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Aug
6
comment Algebraic methods to compute the cohomology ring of the complex topology of a variety?
@MarianoSuárez-Alvarez Thanks. I was wrong. The original theorem is on holomorphic submanifolds of $\mathbb C^n$, not for projective ones.
Aug
5
comment Algebraic methods to compute the cohomology ring of the complex topology of a variety?
@MarianoSuárez-Alvarez For projective algebraic curves, it's fairly easy since the geometric genus $2g-2$ is the only topological invariant.
Aug
2
comment Algebraic methods to compute the cohomology ring of the complex topology of a variety?
@QiaochuYuan Induction method is concerned with the embedding $\mathbb P_\mathbb C^{n-1}\to\mathbb P_\mathbb C^n$, and the spectral sequence associates to $S^1\to S^{2n+1}\to\mathbb P_\mathbb C^n$. Right, $\mathbb P_\mathbb C^n=e^0\cup e^2\cup\dotsb\cup e^{2n}$, but I don't know whether the cup product is easily computed.
Aug
2
comment Algebraic methods to compute the cohomology ring of the complex topology of a variety?
@MarianoSuárez-Alvarez Thanks. If it's a complex manifold, there's a CW decomposition of $V(\mathbb C)$ with only even dimensional cells (cf. Milnor's Morse Theory) and therefore the cohomology is torsion-free, and one can recover $\mathbb Z$-cohomology from $\mathbb R$-cohomology? Thanks for the reference.
Jul
30
comment Cohomology with Coefficients in the sheaf of distributions
@Exterior The fine resolution is similar to $0\to\mathbb R\to\Omega^0\to\Omega^1\to\Omega^2\to\dotsb$, where $\Omega^p$ is the sheaf of $p$-forms, replacing $\Omega^p$'s with corresponding sheaves of distributions of cross-sections.
Jul
30
comment Cohomology with Coefficients in the sheaf of distributions
@Exterior Suppose $\xi\colon E\to M$ is a smooth vector bundle, then we can define sheaf of distribution cross-sections of $\xi$, see Gunning's Lectures on Riemann Surfaces, section 6 (some terms are archaic). Especially, consider exterior products of the cotangent bundle, we obtain a sequence of vector bundles, and we can consider the corresponding sheaves.
Jul
18
comment How can I prove Dini's theorem using the Baire Category theorem?
Fix $\epsilon>0$, and set $E_n=\{f(x)\ge\epsilon\}$, which is decreasing and $\bigcap_nE_n=\emptyset$, then there exists $N$ s.t. $E_N=\emptyset$.
Jul
18
comment Approximation of irrational numbers?
The original problem from the interview (finished but maybe inappropriate to propagate) is that when $\theta$ is the unique root of $P\in\mathbb Z[X]$ such that $\lvert\theta\rvert\ge1$, where $P$ is monic and $d=\operatorname{deg}P\ge2$, then there's a constant $C>1$ independent of $d,P$ such that $\lvert\theta\rvert\ge C$. They gave me a hint that assuming $P(X)/(X^dP(1/X))=\sum_jb_jX^j$ and the identity $1+\theta^2=b_0^2+\sum_j(b_j-\theta b_{j-1})^2$. I have no clear idea to do next. Now your post tells me that my conjectures are all wrong. Do you have any idea? Thanks!
Jul
17
comment Approximation of irrational numbers?
@Théophile I found number theorists usually call algebraic integers as integers, thus to disambiguate, I added the adjective rational.
Jun
7
comment Quotient manifold theorem provides a fibration?
Okay, the submersion, in fact, furnishes local sections, and therefore bundle structure. I would rather refer to the bundle structure theorem in section 7.4 of Steenrod's Topology of fiber bundles.
May
9
comment Cohomology with Coefficients in the sheaf of distributions
@ZhenLin Ugh, but why can't we restrict a Schwartz distribution to an open set? Even more, it constitutes a fine sheaf, so are sheaves of currents, which forms a fine resolution of the constant sheaf $\mathbb R$, say.
May
9
comment Why does the Residue Theorem still hold, when I let my contour get infinitely large?
You cannot make the contour infinitely large. Instead, you choose a family of contours, deriving an equation from each contour, then take limit.
May
9
comment Long exact sequence for a triple follows from long exact sequence for a pair?
Unfortunately, I had no time to check the proof, therefore I directly accepted the answer. Now it seems to me that complexes per se are more substantial than homology groups, from a viewpoint of, say, derived category or homotopical algebra. I will return this topic after pursuing homotopy theory.
May
9
comment Comments on Eilenberg and Steenrod's “Foundations of algebraic topology” and other similar books for recomendation
And I don't know to what extent by homological algebra you mean. Just that of Cartan & Eilenberg era or of more modern stage such as derived category? Related: MO thread and this.
May
9
comment Comments on Eilenberg and Steenrod's “Foundations of algebraic topology” and other similar books for recomendation
I'm still a novice, but I think if you want an abstract, axiomatic and conceptual way towards algebraic topology, maybe homotopy-first books are more appropriate. Perhaps J.P.May's A Concise Course in Algebraic Topology works for you. It's free online.
Apr
17
comment Books on locally convex topological vector spaces
It seems to me that it's flawed that Rudin uses little categorical language, and maybe viewpoint.
Mar
28
comment Rigorous pre-calc book with answers
@crash I didn't really understand what's pre-calculus, but I suggested Concrete Mathematics because most part of the book isn't relied on calculus and should prepare readers techniques of manipulating $\sum$, binomial coefficients, etc. Well, on the other hand, I started my first systematic study of calculus from Rudin's Principles of Mathematical Analysis. It's concise but terse and without solutions.
Mar
28
comment Does differentiability have a geometric interpretation for high dimensional functions?
It's still a linear approximation. Instead of a linear function when $m=1$, it's approximated by a linear transformation represented by the matrix.
Mar
28
comment Prove that $Ker(g \otimes k)= Im(f \otimes 1_{N}) + Im (1_{M} \otimes h)$
@LuisVera Edited. Hope it's clearer now.
Mar
28
comment Rigorous pre-calc book with answers
Take a look at Knuth's Concrete Mathematics to see whether it's what you like?