/sci/ - Science & Math

>>11559477
>harmonic maps, flatness of projective connections, HYM vortices etc are trash and boring
Haha ok kid, don't get uppity.

>>11552288
Yeah, but given a choice of a Riemannian metric they're isomorphic [math]TM \cong T^*M[/math] as each others' duals. This isn't canonical but me writing [math]TM\cong V[/math] isn't canonical either so whatever.

Just know that I literally don't care what you think of me.

>>11494427
The blackhole is a point singularity. Its event horizon is a 3-sphere at past/future infinity, namely any geodesic near it evolves to [math]t=\pm\infty[/math] in finite proper time.
>>11495506
That sounds interesting; can you give an example where people model spin-glasses, or some other long-range order, with cellular automata?

>>11455722
String theory is a CFT, not a QFT. The observables are constructed from chiral Virasoro algebras (glorified ladders) instead of generic field operator nets, which behaves much nicer both algebraically and analytically. This is in essence due to the much larger underlying conformal symmetry CFTs enjoy, and string theory takes full advantage of that. In fact, you can leverage modular invariance to show that the OPE coefficients (conformal blocks) are all holomorphic on [math]S^2\setminis \{0,1,\infty\}[/math] so you can use Laurent expansion techniques to solve the KZ equation, for instance.
You would not be able to prove most things you encounter in CFT, such as the Ward-Takahashi or Yang-Baxter, as easily in general QFTs, but drawing inspiration from CFTs at least allows us to understand what algebraic structures for the local operator nets are necessary in order to yield a consistent QFT. They don't solve the regularity/summability issues of [math]S[/math]-matrices however.

Bump.
https://arxiv.org/abs/1712.02952

>>11320925
11p, double column, alpha citations.

>>11268846
It's either a pinched nerve or inguinal hernia; prostate problems won't just happen when you DL. Go see a physio.

>>11156924
>Moduli spaces and arithmetic statistics
Might be interesting.

>>11110994
arxiv.GM

>write very neat and tight notes
>friends and colleagues keep calling my handwriting girly

>>11065997
See, I knew something was weird.
First of all, does [math]V[/math] have a norm equipped so you can talk about convergence? Or is it formally arbitrary-ranked? If it's the latter then testing linear independence is by definition the same as testing linear independence on every finite subset.

>>11032990
Don't talking badly about Dror please. His graphics are very helpful.
>tfw he called me a retard once

>>10984372
>math.GM

No recap this time huh?
>>10983259
the hint gives you the solution.
>>10983429
Use faltung theorem.

>>10918896
Wake me up when you get to TMFs.

>>10864100
>string theory has beautiful mathematics and has profound implications in analytic, algebraic and arithmetic geometry and may even help the geometric Langlands program progress
Yes.
>reality has inherent hidden degrees of freedom described by a degree-3 algebraic hyperkahler curve embedded in Kaluza-Klein-esque compact dimensions
Lmao no.

>>10382333
>been on HRT since prepuberty
>still have 5 o'clock shadow
Genuinely feel bad for this kid desu

>>10341169
>you flirt with LQG
But I don't? I've barely done any quantum gravity to begin with

>>10150929
[math]p=2[/math] is nice.

>>10125749
In general an extension is a lift along the inclusion homomorphism [math]\iota: A\hookrightarrow B[/math]. Suppose [math]f:A \rightarrow C\in \operatorname{Hom}(A,C)[/math] as a morphism in a category then an extension is a morphism [math]g\in \operatorname{Hom}(B,C)[/math] such that [math]g = f \circ \iota[/math]. This doesn't just mean what this anon >>10125789 said, but [math]g[/math] must also inherit the same properties as [math]f[/math], such as continuity.
>>10126741
No. The energy density determines the overall curvature of spacetime via GR and nothing more. What a unit (i.e. critical) energy density means that the universe is globally flat, which means that it does not have any tendency towards expanding or contracting. This doesn't contradict the fact that the universe is locally expanding.

>>10107950
>wants to do astronomy
Yikes, just be honest and say that you failed out of your first year calc class lmfao

>>9061550
Witten does so we don't have to.

>>9057709
[math]\mathbb{C O N T O U R S}[/math]
[math]\mathbb{O}[/math]
[math]\mathbb{N}[/math]
[math]\mathbb{T}[/math]
[math]\mathbb{O}[/math]
[math]\mathbb{U}[/math]
[math]\mathbb{R}[/math]
[math]\mathbb{S}[/math]

>Classical conformal field theory is defined as a conformal field theory where the conformal weights of all primary fields vanish. In this case, conformal field theory is nothing but group theory.