/sci/ - Science & Math

>>11542698
Yes, disappointing.
Allow me to be the first to post something of actual interest.
https://arxiv.org/abs/1012.2635
>We outline a proof of a remarkable conjecture of Labastida-Mariño-Ooguri-Vafa about certain new algebraic structures of quantum link invariants and the integrality of infinite family of new topological invariants. Our method is based on the cut-and-join analysis and a special rational ring characterizing the structure of the Chern-Simons partition function.
The LMOV conjecture in essence states that, in the large-[math]N[/math] limit, [math]U_q(\mathfrak{sl}_2)[/math]-coloured Turaev-Viro TQFT encodes the same physics as topological strings. The latter theory is a CFT with geometric datum, hence this means that we can in a sense "assemble" topological invariants (e.g. Maslov, rational linking form) into geometric invariants (e.g. Gromov-Witten, quantum traces) through this duality by algebraically (e.g. Khovanov homology) organizing the data on either side.

>>11518981
>no Kahler nor hyperKahler
Won't be voting

>>11420128
I'm not transphobic simply for the fact that it is not conducive to research.
>>11420133
This general isn't any better hun.

>>11387886
First of all the problem is asking about the ring structure of [math]H^\ast(\mathbb{R}P^n,\mathbb{Z})[/math]. The stuff aout pizza's and bananas is about [math]{\bf Ext}[/math] and [math]{\bf Hom}[/math], which isn't useful here since they're for computing the spectral sequences for the stable cohomotopy theory.
Let us consider [math]H^\ast(\mathbb{R}P^n,\mathbb{Z}) \cong H_\ast(\mathbb{R}P^n,\mathbb{Z})[/math] as a polynomial ring over [math]\mathbb{Z}[/math]. We are done if we can find all generators and their relations. Toward this end we write [math]\mathbb{R}P^n \cong S^n\mathbb{Z}_2[/math] where the antipode map [math]\mathbb{Z}_2[/math] acts transitively. This means that we pick up
1. a free generator [math]e_0,e_n[/math] at degree [math]0,n[/math]
from the sphere [math]S^n[/math], and
2. a 2-torsor [math]e_1\in \mathbb{Z}_2[/math] at degree 1 since paths linking antipodal points descend to a non-contractible loop, hence [math]\pi_1(\mathbb{R}P^n)\cong\mathbb{Z}_2[/math] and [math][\pi_1]_{\operatorname{ab}} \cong H_1[/math] by Hurewicz.
To understand what happens at other degrees we decompose [math]\mathbb{R}P^n[/math] into a finite CW complex, whose [math]k>0[/math]-skeletons are endowed with a monodromy action by [math]\pi_1[/math]. Due to the CW structure, a map [math]f:S^k \rightarrow \mathbb{R}P^n[/math] is determined by its value on the [math]k[/math]-skeleton, and hence by its degree [math]\operatorname{deg}f[/math]. Due to the monodromy action this degree is again 2-torsion.
From basic algebraic topology the degree is only non-trivial for [math]k[/math] odd, hence we get 2-torsors at each odd degree, while anything at all even degrees (except 0) are killed off. Hence [math]H^\ast(\mathbb{R}P^n,\mathbb{Z}) = \mathbb{Z}[e_0,\dots,e_n]/\prod_{m \text{ odd}\leq n}\langle e_m^2-1\rangle[/math].

>>11156988
>NOBODY
>CLEARLY
Were you the original anon that asked the question? Note I didn't mention Lebesgue spaces at all when replying to him.
There is only one sense in which "linear independence" is asked [math]specifically[/math] for Chebyshev polynomials, and that's the Sturm-Liouville sense for ONBs of some[math]L^2[/math] space. Why would an elementary linear algebra class ask for linear independence for them instead of just monomials? Why add the extra fluff if they're not even relevant to the basic argument? Nameone single lin alg book that has a question like this.
It is obvious that not everyone who's taken mathematical methods class knows about Lebesgue-Hilbert spaces but that's the setting in which it is phrased. And no, in this setting the linear independence of the polynomial kind do not at all carry over to that of the [math]L^2[/math] kind.
Your post is pure and applied garbage. Please don't reply to me ever again.

>>9028199
>Gorilla has been so talkative that I already know who both of you are.
Truly a suboptimal situation. Well, I'm leaving town in a month so I guess I don't need to worry too much.

>>9018855
Right, the produced results aren't constructive is it? It would only state that such-and-such statements are provably true but wouldn't give you any hints as to how to actually prove it, right?
Why even do this again?