/sci/ - Science & Math

cause im a brainlet okay

Cause insecure nu-mathematicians think complexity lies in abstraction, and generalizations to avoid actual complex and wild subjects.

>>9769287

differential equations are easy and I'm a brainlet

Category theory and algebra are for brainlets who can't into hard analysis.

I don't really see the hate here, they're just not posted much. I do PDEs. AMA.

>>9769533
In my QM class, it seems that the hilbert spaces comes from the fact that dynamics follow a linear PDE and not a an ODE, but it seems that to define properly a space of solutions, boundry conditions are needed, so how exactly hilber spacea come to play?

>>9769533
Nice. Any recommendations for books to look at? Late undergraduate/early graduate level student looking to get into some hard analysis.

>>9769567
Not the original anon, but I do mathematical physics.
>Schrodinger equation
>linear
Only in mean field approximation my child, and this approximation can often miss phase transitions in strongly-correlated systems.
>so how exactly hilber spacea come to play?
In general the Hamiltonian [math]\hat{H}[/math] is defined as the generator of time evolution along the physical trajectories on a symplectic Hilbert space [math]\mathcal{H}[/math]. The Schrodinger's equation [math]i\dot{\psi} = \hat{H}\psi[/math] is then literally just the usual flow equation due to time evolution. The fact that it's the total energy comes from Noether's theorem (time translation symmetry = conservation of energy).
Now what exactly that Hilbert space is comes from Born's postulate: i.e. that wavefunctions are sections of a Hilbert ([math]L^2[/math] Hermitian) line bundle [math]L\rightarrow M[/math] with a structure group [math]U(1)[/math], which implies probability conservation (again, by Noether's). If we denote [math]\rho: U(1) \rightarrow S^1[/math] the representation and [math]L^2[/math] the representation space, the associated vector bundle is [math]L\times_{U(1)}L^2 \rightarrow M[/math], where the first factor (Hermitian line bundle) gives the [math]\mathbb{C}[/math]-valued-ness of the wavefunction while the second factor gives the square integrability over your position space [math]M[/math].
The fact that PDE's arise from, for instance, Schrodinger's equation is precisely due to this representation. The Banach algebra of bounded operators [math]\mathcal{B}(\mathcal{H})[/math] on the Hilbert space admits a representation as differential operators on [math]L^2[/math]. The Hamiltonian [math]\hat{H}\in\mathcal{B}(\mathcal{H})[/math] is hence also represented as a differential operator, so Schrodinger equation in general is a differential equation, which is not necessarily linear.
There, you now know the entirety of QM.

Why can't we solve all PDE using numerical methods?

>>9769287
I mean, how much different are PDE's from ODE's? I know ODE's was cake

>>9769826
they're pretty much the same
assume solution then solve for coefficients
it's more work and you have to keep track of slightly more but it's also more fun imo

>>9769823
Have you tried?

>>9769823
Because solutions have to exist before you can solve it, whether through analytical methods or numerical methods.
Also numerical methods have problem with non-[math]C^1[/math] boundary conditions since they can't chop up the domain into infinitely many small pieces (hence the name [math]finite[/math] element).
>>9769826
PDE's are beautiful geometric optimization problems while ODE's are monstrocities (at the research level). E.g. Atiyah-Singer sindex theorem in PDE's vs. the Airy equation having non-[math]L^2[/math] solutions in ODE.
>>9769828
>mfw an undergrad answers questions he's not equipped to answer

>>9769287
PDEs are cool when they intersect geometry.

The space of solutions to the Hermitian Yang-Mills equations on a Hermitian vector bundle over a compact Hodge surface can be compactified to a projective variety.

>>9769847

no offense, but you sound like you have no clue what you're talking about.

>>9769690
Wait, the Hamiltonian isn't generally bounded is it?

>>9769571
Evans or Foland.

>>9769567
Hilbert spaces can have boundary conditions built into them, yes. For example, the space of [math]L^2[/math] functions with zero on the boundary form a Hilbert space.

>>9769823
>>9769847
It's not only about existence of solutions, but also stability of solutions. Sure, you can have a solution to a PDE, but if you perturb the parameters just a little bit, maybe the solution is lost. Numerically, you wouldn't see the solution, because those perturbations would indeed be realized since it's an approximation in the first place.

>>9769828
lmao no. That only works in a few linear cases.

Because nonlinear PDEs are a bitch with minimal useful theory existing to say anything about them.

Like the three basic linear PDEs are about the only thing a nonspecialist should kbiw. Anything beyond is largely fuckery at this point, and who the fuck came up with distribution theory? Why?

>>9770449
>Because nonlinear PDEs are a bitch with minimal useful theory existing to say anything about them.
And that's why it's interesting and worthy of study. If there was one theory to sum it all up, the field would be dead. But there are certainly techniques to approaching certain aspects of PDE, for example, bifurcation theory or perturbation analysis.

>>9770460

It's interesting only in so far as publishing obscure papers that most people will never read with minimal if any impact.

There exist books on Navier-Stokes equations considering only specific boundary conditions.

It's the ultimate specialists field but will what you do matter in general? Probably not.

If there was a general theory for PDEs I may be more impressed/interested but it just seems too problem specific.

>>9770465
You are what's wrong with modern math.

>>9770465
You know all those specific problems, though? They often have some niche (yet important) application to finance, physics, engineering, biology, etc. Not that I should need to give reasons for the problems to be interesting in their own right,

No, it's not some unified, over-arching field. If there was, it wouldn't be meaningful to study.

Unlike the theory of ODEs, the theory of PDEs is highly non-constructive. Look at the Perron's """"""""""""""""method"""""""""""""""". It's laughable. Whoever uses compactness arguments, doesn't have balls

>>9770465
>It's interesting only in so far as publishing obscure papers that most people will never read with minimal if any impact.

Ricci Flow, Monge–Ampère are both examples of nonlinear PDE with extremely significant applications in geometry.

>>9769287
bad (bag of tricks + unnecessary technical) teaching in my case

>>9769690
Do you have any mathematical phys book recs for brainlet UG who more-or-less knows Shankar QM and basic diff geo? I've heard Arnold is good, anything besides that?

>>9769690
Not the same anon but.
I understood literally fucking nothing.
Fuck.

>>9770330
>no offense, but you sound like I have no clue what you're talking about.
Fixed your post for you.

>>9770355
It depends on both what the 1-parameter (semi)group of time evolution is and what you consider "physical" trajectories. If the 1-parameter group is not Lie or if you have caustics along the physical trajectory then [math]\hat{H}[/math] in general need not be complete or bounded.
In the GNS construction of the Fock space [math]\mathcal{F}[/math], there is a theorem that states the number operator [math]\hat{N}[/math] is bounded iff a unique vacuum [math]|0\rangle \in \mathcal{F}[/math] exists. So once you can find a distinguished vacuum vector, then you can build up the usual SHO Hamiltonian without worrying about boundedness.
>>9770415
That's right, and also it'd be a problem if solutions exists in a nice space, e.g. [math]H^1[/math], only for some range of values of a parameter in the PDE. I've had a colleague asking me about why he sees finite-proper time blow-ups in the numerical solutions to his Einstein field equations at certain values of the cosmological constant, even if he fixed a nice initial condition.
>>9770581
Guillemin & Sternberg, Woodhouse, von Neumann

>>9771513
>not Lie
Compact Lie*. Also you could pick up some cool holonomies like the Berry phase if it's not simply connected.

>>9769287
nice field

>>9769690
>>9769847
>>9771513
nobody cares u dumb ass fuckin nerd go make youtube tutorials and grade tests

>>9769287
Phospodiesterases are kind enzymes.
The deserve more respect.