/sci/ - Science & Math

>>11485032
You can treat [math]f[/math] as a cocycle representing a class in [math]H^*(BG,\mathbb{Z})[/math]. Then you force equivariance under the [math]G[/math]-action and the cocycle condition [math][g\cdot,\delta] = 0[/math] gets you an alternating sum.
>how in hell can you get H*(G,A) with nontrivial action of G on A from a functor from Top
We have [math]H^*(BG) \cong H^*_G(\bullet,\mathbb{Z})[/math] for which you can then apply UTC to send [math]H^*_G(\bullet,\mathbb{Z})[/math] into [math]H^*_G(\bullet,A)[/math] for Abelian [math]A[/math]. These maps are classified by [math]G[/math]-fixed points of [math]\operatorname{Hom}^{s,t}(\mathbb{Z},A)[/math] and the derived objects thereof.
>no way that K(G,1) or any space remembers about the group structure of G
But [math]K(G,1)[/math] remembers [math]G[/math] through its loops? Or are you asking for something else?

>>11476957
"Spins" are irreps of [math]\mathfrak{su}(2)[/math], nothing more and nothing less. The analogy with angular momentum comes from the fact that [math]SU(2) \cong \operatorname{Spin}(3)[/math] where [math]\operatorname{Spin}(3)[/math] is the double cover of [math]SO(3)[/math].

>>11246003
Naively speaking group theory would definitely be relevant in studying graph isomorphisms; e.g. vertex/edge permutations. I'm not an expert in combinatorics though so I'm not aware of any deeper relationships.

>>11043628
Science says that you have a female brain and should probably speak to a gender therapist.

>>10897982
>most results about vector bundles (especially Serre duality and Riemann-Roch) rest on analytic results (Fredholm theory and elliptic PDE iirc)
It's the other way around really. Riemann-Roch doesn't require any anal beyond [math]\overline{\paritla}[/math]-Dolbeault diff alg's, while it can say something about the rank of holomorphic sections. Things like existence and index problems in anal can be tackled generally with alg top/geo.
An extreme case is Gromov-Witten, where point-counting in a Schubert variety tells us the rank of the [math]D[/math]-module of some exactly solvable DE, including KdV and NLSE.

>>10835112
> vixra
> Mohamed

>de facto

>>10743447
>thought I could escape reductionism in math
>see this bullshit

>>10713975

>>10377822
In general if [math]T\in B(\mathcal{H})[/math] on a separable Hilbert space [math]\mathcal{H}[/math] is Toeplitz then [math]|T| \leq |\operatorname{Spec}T(W)|[/math], where [math]W[/math] is the Wiener-Hopf symbol (math)/kernel (physics) of [math]T[/math] and the norm [math]|\cdot|[/math] here is the (virtual) Lebesgue measure. Equality is saturated by diagonalizable [math]T[/math], i.e. if [math]X = L^2(S^1,\mathbb{C})[/math] with a [math]L^1[/math] kernel [math]W[/math]: Fourier transform leads to [math]T[/math] being Toeplitz and unitary.
See Bottcher's book for a proof.

>>10267729
>he panic uploaded to arXiv.GM
>put him to sleep then pump Atiyah's Alzheimer's for 5 hours

>to prove the lemma, consider the following algorithm

>be at conference
>attended every talk I came for the previous days
>bored, attend string theory seminar
>ask how these string ground states are characterized
>"oh, different geometric configurations give rise to a distinct ground states!"
>speaker talks about how one single isometry class of strings leads to a whole family of ground states as if it's a good thing
>mfw they're not topologically characterized
>mfw not even isotopic configurations of strings do not give rise to the same ground state rays
And that's when I realized that string theory is bullshit.

>>10160645
Like it or not tools inspired by physics is being used to solve problems in pure topology (c.f. monopole Floer cohomology & Seiberg-Witten theory).

>>10140064
"Particle in a box' typically means we apply Born-von Karmen boundary conditions, or in other words we study quantum mechanics on the Pointrjagyn dual of the box instead.
Let [math]H[/math] be a Hamiltonian that acts on the Hilbert space of sections [math]\Gamma(L^d,F) \cong L^2(L^d,\mathbb{C})[/math] of a FLAT Hermitian line bundle [math]B \rightarrow L^d[/math]. Let us suppose the Weil integrality condition is satisfied and a proper holomorphic polarization [math]P[/math] on the tangent bundle [math]TL^d[/math] is given to simply our discussion.
For each Hilbert section [math]\psi: L^d \rightarrow \mathbb{C}[/math] covariantly constant on [math]P[/math] (i.e. a [math]P[/math]-wavefunction), Born-von Karmen periodic boundary conditions means that [math]\psi[/math] descends to the quotient space [math]L^d/\sim \cong \mathbb{T}^d[/math], where [math]x \sim y[/math] iff [math]y_i = x_i + L[/math] for some [math]1 \leq i \leq d[/math] and [math]y_j = x_j[/math] for all [math]j \neq i[/math]. Now since [math]L^d[/math] is compact, the Fourier transform [math]\mathcal{F}[/math] is an isometric, fibre-preserving automorphism on the Hilbert sections of the Hermtian line bundle [math]B\rightarrow \mathbb{T}^d[/math], hence [math]\mathcal{F}\psi[/math] is a Hilbert section on the Hermitian line bundle [math]B' \rightarrow \mathbb{Z}^d[/math], where [math]\mathbb{Z}^d[/math] is the Pontrjagyn dual of [math]\mathbb{T}^d[/math]. If [math]\operatorname{Spec}H[/math] is bounded from below, then spectrum of its "Fourier pullback" [math]\mathcal{F}^*H[/math] is isomorphic to [math]\mathbb{N}^d[/math]. These give rise to the discrete/quantized energy levels.
Hope this is clear enough.

>>10131604
In general the ladder operators are generators of the Heisenberg algebra [math]H[/math], which is a (possibly infinitely-generated) free [math]*[/math]-algebra. A unitary representation [math]\rho: H \rightarrow \mathcal{U}(\mathcal{H})[/math] on a Hilbert space [math]\mathcal{H}[/math] is possible if the number operator [math]N = a_i^\dagger a_i[/math] is well-defined, and this is equivalent to the existence of a cyclic vacuum state [math]|0\rangle \in \mathcal{H}[/math] such that [math]a_i |0\rangle = 0[/math] for every [math]i[/math]. Hence for ever ladder operator problem you should think to yourself: what is this vacuum state? Can you make it from [math]\hat{x}_i[/math] or [math]\hat{p}_i[/math]? If the answer is yes then the ladder operators are well-defined.
>>10133084
Let [math]A \in \operatorname{Hom}(V,V)[/math] and suppose [math]\operatorname{Spec}A = \{\lambda,\mu\}[/math] with [math]\lambda \neq \mu[/math]. Define projectors [math]P_{x} = A - x I[/math] for [math]x = \lambda,\mu[/math], we have subspaces [math]W_{\lambda,\mu} = \operatorname{ker}P_{\lambda,\mu}[/math], then it suffices to show [math]W_\lambda \perp W_\mu[/math]. Suppose [math]{\bf a} \in W_\lambda \cap W_\mu[/math], then [math]P_{\lambda,\mu}{\bf a} = 0[/math] so [math]\lambda {\bf a} = \mu{\bf a}[/math]. But [math]\lambda \neq \mu[/math] so [math]{\bf a} = 0[/math], which implies [math]W_\lambda \cap W_\mu = 0[/math] and hence [math]W_\lambda \perp W_\mu[/math]. Since [math]{\bf x} \in W_\lambda[/math] and [math]{\bf y}\in W_\mu[/math], they must be linearly independent.

>>9918853
Let [math](\mathcal{H},D)[/math] be a Dirac theory on a Reimannian manifold [math]M[/math]. Suppose [math]\operatorname{dim}M[/math] is even, then the spinor bundle [math]\mathbb{S}\rightarrow M[/math] admits a skew-symmetric "chirality operator" [math]\gamma = i \prod_{i=1}^{n}\gamma_\mu \in \operatorname{End}(\mathbb{S}) \cong \operatorname{Cliff}^{n,1}(\mathbb{C})[/math] which satisfies [math]\{\gamma,\gamma_\mu\} = 0[/math], and hence [math]\{\gamma,D\} = 0[/math]. This allows the decomposition of the Hilbert space [math]\mathcal{H} = \mathcal{H}_+ \oplus \mathacl{H}_-[/math] into chirality eigenspaces [math]\mathcal{H}_\pm[/math] such that [eqn]D = \begin{pmatrix} 0 & D_+ \\ D_- & 0 \end{pmatrix}[/eqn].
For any principal gauge [math]G[/math]-bundle [math]P\rightarrow M[/math] compatible with the spinor bundle, its representation in [math]\mathcal{H}[/math] then admits a chiral central extension [math]\tilde{G}[/math] by [math]\mathbb{Z}_2[/math] (this is why models with chiral symmetry have universality classes labeled with [math]\mathbb{Z}_2[/math]), and the resulting associated vector bundle [math]P\times_G \mathcal{H} \rightarrow M[/math] acquires gauge transformations [math]\operatorname{End}(P\times_G \mathcal{H}) \cong \operatorname{Map}(M,\tilde{G})[/math].
For instance, if [math]G = U(1)[/math] corresponds to the EM gauge group that mediates local charge conservation, then the chirality operator allows a central extension [math]\tilde{U}(1)[/math] that acts as [math]e^{\pm i\varphi}[/math] on [math]\mathcal{H}_\pm[/math], meaning [math]\mathcal{H}_+[/math] is the "particle Hilbert space" while [math]\mathcal{H}_-[/math] is the "hole space". On the other hand if [math]G = SU(2)[/math], then we may make [math]U^{(\dagger)} \in G[/math] act on [math]\mathcal{H}_\pm[/math]; this means that [math]\mathcal{H}_+[/math] is the "parallel-spin" space while [math]\mathcal{H}_-[/math] is the "antiparallel-spin" space.
Hope this clears up the confusion.