The h Bar

DanielSank

12:00 AM

Alright, so $\int_S P_Y(y)dy = \int_Q P_X(x)dx$.

Now just apply the change of variables formula :-)

You'll see the result drop right out!

0celo7

I see

Something broke Chrome

DanielSank

Physicists are never taught this, so we languish in ignorance of how to deal with noise in our experimental data, etc. (unless you're an astrophysicist, in which case your whole field is so data starved that you learn every damn signal processing trick in the book).

Although sadly, even astrophysicists don't usually get a good education in stochastic processes.

0celo7

@DanielSank brb have to switch OSs

user54412

@DanielSank Are you sure? For concreteness, say your 2D distribution is $f(r) = (1/2\pi\sigma^2) \mathrm{e}^{-r^2/2\sigma^2}$. Then this projects to the $x$-axis as $g(x) = 1/\sqrt{2\pi\sigma^2} \mathrm{e}^{-x^2/2\sigma^2}$ as expected. But the probability of being within $d$ of the origin in 2D is $2\pi \int_0^d f(r) r \, \mathrm{d}r = 1 - \mathrm{e}^{-d^2/2\sigma^2}$.

0celo7

Chrome is broken in OSX as well.

user54412

12:04 AM

@DanielSank I'd expect an error function for 100% correlated variables, but for 0 correlation you're integrating the Gaussian with the extra r factor for 2D and the antiderivative is elementary.

DanielSank

@ChrisWhite You're right. Sorry I should've said: the result I get is $(1 - \exp (-R^2 / (2 \sigma^2))$ where $R$ is the threshold radius.

That function is the fraction of points within the threshold.

user54412

Ok we agree then.

DanielSank

\(^.^)/

0celo7

integral masters

user54412

@DanielSank When you only have 10 or 20 neutrinos to work with...

FenderLesPaul

12:09 AM

newtrainees

0celo7

@FenderLesPaul huh?

DanielSank

@ChrisWhite Yah.

FenderLesPaul

@0celo7 I was just trying to be cute

jesus

no one understands me T_T

0celo7

@FenderLesPaul you ready for the 8.1.1 proof

ACuriousMind

@FenderLesPaul But:

7 hours ago, by FenderLesPaul

Why am I so loved T_T

FenderLesPaul

12:13 AM

I'm so loved AND misunderstood

0celo7

@ACuriousMind Why can't I have HDR and AA on at once?

FenderLesPaul

@0celo7 yeah gimme like an hour just gunna finish my food and satanic rituals

ACuriousMind

@0celo7 IDK

0celo7

@ACuriousMind You're the resident gamer!

ACuriousMind

That doesn't mean I know why the graphics options of a game don't admit a specific combination! :D

0celo7

12:15 AM

I'm not sure if I even notice a difference when AA is on.

@ACuriousMind Do you?

ACuriousMind

AA? It smoothes lines that otherwise look a bit blocky, but it's been a while that I looked at a game without it

I know that I noticed the difference back in Neverwinter Nights :D

0celo7

You people and your gaming PCs

DanielSank

@0celo7 I know, right?

My strategy is to stay ten years behind modern video game technology.

0celo7

I think my setup was decent in 2011

DanielSank

This not only saves tons of $$, but I'm guaranteed to only ever play games that are so good that people remember them after ten years :-)

0celo7

12:19 AM

@DanielSank Yeah, I have a bunch of old games

I swear if I can't run Fallout 4...I'll be really upset

user54412

@DanielSank That's my strategy with movies.

0celo7

@ACuriousMind Btw, my Fallout NV is broken. I get stuck in an infinite loading screen when coming from the main menu. I found a solution: start new game and load save. Do you have a better one?

DanielSank

@ChrisWhite :D

ACuriousMind

@0celo7 Never heard of that error, sorry

vzn

12:40 AM

Is our universe FAKE? Physicists claim we could all be the playthings of an advanced civilisation / dailymail

0celo7

@vzn I've always said this

QM is an artifact of being in a larger, classical simulation

vzn

@0celo7 lol, joking? whats your take on bohmian mechanics? to me its the closest theory to the simulation idea, and yet among physics conventional wisdom, its nearly a laughingstock or worse...

0celo7

@vzn don't know the first thing about it

vzn

the article is in a tabloid, but its not all that bad. esp enjoyed the obscure link to GZK cosmic ray cutoff.

Jan 26 at 21:59, by vzn

in toy models of QM, 51 secs ago, by vzn

superclassical/ emergent QM, recent developments, rough outline/ overview/ leads / Tmachine blog

ACuriousMind

@vzn These cutoff are generic consequences of spacetime being a lattice, though - but spacetime being discrete does not necessarily imply that we live in a simulation.

vzn

12:51 AM

@ACuriousMind agreed. actually the very definition of simulation is probably difficult to define. the article proposes there might be "inconsistencies" or other artifacts. but what about a "perfect" simulation? etc...

re that long recent chat thread/ argument about "defns" lol...

0celo7

From a purely aesthetic standpoint, the simulation idea is second to none

The hierarchy problem is solved instantaneously

vzn

@0celo7 yes, and it is many decades old. it goes back to early musings by fredkin about "digital physics". & is tightly coupled to wolframian "new kind of physics" ideas. etc

what is the "hierarchy problem"?

0celo7

@vzn @ACuriousMind recently did a lecture on it

vzn

one could argue the "simulation" idea goes back thousands of years even into hindu concepts of "purusha vs prakriti" etc

0celo7

in a nutshell: why does gravity suck

ACuriousMind

12:56 AM

Not really, it was just the motivation for the model

0celo7

@vzn in more techincal terms: why is the Higgs mass so much smaller than the cutoff?

(I think)

ACuriousMind

@vzn In one formulation, it's the "puzzle" why the Higgs VEV is so small - by renormalization arguments, one would expect it to be of order of the cutoff of the theory, which is believed to be around the Planck scale for the Standard Model, yet it is about 17 orders of magnitude smaller.

In more popularized terms: Why is gravity so much weaker that the electroweak/electromagnetic force?

0celo7

@vzn you can see @ACuriousMind is more eloquent

user54412

You know, in most fields of empirical science, when a line of reasoning produces clearly wrong results, the response is to abandon the line of reasoning, not ask why nature isn't as sensible as us.

vzn

have long felt that maybe gravity is some "shadow force" of emergent properties of the universe...

0celo7

12:59 AM

@ChrisWhite ...where did that come from?

ACuriousMind

@ChrisWhite Well, the result isn't wrong, it just seems...very specific that the natural constants should have the values that exactly cancel the renormalization contributions to the Higgs mass

vzn

@ChrisWhite not following either :|

ACuriousMind

Some people don't think the Hierarchy problem is a problem at all - fine-tune the input parameters, get the correct Higgs mass, no problem.

user54412

^ that

user54412

or rather, I wouldn't use the term "fine tuning" at all

ACuriousMind

1:01 AM

Yeah, I'm uneasy with that terminology, too

user54412

it implies not only existence but also knowledge of some a priori distribution of parameters that the universe is drawn from

0celo7

@ChrisWhite sounds like a simulation

vzn

saw recent video (posted here) by michiu kaku denigrating std model. do not disagree with std model, but it does seem "rube goldberg" sometimes. way too many separate constants. think (crazy sounding-soundbite follows...) the entire std model is an emergent property of 3d solitons.

0celo7

@ACuriousMind loves Kaku

ACuriousMind

@ChrisWhite Totally agree with that. I tend to not think of the Hierarchy problem as an actual "problem" either.

user54412

1:03 AM

The Hierarchy Curiosity, perhaps?

0celo7

@ACuriousMind So you think the constants have their values for no reason at all?

@ChrisWhite I agree with this.

user54412

@0celo7 The word "reason" has a load of metaphysical baggage. This isn't necessarily a bad thing, but it is not a word to be slung around lightly.

0celo7

@ACuriousMind Where is the QFT vacuum energy?

@ChrisWhite I'm not slinging it "lightly", I think this is a fundamental question.

user54412

It may be a fundamental question. But before you look to answer it, you (and everyone else) should know what the question is and what would even constitute an answer ;)

vzn

std model seems an apex of reductionistic physics. think we are on the cusp of its "wholesale revision"

ACuriousMind

1:07 AM

@0celo7 I think there are two possibilities: Our final fundamental theory of nature will have input parameters, or it won't. If it will, then people will forever question why the input parameters have the values they have (note that the answer to this would essentially be a "more fundamental" theory). If it won't, then they will forever ask why exactly that structure is what is realized in nature. Either way, that's a question the most fundamental theory, and hence physics, can't answer.

0celo7

@ChrisWhite Do you understand the Anthropic Principle?

vzn

yes not following what "line of reasoning" is supposedly incorrect. advanced theories (eg string theory) are all nearly indistinguishable "lines of reasoning".

0celo7

@ACuriousMind Physics can't describe the most fundamental theory?

vzn

input parameters = constants. am ok with constants, agree they will always be there, but only a few. far/ dramatically fewer than we have now.

user54412

@0celo7 Understand, yes. Believe it is a solid method of reasoning... not so clear. Again, it is usually described in reference to probability distributions outside the universe per se.

user54412

1:09 AM

...which is like doing GR by embedding in R^2n

ACuriousMind

@0celo7 No, I'm saying this "why" question is either a process that never terminates (if every theory has inputs), or that terminates without a satisfactory answer (if there is a theory without input, of which we then ask why it is that theory that is realized).

0celo7

@ChrisWhite Does it imply multiple universes? So each universe is a point in parameter space and we happen to be at the point where humans can exist?

vzn

@ACuriousMind "its elephants all the way down..."

ACuriousMind

@vzn Nah, it's four elephants on a turtle swimming through space, everyone knows that.

0celo7

@ACuriousMind Hmm. I don't think it terminating would be unsatisfactory for me.

user54412

1:12 AM

@0celo7 Usually it does. It's one thing to use AP to ask "why are we on Earth, rather than 51 Peg b?" But it's quite another to ask "why is everything the way it is, rather than something else?"

vzn

@ACuriousMind so the magic constants are 4 + 1 = 5!

0celo7

@vzn 4 + 1 != 120

idiot :P

vzn

← oops/ ouch/ touche way out of my league here gotta go take some advanced math/ physics classes asap!

0celo7

@vzn Read HE

ACuriousMind

@0celo7 Okay, then I declare all "inputs" to the Standard Model as intrinsic part of it. You can't adjust them, else it's a different theory. That satisfactory? (What I'm getting at is that "a theory has inputs" is not a necessarily well-defined notion)

user54412

1:14 AM

Undergrad abstract algebra for me was Math 5. Graduate abstract algebra was Math 120, aka Math 5!. This couldn't have been a coincidence.

vzn

@0celo7 HE?

ACuriousMind

@vzn You're on to something there ;P

user54412

@vzn Hawking and Ellis. The acronym is used nowhere but this chat room.

0celo7

@vzn amazon.com/…

@ChrisWhite Not true.

ACuriousMind

@ChrisWhite Others don't call HIM that?

Kyle Kanos

1:16 AM

It's an entirely bogus book too. Filled with all sorts of lies about space-time and changing coordinate systems

ACuriousMind

(No, I'll never think the pronoun joke is unfunny :P)

0celo7

@ACuriousMind No. I'd be satisfied if the input was like $3\pi/\mathrm{e}^{\sqrt{\gamma}}$. But some random numbers, that's BS.

vzn

interesting though that it points to the key problem of physics of resolving GR with tiny scales all the way back to 1975. arguably quite prescient.

0celo7

@ACuriousMind So where is all the vacuum energy?

ACuriousMind

@0celo7 Huh?

(I really don't understand the question)

0celo7

1:18 AM

@ACuriousMind Does QFT not predict a huge vacuum energy?

ACuriousMind

@0celo7 AFAIK, it doesn't predict a vacuum energy at all - the vacuum energy has to be renormalized to be rendered finite, and so it's an input to the theory, not a prediction

0celo7

@ACuriousMind Yes but any reasonable cutoff leads to a huge vacuum energy.

ACuriousMind

@0celo7 If the bare parameter is not "fine-tuned" to cancel that contribution :P

All these things depending on the cutoff are fine-tuning arguments, not actual predictions

0celo7

@ACuriousMind So one can solve the cosmological constant problem by simply fine-tuning crap?

Why have I never heard of this before.

ACuriousMind

@0celo7 I may be that my understanding is too simplistic, but I believe simply tuning the bare cosmological constant achieves this, yes

You might consider asking a question about that, though!

0celo7

1:23 AM

Hmm, @ChrisWhite do you know of any good recent articles on the cosmological constant problem?

user54412

@0celo7 Because most people who write about the cosmological constant "problem" intend to "solve" it by some other means.

0celo7

@ACuriousMind I guarantee someone has written about this. Modern GR books make a huge deal out of it.

@ACuriousMind I know (as of 2007 = BBS) there are two promising string theoretic solutions.

vzn

isnt cosmological constant so called "einsteins biggest mistake"?

user54412

@0celo7 Recent? Not really.

ACuriousMind

@0celo7 Yeah, the string theorist are almost always people who think fine-tuning is a big problem :P

dmckee

1:24 AM

@vzn Yeah, but we may have to amend the point o that removing it is the big mistake.

0celo7

@ACuriousMind Exactly, because the theory has no free parameters.

dmckee

Depends on exactly how the dark energy behaves.

ACuriousMind

@0celo7 String tension? Choice of compactification?

0celo7

@ACuriousMind I take that back. It has one, namely the tension.

user54412

@0celo7 It's worth pointing out (re: what dmckee just said) that there is a lot more to the cosmological constant that the theoretical physics attempt to derive it. For example, a large number of cosmologists are involved in observations to try to determine if it really is a constant over the history of the universe.

0celo7

1:26 AM

@ACuriousMind Is choice of compactification really a "parameter"?

We need a string theorist in here...

vzn

cosmological constant governs universe expansion right?

user54412

We joke that they're not satisfied knowing just w, but they want to be sure w' = w'' = w''' = ... = 0.

user54412

($p = w\rho$ for dark energy)

0celo7

w=-1

user54412

@vzn yes

0celo7

1:29 AM

@ACuriousMind Oh do you mean the radius of the T-dual thingie? Shit, I dunno about that

@ACuriousMind That radius is the vev of a scalar field in the $n$th dimension -- not a parameter.

Kyle Kanos

24

Q: Is the Xenomorph life cycle based on the life cycle of a real world species?

The Xenomorph life cycle is quite complex. It basically boils down to this1: Queen Xenomorph lays eggs. Eggs hatch into facehuggers. Facehuggers implant eggs in a host organism's throat. Implanted eggs hatch into chest-bursters. Chest-bursters grow into adult Xenomorphs. Their primary fu...

That kinda creeped me out a bit

ACuriousMind

@0celo7 What "T-dual thingie"? You need to choose a 6D compactification manifold to get to an effective 4D theory, do you not?

And yes, the size of the compactification will/can be controlled by moduli fields, but what sets the VEVs for these?

0celo7

@ACuriousMind I'm not that far yet, still working on T-duality

@ACuriousMind But is this really "chosen" or is there one that corresponds to reality and the rest are toy models?

Wait

Oh my god that's a parameter

My life is a lie

ACuriousMind

:D :D :D

0celo7

::throws string theory books around::

Argh you make my head hurt!

@ACuriousMind Have you learned more ST since your course? I don't recall 6D compacifications in those notes...

ACuriousMind

1:38 AM

@0celo7 Not really, but I've talked a bit too much with string theorists, I guess :D

0celo7

@ACuriousMind You should ask them about free parameters

0celo7

2:02 AM

particle physicists

I wonder if @Qmechanic knows any physics jokes

Qmechanic

6:44 AM

@0celo7 : Sure. For starters, there are a few joke posts on Phys.SE, e.g. this.

Danu

@Qmechanic Now I'd like to hear your best physics joke!

@HDE226868 Question above

Huy

6:57 AM

@Danu: Do you know the one from Big Bang Theory?

Slereah

Hello

Qmechanic

@Danu : Nice try :)

Danu

@Qmechanic Oh come on

@Huy I hate the Big Bang Theory, so I don't watch it :P Was it any good?

@Qmechanic Are you afraid it'd... damage your reputation? ;)

skull patrol

7:20 AM

But @Danu do you hate the theory of the Big Bang? :P

Huy

@Danu: It doesn't require a lot of knowledge of physics but I still found it funny.

@Danu: "There's this farmer, and he has these chickens, but they won't lay any eggs. So, he calls a physicist to help. The physicist then does some calculations, and he says, um, I have a solution, but it only works with spherical chickens in a vacuum."

Danu

7:38 AM

@Huy I've seen a better version of that :P

Slereah

8:09 AM

One thing that annoys me in The Big Bang Theory is

The whiteboards

They are touted as being complicated physics

But they are always textbook shit

Sometimes they just write down gamma matrices identities

DanielSank

8:23 AM

I'd like to understand something about comment flags. I recently flagged John Rennie's comment on this question because I thought it was the source of much confusion.

The comment suggested that the question is a duplicate of another one, however inspection of the supposed duplicate and reading of the present question without preconceived bias reveal that there is definitely no duplication.

The question was closed rather quickly after it was posted. I then pointed out in the chat that it's not a duplicate (at least not of the questions it was accused of duplicating).

I also left a comment pointing this out.

The question now has 4 reopen votes pending, and my comment pointing out that the question isn't a duplicate has 4 upvotes.

I'd like to know why the flag on the original comment accusing the question of being a duplicate was declined.

I understand flags to be for marking inappropriate content. I think I may have overextended the meaning of "inappropriate" to include "content which lowers the value of the site." Do I misunderstand or is it that the mods who reviewed the flag just don't think John's comment is problematic?

Huy

@Slereah: It's not a tv show aimed at professional physicists, and even as a mathematician with some physics background, I am not familiar with all the stuff on the whiteboards. I know it's not "complicated physics" for professional physicists, but I can't understand how one can be annoyed at that.

Slereah

Well they always brag about how they have a physicist on the show to write the boards

Physicist, you are slacking

Huy

@Slereah: I've never heard them bragging about that.

Slereah

But have you looked :V

Huy

@Slereah: Maybe you should direct your concerns directly to Prof. Saltzberg?

@Slereah: I know they do have a physics professor who helps them, but stating that isn't bragging to me.

Slereah

8:33 AM

I will send him a sternly worded letter

I'm sure you will.

(I will not)

I know.

(I will)

(I will not)

$\frac{1}{\sqrt{2}}$ I will + $\frac{1}{\sqrt{2}}$ I will not

Huy

Please don't overwhelm me with your complicated physics.

(I'm not overwhelmed.)

Slereah

8:36 AM

You know what had good whiteboards?

Black Mesa.

Huy

Why had? Did they remove them?

Slereah

Huy

(I never really delved into the HL series or any mods so I wouldn't know)

Slereah

It's written a bit too neatly for a whiteboard though :p

No giant stains or mysterious arrows

But the content is pretty good

user54412

The only thing to be annoyed about at all is that people think real physicists use whiteboards. Real physicists use blackboards. Whiteboards are for java-school CS sorts to pretend to look smart at silicon valley startups or something. (That's right, I said it.)

7

Slereah

8:39 AM

I had a whiteboard back when I was in a lab

Huy

@ChrisWhite: Why do real physicists use blackboards? There's plenty of physics profs at my uni who are using whiteboards in their office.

user54412

@Huy Universities often don't give people a choice. Something about "dust is bad for computers" and "we're all modern"

Slereah

Whiteboard versus blackboards aren't really a discipline thing

Huy

@ChrisWhite: You can be very sure every prof here has the choice.

Slereah

It's more of an era thing

Blackboards are being replaced by whiteboards

user54412

8:40 AM

We had a technology that (1) always worked and (2) always erased. And then we regressed.

Slereah

Because they are more practical

Huy

@ChrisWhite: There's offices right next to the ones with whiteboards but with blackboards instead, I'm pretty sure everyone could use whatever they preferred.

@ChrisWhite: There's also offices with both where I've seen both or just the whiteboard used.

skull patrol

We all should have a choice :P

Slereah

Soon we will all use wall tablets!

Huy

Hopefully!

user54412

8:47 AM

I'm trying to imagine how hot it would feel to stand in front of one of those.

Huy

Literally hot or ?

user54412

yeah

Huy

@ChrisWhite: I actually worked at a small company that was working with huge wall tablets. It wasn't literally hot to stand in front of them, even after a whole day.

user54412

I wonder what display technology that was then

user54412

large tvs are definitely warm to be within arm's reach of

Huy

8:49 AM

@ChrisWhite: I didn't engineer it, but you can check out rentouch.ch, maybe you can find out.

Slereah

9:25 AM

So do we know why perturbative QFT works

Since the interacting Hilbert space isn't the same at all as the free Hilbert space

Are they still somewhat similar

Danu

10:03 AM

No we don't know, at all.

Slereah

It is unfortunate

Danu

Also @ChrisWhite couldn't agree more <3

Slereah

Don't we have some totally solved QFTs, though

Like supersymmetric ones and such

Danu

Totally solved? What does that mean? Rigorously constructed qft's are so far only known in lower dimensions afaik

Slereah

Those too, I suppose

But I recall hearing that in 4D, some supersymmetric ones were as well

Don't recall the details, though

Do we have the details for the low dimensional ones, though, I suppose

Like the difference between the states of the Dirac equation and the Thirring model

user54412

10:07 AM

it's so strange that something can seem to work in arbitrary dimensions yet we can only find proof that it works in low dimensions

Slereah

Working in arbitrary dimensions is fine until you need to actually solve the equation

user54412

numerical PDEs have this problem with Lax-Wendroff -- all the theorems that tell us we're converging to the right answer have only ever been shown to work in 1D

user54412

but of course we simulate 3D fluids all the time, and it seems to work

Slereah

GR in n dimensions are always a nightmare

Because not two are alike

For every n the theory is wildly different

At least from 0 to 4

Sokhotski–Plemelj theorem

The Sokhotski–Plemelj theorem (Polish spelling is Sochocki) is a theorem in complex analysis, which helps in evaluating certain integrals. The real-line version of it (see below) is often used in physics, although rarely referred to by name. The theorem is named after Julian Sochocki, who proved it in 1868, and Josip Plemelj, who rediscovered it as a main ingredient of his solution of the Riemann-Hilbert problem in 1908. == Statement of the theorem == Let C be a smooth closed simple curve in the plane, and φ an analytic function on C. Then the Cauchy-type integral defines two analytic functions...

Why do QFT textbooks never mention this theorem

yet use it all the time

user54412

10:25 AM

I feel like that's been brought up before in this room, but I can't seem to locate the reference.

user54412

Either I'm misremembering or google's indexing of this room is failing

user54412

aha!

user54412

Dec 24 '14 at 21:07, by DanielSank

@DavidZ: Found it. It's the Sokhotski–Plemelj theorem: http://en.wikipedia.org/wiki/Dirac_delta_function

David Z

10:38 AM

@DanielSank being a source of confusion isn't really a valid reason to flag a comment.

ACuriousMind

12:07 PM

@DavidZ Oh, how many I'd flag for that :P

Slereah

12:19 PM

desy.de/~jlouis/Vorlesungen/QFTI10/QFTI.pdf

This actually has a decent derivation of the quantization of the scalar field!

That doesn't rely on vague clues or "Just remember the SHO"

I liek it

It's pretty rare to find a derivation of a thing "from scratch"

ACuriousMind

0

Q: What is the freezing/melting point tap water?

Yes, I know purity is not a constant and depends on area etc. Does anyone have any experimental results on freezing/melting point of any tap water, as opposed to distilled, de-ionized, saline etc? I would like some figures to within 30mK ideally.

experimental-physics

^too broad?

I know it's asking for experimental results, which may vary, but "tap water" seems like a really vague und unuseful category

Slereah

It's fine I guess?

You could just have like

a range of results

Danu

@ChrisWhite Why is it so strange? Degrees of freedom... :\

ACuriousMind

@Slereah I'm not asking if it's an acceptable question for physics. I'm asking if it is an acceptable question for this site.

user54412

@Danu But R^n is trivially just copies of R. We didn't even break a sweat constructing it, so why does it affect theories about things taking place in it (i.e. functions from it to other spaces?) so much?

Danu

12:27 PM

@ChrisWhite There was an excellent math.se/mathoverflow post about surprising "discontinuous" changes in higher dimensions... I can't find it right now though :(

@ChrisWhite What I find surprising is how special 4 dimensions are in terms of manifolds.

ACuriousMind

@ChrisWhite Divergences of integrals, for instance. The 2D and 3D divergences are almost all logarithmic, but because the measure (in spherical coordinates) has a $r^{d-1}$, these get worse (quadratic and higher) in higher dimensions

user54412

@Danu That too is really weird.

Slereah

4 dimensions isn't extremely special

All dimensions are special

Although it's hard to say for sure

I don't think GR has been done in more than 6 dimensions or so

user54412

hmm, time to relocate to the office...

Danu

48

Q: Dimension Leaps

Many mathematical areas have a notion of "dimension", either rigorously or naively, and different dimensions can exhibit wildly different behaviour. Often, the behaviour is similar for "nearby" dimensions, with occasional "dimension leaps" marking the boundary from one type of behaviour to anoth...

soft-question examples big-list

@Slereah No, I think it is.

Mathematically

Slereah

12:29 PM

Well yes, but the point is that all dimensions have their peculiarities

Danu

4 really are special, see e.g. here:

4-manifold

In mathematics, 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure and even if there exists a smooth structure it need not be unique (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic). 4-manifolds are of importance in physics because, in General Relativity, spacetime is modeled as a pseudo-Riemannian 4-manifold. =...

ACuriousMind

I've heard someone say that 3D and 4D seem special because only in these we know how to ask the interesting questions.

Danu

^maybe

ACuriousMind

And that's kinda plausible - we have no intuition for higher dimensions, so we have no intuition for what is "trivial" and what is "interesting" there.

Slereah

Yeah, seems a bit biased

Danu

12:30 PM

Exotic R4

In mathematics, an exotic R4 is a differentiable manifold that is homeomorphic but not diffeomorphic to the Euclidean space R4. The first examples were found in 1982 by Michael Freedman and others, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's theorems about smooth 4-manifolds. There is a continuum of non-diffeomorphic differentiable structures of R4, as was shown first by Clifford Taubes. Prior to this construction, non-diffeomorphic smooth structures on spheres — exotic spheres — were already known to exist, although the question of the...

Slereah

Because you have a list of things that only apply to R^4

Danu

Only $\mathbb R^4$ is strange!

Slereah

But it does not specify if there are a lot of things that apply only to R^n

Which I suspect there are!

There's a finite number of connected 1 manifolds and they are all diffeomorphic to flat space, it is amazing!

Only in compact 2-manifolds can you classify them by their Euler characteristics!

Also the uniformization theorem

Danu

No, not really at all.

Slereah

etc etc

Danu

12:32 PM

Low-dimensional things are not so amazing, because they follow from lower degrees of freedom

4 is right in between low and high dimensions, apparently, and that's interesting.

Slereah

But 5-manifolds also have amazing properties!

Huy

Can you list some?

Slereah

5-manifolds are the lowest dimension where you cannot classify their topology

(at least where it is proven)

ACuriousMind

A "bias-free" argument would indeed be that we have variants of the uniformization theorem for 1,2,3 dimensions, but not for 4.

Danu

@Slereah If you'd look at the thing I linked, it gives many examples of things that are proved by low-dimensional methods for $n\leq 3$ and high-dimensional for $n\geq 5$ and that's so nice: 4 falls inbetween!

Slereah

12:34 PM

(That's mostly all I know about 5-manifolds)

Danu : Yes, but there might be things like that for many n's, though

Is my point

ACuriousMind

@Slereah "might be"...name some ;)

Slereah

I'm no topologist, unfortunately

Danu

We know there are for 4, and it's not like 4 has been a focus for mathematics. It really hasn't.

Slereah

The focus decreases as dimensions increase, though

Danu

Also I'm talking about smooth manifold (so diff geo) things, not pure topology.

Slereah

12:35 PM

I doubt there's a lot o work on 53D manifolds

ACuriousMind

@Slereah Because there's no indication that they're interesting

Slereah

Yes, but then you will not know that until you study them

ACuriousMind

Quite a bit of work is done to understand the homotopy groups of spheres in high dimensions, and that's only interesting in large dimensions (like >10)

So people there are löoking at high-dimensional spheres

Slereah

Any amazing results?

Danu

(guess what the most interesting case for smooth structures on the sphere is?)

Slereah

12:38 PM

" A highly dense sphere-packing is possible in 24 dimensional space, which is related to the unique qualities of the Leech lattice."

neat

Danu

Sure, the Leech lattice gives some other cool stuff. Many results come from that thing.

ACuriousMind

@Slereah The most amazing result ever: We don't have a damn clue what's going on there. :D

@Danu It's even a Lie group!

Slereah

Also 5D GR has a lot of weird black holes, from what I recall

Like ring holes

Danu

Those occur in any higher dimension

Slereah

I do hope that 4D manifolds are classifiable

Danu

12:41 PM

(the fact that they don't in 4D, however... (jk))

Slereah

Would be nice for some QG theories

"Four is the dimension where a maximum number of regular polyhedra exist. (Apart from polygons in the plane, of course. But those are "abelian", hence boring :) )"

Huy

@Danu: From your MO link I actually find the top voted answer the most mind-blowing one.

ACuriousMind

That one hurts my brain

Huy

+1.

I can't even begin to picture it.

Slereah

"The Euclidean ball takes up the most space in dimension 5."

See? 5D can be special too!

Huy

12:58 PM