So many off-by-one index errors. It was kind of embarrassing actually
Library I use is in C++ so it's 0-based, my code is in Fortran so it's 1-based. But C++ library reports all indexes in global coordinates while my code stores them in local coordinates for each processor
@Slereah maybe these videos on algebraic topology are good? syllabus
Looking at the syllabus it seems the overarching point is discussing connectedness of a topological space in terms of paths and other surfaces embedded in the space allowing some use of algebra
First a warning: I don't know much about either algebraic topology or its uses of physics but I know of some places so hopefully you'll find this useful.
Topological defects in space
The standard (but very nice) example is Aharonov-Bohm effect which considers a solenoid and a charged particle. Id...
"In summary: there cannot be a fiber bundle such that its sheaf of local sections is the sheaf of configurations of the Yang-Mills field. But there is a fiber 2-bundle whose stack of sections is the stack of configurations of the Yang-Mills field."
aaaah
@ACuriousMind Well yes, but try learning category theory without learning algebraic topology first
The foundational questions in al. top. are often very mundane and intuitive. "How many ways can I wind stuff around other stuff?", "Is this knot really a knot?", etc.
category theory's definitions are often a bit strange because they are often taking as definition what one usually would derive from more "intuitive" definitions
just take the basic morphism definitions: "monomorphism" and "epimorphism" are in many settings just injective and surjective functions, and you'd more naturally just define them as that. But because you're in a generic setting where the objects don't have underlying sets you can't talk about injectivity or surjectivity.
I think the best way to think about sheaves is to first work through why the sections of bundles are a sheaf. Then a sheaf is a generalization of bundle - local data that's compatible with each other on overlaps, but that doesn't necessarily glue to a nice space.
if you accept that bundles are worth caring about, it's not so large a leap to accept sheaves might be worth caring about
yeah, as I said, if you have guaranteed underlying sets with points then you usually can find a much simpler way to say the things category theory wants to say
The abstraction is there so that you can care about objects with "too few points" to be e.g. a manifold in the usual set-theoretic setting
the point (ahem) is that there's "structure" there not captured by anything to do with the points themseives. iirc there is a generalized smooth space of differential forms (if you will the platonic ideal of differential forms of which all concrete forms are literally images), but if you forget the structure and just look at the set it's just a point :P
this isn't about whether or not you take categories or sets as the foundations. The foundational issues are mostly irrelevant if you don't go looking for them
Pages 1 - 2 of this define a sheaf as a triple $(S,\pi,X)$ such that $\pi : S \to X$ is a surjective local homeomorphism, so that $S$ is locally homeomophic to $X$, while a bundle is a triple $(E,\pi,B)$ such that $\pi : E \to B$ is a surjective continuous mapping, and this implies the direct product thing in a bundle, hmm
First remark, there is the definition of sheaf from wikipedia (which by the way talks about étalé spaces and that adjunction business) and the équivalent one 1.2. p. 3 of Bredon, Glen E. (1997), "Sheaf theory" which looks much more like that of a bundle (the A in that definition is the étalé spac...
I have to say I have a serious bias towards the idea of category theory as unifying subjects, but in practice it's not like that at all and is super off-putting
Lets put it this way, people were hesitant to accept the gruppenpest, but it convinced everyone through the strength of the results, has this stuff produced anything comparable
Classical mechanics is concept 47295 from generalized unified subject area A combined with concept 87364 from unified area B with a smattering of sub-concepts from 8 other unified areas. (Note all of this is obvious in terms of lagrangians...)
Imagine how the transition from the Heisenberg picture to the interaction picture is described by them
Yes, there is a "u" in his profile picture. It's hidden in the word "Gauss".
user434058
I am quite sure that this answer is just a bluff (most probably non mainstream), but I am not quite sure because I am not knowledgable enough to judge. It would be nicer if any of you voted to delete it, and/or explain the OP why their answer's wrong.
user434058
BTW I lost it when the OP said "light moves faster in a darker environment" :P
@JohnDoe It's not too bad :) I'm plenty busy in these times trying to get a couple of simulations to do what I want them to do (they were supposed to be experiments, but a virus got in the way of that, short term at least :/ ) while also trying to write stuff up into draft paper form... How're things going with you?
@Slereah Well, in principle one could hope that a true QG theory also explains dark matter/energy, but I think the position of the critics is more that precisely because there's no data to predict it's a bit pointless to go into such detail about these speculative theories
I meant more that in principle it would be possible that true QG contains some effect that explains it away without it just being, well, really dark matter :P
@Mithrandir24601 Busy with some coding simulations as well, graph state optimization stuff mostly. Was quite productive until recently, but now probably watching too much news...If you have a chance please take a look at my recent post. Should possibly move it to the quantum computing forum...
In closing questions to transfer to another site of SE,
so the "Belongs to another site of SE" option,
there is no ready option for the HSMSE, a really crying need: cf. this question, a poster child for the need.
Is there an easy way to recommend HSM transfer, or a workaround?
Would anyone know a bound for the largest value of an (infinite-dimensional) positive definite Gramian matrix $M$ with $1$ on the diagonal and $M_{ij}<1$ elsewhere? I suspect the bound is related to the sum of the entries on any row but I cannot remember the details, much less the source of this random bit of knowledge.
@Charlie Just above the event horizon, a radially-directed photon can escape, but a horizontally-directed photon will be captured. The cone of photons that can escape gets wider as you climb. The photon sphere is where horizontally-directed photons make stable orbits, so it has to be above the horizon.
In closing questions to transfer to another site of SE,
so the "Belongs to another site of SE" option,
there is no ready option for the HSMSE, a really crying need: cf. this question, a poster child for the need.
Is there an easy way to recommend HSM transfer, or a workaround?
