/sci/ - Science & Math

>>12495815
>YTM 2021
Tempting...

>>11559347
>physicist
>doesn't even know that solving EFE/optimizing EH action requires solving a bunch of tensor PDEs
>calls PDE trash while GR great
Yeah, not convinced.

>>11522482
Looks like second-order GR. In particular if you treat [math]F = [D,D][/math] as the curvature of a principal [math]SO(1,n)[/math]-bundle on a 4-fold [math]M[/math] with spin connection [math]D = d + [\omega,\cdot][/math] acting on the vierbeins [math]e\in T^*M[/math], first order GR is [math]\int_M \operatorname{tr}(F\wedge\ast F) = \int_M R[/math] while second order GR is [math]\int_M \Lambda\left[\operatorname{tr}(F\wedge \ast F)^2 - (\operatorname{tr}F\wedge\ast F)^2\right] = \int_M \Lambda[R\wedge \ast R - R^2][/math], with [math]\Lambda[/math] the map dual to multiplication by the Kahler/metric 2-form [math]\lambda[/math]. These are Kahler [math]SO(1,n)[/math]-Yang-Mills theories with topological vacuua [math]c_1(TM) = \int_M \operatorname{tr}R[/math] or [math]c_2(TM) = \int_M \left[\operatorname{tr}(R^2)-(\operatorname{tr}R)^2\right][/math] bounding them from below, respectively, so Nash was probably looking at the compatibility of the Riemannian structure on [math]TM[/math] with the spin structure on a principal [math]SO(1,n)[/math]-bundles on homologica 4-spheres [math]M[/math].
In particular you can see in the slides the requirements [math]c_1(TM) = 0[/math] in dim-4 and [math]c_2(TM) = 0[/math] in dim-2 from the field equations, which are obvious from a first course in GR.

>>11509528
Close; UCT says that only the free part is identical, while the torsion part is shifted by one. [math]\operatorname{tor}H_k(X,\mathbb{Z})\cong \operatorname{tor}H_{k-1}(X,\mathb{Z})[/math], so their [math]\mathbb{Z}_2[/math]-torsors differ (up to a shift in grading).
>>11509555
Poincare duality relates [math]H^k[/math] and [math]H_{n-k}[/math] honey.

>>11503986
3 minutes.
>>11504038
I can try I guess?

>>11490649

>>11461124
Notice that [math]K(s) = \kappa_1(s)\kappa_2(s) = \operatorname{det}df[/math] is the determinant of the differential the Gauss map. Using cylindrical coordinates along [math]x[/math], we can write a point on the surface as [math]s= (y(x),\phi,x) \in S[/math]. We define the Gauss map [math]f: S\rightarrow S^2[/math] by sending [math]s\mapsto \hat{\bf n}(s)[/math] to the unit normal at [math]s\in S[/math]. The Gaussian curvature is then defined by [math]K = \operatorname{det}\nabla_i \hat{n}_j(s)[/math].
Explicit computations are left to the reader.

>>11431957

>>11381120
Well let's get it started on the right track then. Post interesting articles you're reading.
https://arxiv.org/abs/1712.02952
>We study topological phases in the hyperbolic plane using noncommutative geometry and T-duality, and show that fractional versions of the quantised indices for integer, spin and anomalous quantum Hall effects can result. Generalising models used in the Euclidean setting, a model for the bulk-boundary correspondence of fractional indices is proposed, guided by the geometry of hyperbolic boundaries.

>>11362187
>so I can't add you
Nice try hun but I'm already taken.

>>11353950
Wannier states are by definition the filled states in real space, and they form irreps of the symmetries on each fundamental domain [math]\Gamma[/math]. They in general define a Hilbert bundle [math]\simeq \mathcal{F}^{-1}P_\text{occ}\mathcal{H}[/math] in real space, the phase of which as you move about [math]\Gamma[/math], or equivalently the Berry phase of the Bloch states within the first Brillouin zone, will determine exactly the Chern number in your TI/TSC. Specifically, it is [math]c_1 = -\operatorname{arg}\operatorname{det}P_\text{occ} \mod 2\pi \in\mathbb{Z}[/math] where [math]P_\text{occ}[/math] is the Fredholm projector on [math]\mathcal{H}[/math] onto the occupied states.
The unfilled states will also form irreps, but as I've said before these irreps will in fact trivialize the topology of the system. This is why we don't call them Wannier states. Since, as you know, the topology determine the presence of gapless edge states, we essentially only want to count the chirality on the occupied bands, otherwise [math]c_1[/math] will always be zero since gapless states always connect occupied and unoccupied bands, acquiring chiralities [math]\pm 1[/math] resp. for each gapless state, whence they cancel.

>>11321392
>amplituhedron
https://arxiv.org/abs/1612.04378

>>11308493
Con: she eats you.
Or it might be a pro if you're into that.

>>11288815
Did you mean what you typed out? Did you mean that [math]\chi(\emptyset) = 0[/math] as the number [math]0\in \mathbb{R}_{\geq 0}[/math] or as the zero function in [math]X\rightarrow \mathbb{R}_{\geq 0}[/math]? And by [math]\cdot[/math] and [math]+[/math] you mean point-wise multiplication/addition? If this is the case then how did you define the partial order [math]\leq[/math] in >>11288835?

Anyways, it may be worthwhile for you to look at spectral projections [math]P_A[/math] of a densely defined operator [math]A[/math]. They define operator-valued measures [math]dP_A(x)[/math] on a Banach space for which [math]f(A) = \int_\Omega dP_A(x) f(x)[/math] where [math]\Omega[/math] contains the spectrum of [math]A[/math]. In particular, we can define [math]\chi_A:P(\Omega)\rightarrow \mathbb{R}_{\geq 0}[/math] by [math]S \mapsto \int_S dP_A[/math]; basically [math]\chi_A(S)[/math] measures the "volume" of the continuous spectrum of [math]A[/math] contained in [math]S\subset \Omega[/math]. I haven't looked at your axioms closely but I think they are satisfied for [math]\chi_A[/math] for sufficiently nice operators [math]A[/math].

>>11267460
I shill working for physicists not physics itself.

>>11182499
Yes, with applications to twistors, [math]tt^*[/math]-geometry and Ginzburg-Landau theory. otherwise it's rejected.

>>10946334
Please refrain from relinking questions that ask for resources without providing context (their level, their background, what they wish to get out of the resource, etc.). They add nothing to the thread and is unhelpful to everyone else.
>>10941317
It depends on the boundary condition. Strong max principle states that if a harmonic [math]f[/math] achieves a local maximum in the interior of some region [math]\Omega[/math] then [math]f[/math] is exactly constant, which is what you are referring to. However, if [math]f[/math] achieves a local maximum on the boundary [math]\partial\Omega[/math], then [math]f[/math] need not be constant. In other words, harmonic [math]f[/math]'s merely [math]minimize[/math] the functional [math]I[f] = \frac{1}{2}\int_\Omega dx|\nabla f|^2[/math], not render it [math]0[/math].
If you put [math]\Omega = \mathbb{R}^d[/math] and the boundary at infinity, then of course any local max of [math]f[/math] renders [math]f = \text{const}.[/math] by the strong max principle hence [math]\nabla f = 0[/math] a.e. But if the boundary condition is non-trivial, then you must minimize [math]I[f] = \frac{1}{2}|\nabla f|_{L^2(\Omega)}[/math] subject to these boundary conditions, and this typically renders your space of admissible functions to be much smaller than [math]L^2[/math].

>>10042896
Hi.
I'm trying to get a paper published soon and I got a bf, so I don't spend nearly as much time on 4chan as I used to, though I still browse through once in a while. Just a heads up.