula for the sum of cubic roots of roots of a cubic equation and obtain (2). All cosines in (3) are of course roots of a quintic equation (namely, $32x^5+16x^4-32x^3-12x^2+6x+1=0$)...
Dec
19
comment Explanations of the Euler's continued fractions to compute exponential
(slightly) related: Continued fraction for $\frac{1}{e-2}$
Dec
19
comment How to prove $H^2(\mathfrak{g}, J(\mathfrak{g}))\neq0$, where $J(\mathfrak{g})$ is the augmentation ideal of $\mathfrak{g}$?
repost @ MO: mathoverflow.net/q/83169
Dec
19
comment Proving that $\frac{\pi^{3}}{32}=1-\sum_{k=1}^{\infty}\frac{2k(2k+1)\zeta(2k+2)}{4^{2k+2}}$
I'm not sure I understand — certainly you won't be satisfied $\frac27(1+2\sum\eta(2k)2^{-2k})$ — but what's the question exactly then? Anyway, first question is, what is $\sum\zeta(2k)x^{2k}$ and $\sum\eta(2k)x^{2k}$ — do you happen to know answers?