The h Bar

ACuriousMind

00:00

@Slereah Well, the conical singularity has a local atlas with a coordinate $r$ such that the singularity is at $r=0$ and the metric is $\mathrm{d}s^2 = \mathrm{d}r^2 + r^2\mathrm{d}\Omega^2$ where $\mathrm{d}\Omega^2$ is any $n-1$ dimensional metric.

This gives an isometry to a space that is exactly a cone over the $n-1$-dimensional manifold with metric $\mathrm{d}\Omega^2$

But I don't think you can make that $n-1$ -dimensional manifold into $\mathbb{R}^n$ - just think of the actual 2D cone:

No matter how close you get to the singularity, you always still see that the base was a circle.

Slereah

true

hm

ACuriousMind

There is no neighbourhood that includes the tip of the cone where you would lose that information about the circle.

Slereah

How does GR deal with conical singularities

A few metrics have them

are there any serious treatment of those?N

Or are they treated like other singularities

just removed from the manifold

ACuriousMind

@Slereah I think nothing really blows up at the singularity, so you are probably fine to just ignore it.

Slereah

Not exactly

2D black holes have a conical singularity

you still get interactions if you send something at it

ACuriousMind

00:08

Yeah, but I think you don't need to remove the point since all invariants are finite in the limit $r\to 0$.

It's an allowed point, it's just not very manifoldy there :P

Slereah

Well yes but that's my point

Does it work in basic GR or do you need to extend it to conifolds

ACuriousMind

So it's quite unlike the singularity inside an 4D black hole, where curvature blows up

@Slereah Do you mean spatial 2D or spacetime 2D?

(only one correct answer :P)

Slereah

2+1D

ACuriousMind

@Slereah I think that depends on how you defined "basic GR" in the first place. Strictly speaking, you can't even say "my manifold has a conical singularity", you'd have to say "Oh shit, this isn't a manifold".

Slereah

That is kind of my point, yes

ACuriousMind

00:11

And then you have to define a metric tensor on things that aren't everywhere locally $\mathbb{R}^n$, and so on, and so forth

So, no, rigorously, this doesn't work at all

Slereah

I mean while the curvature doesn't blow up at the singularity

Is it even defined

Since the conical singularity is a kink

ACuriousMind

@Slereah Well, you define it by the limit ;)

Slereah

Probably not differentiable

ACuriousMind

I guess the most straightforward way is to just allow that isolated points have neighbourhood like cones, and then you define any quantity you can't define at the tip due to non-smoothness just by the limit of that quantity towards that point.

Slereah

I'm not even sure there's anything about that

Most conifold papers are string theory

ACuriousMind

00:13

Yeah, I'm coming up with nothing trying to search for a rigorous definition

Those filthy physicists just say "look, it's a cone" and then act as if it was no big deal :D

Slereah

basically

hm

I think looking at 2+1D black holes is not gonna work, since nobody cares

but cosmic strings also have conical singularities

Well better go to bed

busy day tomorrow

ACuriousMind

Ha! The math term for what conifolds are is a pseudomanifold, which is really just a regular manifold together with a singular set of points around which it looks weird.

HDE 226868

Does anybody here know much about the Kozai mechanism?

ACuriousMind

@HDE226868 Classical celestial mechanics? I don't think we have someone for that :/

HDE 226868

I thought perhaps Chris White would know about it.

I know his specialty is astrophysics, but you have to study celestial mechanics to get there (I think).

Not that it's a common topic, but still.

ACuriousMind

00:20

Have a star, maybe someone comes by who knows

HDE 226868

@ACuriousMind Thanks.

0celo7

@ACuriousMind I've got my eye on you, Astronomer Candidate 2...

ACuriousMind

@0celo7 Do you still think "the astronomer" is a single person?

Poor lad.

0celo7

You're in on the conspiracy??

ACuriousMind

I didn't say nuthin'

0celo7

00:30

right...

ACuriousMind

(I think he's on to us, guys)

0celo7

Yeah, you bullies!

What is your motive for starring my stuff? Answer me!

3

Astronomeeeeeeeeeeeeeer!!

@ACuriousMind You know, Weinberg defines the momentum as the variation derivative wrt. the time derivative of the field of the Lagrangian. (Not to be confused with the Lagrangian density.) Is that ill-defined as well? Sect. 7.2 in QTF 1.

ACuriousMind

00:54

@0celo7 If "variation derivative" is the same as functional derivative, then yes.

0celo7

Typo, should be variational, but yes.

Ok.

ACuriousMind

I think this area is also a bit muddy because it is not guaranteed that the physicists really mean the mathematical functional derivative when they write it

Just like they don't mean $\otimes$ when they take the product of groups :P

0celo7

What happens if you ask an algebrist what $\mathrm{SU}(3)\otimes\mathrm{SU}(2)\otimes\mathrm{U}(1)$ means?

Does the average mathematician know what is meant by that?

ACuriousMind

@0celo7 They will try to define a tensor product for non-abelian groups, and then ask you in what situation you would possibly encounter that object.

Secret

that is why the standard model is awkward pasting many isolated bits together

0celo7

01:00

Is ST really less awkward? There's like 10^500 ways of doing it.

ACuriousMind

It's not very natural to think about the tensor product of group because the usual universal property is "twisted".

Secret

I know so little about ST to even able to comment about ST awkwardness

0celo7

Also no ST books that I know of actually mention the standard model.

ACuriousMind

@Secret No. Physicists don't mean the tensor product when they write $\mathrm{U}(1)\otimes\mathrm{SU}(2)$. They just for some reason use the wrong symbol.

They actually mean the direct product $\times$, which is much more natural

0celo7

Like, how on Earth do we get electrons from superstrings?

Secret

01:02

Ah, that cleared a lot of confusion, because I often confuse the direct product with the tensor product

0celo7

How on Earth does stuff like annihilation work on the string level?

How do fermionic strings get mass?

@ACuriousMind Can you answer?

Secret

Whenever I saw $\otimes$ in physics I always thought: What is a tensor producing doing in here

and then over time I am mislead to believe that the tensor product is the direct product

ACuriousMind

@0celo7 No, I couldn't when you last asked me that question, remember?

0celo7

@ACuriousMind I don't remember.

But I probably asked you long ago, you've gotten smarter since then.

ACuriousMind

I think I only get more knowledgeable, not smarter at this point.

0celo7

01:05

Same thing. I seem to get less of each.

ACuriousMind

And you know I didn't learn string theory :P

0celo7

I know you now have a stylish paperweight :D

I use my $200 mechanics book as a laptop stand :D

Figured you could do a similar thing with BBS

ACuriousMind

I have a desktop computer here, no stand required

It currently just lies in the corner accusingly.

0celo7

This summer, the 0celo7 desktop will happen

ACuriousMind

@0celo7 But before that, winter is coming.

0celo7

01:11

@ACuriousMind Ok, I need the magic pedagogy for: "Find the orthogonal projection of the matrix [[3,6][1,7]] onto the space of anti-symmetric 2x2 matrices. "

ACuriousMind

Do you have Christmas markets in the US?

0celo7

@ACuriousMind I'm gonna be in K town...you should come and we can hang out!

ACuriousMind

What is K town?

0celo7

@ACuriousMind We defined the norm on the space of matrices as the dot product of the entries of the matrices regarded as vectors.

@ACuriousMind Kaiserslautern

Yes, the Amis call it K town

ACuriousMind

@0celo7 Just write it as the sum of an undetermined symmetric matrix and an antisymmetric matrix and solve the equations you get

Secret

01:16

I am wondering, would we ever had the concept of vectors in $\mathbb{R}^n$ as arrows if we only have the concept of 1 dimension? (but not 2 dimensions or 3 etc.?)

0celo7

Wait, is the orthogonal projection here a symmetric matrix?

ACuriousMind

@0celo7 I think you're asked to calculate the image of that projection, which is the antisymmetric matrix

0celo7

@ACuriousMind Ok, that's what I thought. Problem is, I got 0 for the off-diagonal entries.

Which is a wrong answer.

Secret

Direction, seemed to me is just how the components are related to each other, and unlike magnitude, cannot be written in some sort of explicit formula

0celo7

If $A$ is the projection into the subspace, I minimized $|M-A|$.

Isn't that a definition of the projection?

ACuriousMind

01:21

@0celo7 That is correct

0celo7

Problem is, $|M-A|^2=(3-a)^2+(6-c)^2+(1+c)^2+(7-b)^2$. Here $a,b$ are diagonal elements and $c$ is the off-diagonal one. Taking the partial wrt. $c$ ~~gives 0~~ $c$ falls out.

And $a=3$ and $b=7$.

ACuriousMind

@0celo7 Wait. $a=b=0$ for antisymmetric matrices anyway

0celo7

Uh

fml

ok

Secret

Matrices as elements in a vector space seemed to have some kind of ordered structure since
$$\begin{pmatrix}a & b \\ c & d\end{pmatrix}\neq\begin{pmatrix}c & d \\ a & b\end{pmatrix}\neq \begin{pmatrix}b & a \\ d & c\end{pmatrix}$$

If we don't think of matrices as made of column vectors, then how mathematically is this order structure is being introduced?

0celo7

ok big fail

ACuriousMind

01:25

And $c$ doesn't drop out, that polynomial is quadratic in $c$

0celo7

@ACuriousMind when I take the partial wrt. c it does

You have to minimize, right?

ACuriousMind

@0celo7 We have: $\lvert M-A \rvert^2 = 2c^2 -10 c + \text{const}$.

0celo7

sigh

just ignore this

I did the chain rule wrong...

see, I really am regressing

ACuriousMind

Why were you doing the chain rule?

0celo7

derivative of $(6-c)^2$ is not $2(6-c)$

forgot the extra minus

ACuriousMind

01:29

Ah. Well, sometimes it is better to do stuff explicitly instead of trying shortcuts

0celo7

indeed

0celo7

01:46

> Find the orthogonal projection of the matrix [[5,7, 6][2,6, 1][2,1, 5]] onto the space of anti-symmetric 3x3 matrices.

Oh you have got to be kidding me.

Galois in the Field

03:05

Hi

Galois in the Field

03:58

Does $\sum_{n\in \Bbb Z}a_n\left(-n\epsilon_{jkl}q_kq_l^{n-1}\right)=0$ for some reason?

Sorry ignore the $a_n$

0celo7

Amazon is out of stock of the GSW anniversary edition

well, I probably don't need to get more books anway

Galois in the Field

@0celo7 Buy me some ;)

D&F's abstract algebra & A&M commutative algebra for starters

0celo7

how about no

I need to get...

how about I finish Arnold

Galois in the Field

How about no, you do buy me the texts

And I give you <3

0celo7

I have a lot of physics books

but no math books

I should get...something that doesn't need analysis

Galois in the Field

04:03

That sounds like a sad life :(

0celo7

indeed, considering I'm a math major

so I need more math books!

Galois in the Field

Oh wow yes then

Do you want actual recommendations? Are you doing any algebra?

0celo7

not yet

dunno when I will

pretty sure the first course uses Artin

I doubt I'll get further than one or two semesters of undergraduate stuff

Galois in the Field

Group theory -> ring theory -> field theory -> galois theory

Use D&F for exercises, and Artin for learning on the first pass, and A&M when you get through the intro stuff

Algebra = life

0celo7

cambridge.org/us/academic/subjects/physics/…

algebra is boring

Galois in the Field

04:08

But you haven't done any ':<

0celo7

because it's boring

Galois in the Field

Do you know any?

0celo7

no, it's boring

Galois in the Field

Do you know what an algebra is?

You think an associative $R$-algebra is boring?

Exactly! Googler!

0celo7

@GaloisintheField yes

Galois in the Field

04:10

@0celo7 What is it concisely

0celo7

@GaloisintheField yes

@GaloisintheField vector space with a map

Galois in the Field

@0celo7 NOOOOOOOO

Generally its base 'field' is a ring

0celo7

module with a map then

same thing

Galois in the Field

Not the same thing :P

0celo7

either way

boring

Galois in the Field

04:11

:'( my life is boring :''(

0celo7

yeah

probably

Galois in the Field

:'''(

0celo7

still dunno what I want to do math-wise

Galois in the Field

Have you done any functional analysis?

0celo7

TIL algebraic geometry has applications in engineering

no, I don't know any analysis

Galois in the Field

04:13

Algebraic geometry is pretty heavy. You have to do A&M's commutative algebra :P

0celo7

well, I'm a double major

so I have to pick one branch

Galois in the Field

What are the options?

0celo7

I don't really want to spread out too much

pretty much everything besides graph theory

unless the combinatorics classes do that

Galois in the Field

I don't know what everything is sorry

0celo7

but I have no desire to do graph theory

Galois in the Field

04:14

Algebraic geometry is a path for some reason?

0celo7

algebraic geometry is a class at the end of the algebra path

Galois in the Field

4th year?

0celo7

graduate

I could get to it in my...5th?

Galois in the Field

But algebra is boring :''''(

0celo7

I could do it in the 4th if I didn't do anything but math

Galois in the Field

04:16

Algebraic geometry is really hard honestly, and unless you are planning on doing pure fields, I don't see how useful it would be

0celo7

I was thinking of going the PDE/functional analysis route

Galois in the Field

That would make much more sense

0celo7

indeed

but I also want to learn algebraic topology

dunno if I have time for that

Galois in the Field

Why the heck do you want to learn algebraic topology lol

0celo7

K theory is cool

number theory is boring

Galois in the Field

04:20

@0celo7 I think so too

But any deep subject is interesting if you work on it enough tbh

0celo7

ehhhh

not sure if I'd ever find bridges interesting

or robots

robots are booooooooring

Galois in the Field

Robots are boring?

Surely there are interesting robots?

...

Did you delete that or reported lol

0celo7

deleted

Galois in the Field

Oh haha, I have to go now, best of luck :P

0celo7

I'm only in my first semester

I have like a year before I have to choose

everything until then is just analysis, algebra and topolgoy

all required

@GaloisintheField I do have Hatcher (Algebraic Topology) next to me

just...not motivated to read it

not really prepared for it

@GaloisintheField you should find me a nice book on Springer on...GR or some applied diff geo

maybe Morse theory?

dunno

@ACuriousMind @Slereah Do you know of any Springer books that are applied differential geometry?

link.springer.com/book/10.1007/978-3-319-18573-6

Hmm!

skull petrol

04:46

@0celo7 what ever happened to that Fed guy with the guitars avatar?

The GR room you guys opened "spacetime has curves?" is frozen

David Z

05:15

0

Q: Propogation of error using Euler's first order method

I was estimating a falling object's position versus time by using a simple first order step function, where for i=2:length(t) % We're using Euler, so we need an initial previous point to start t=t(i); V=V+dv; rho_i= % Calculate our height and then pull rho from data K=(m/(mg-.5*C_d*A*...

computational-physics

off topic? I'm not sure, since it's asking about error propagation instead of implementation details... but I tend to think of error propagation in numerical algorithms as a computing question, not a physics one

user685252

05:42

Indeed, there is very little Physics there.

yuggib

07:25

0

Q: Relation between Quantum Field Theory and consciousness

Question is simple...are there any interesting theories on this topic? Is there anything to it? After all, brain is some sort of solid state, and there are many theories based on QFT that explain properties of some solids, like BCS theory of superconductivity.

quantum-field-theory biophysics perception spin-model network

quantum mysticism seems a recurring topic on phys SE

Slereah

08:55

One thing I wonder is

Are not all manifolds triangulable?

What's an example of a non-triangulable manifold

skull petrol

10:01

the genetics of a raider

user116211

10:15

Quantum entanglement is proved again; sorry Einstein :P nytimes.com/2015/10/22/science/…

Slereah

You can't deny Einstein and the Evidence

skull petrol

10:30

thE Evidence

or

thĒ Evidence

and Einstein

@user36790 no apology necessary pal :P

yuggib

@Slereah "In dimension 4, however, the E8 manifold does not admit a triangulation, and some compact 4-manifolds have an infinite number of triangulations"

from wikipedia...

Triangulation (topology)

In mathematics, topology generalizes the notion of triangulation in a natural way as follows: A triangulation of a topological space X is a simplicial complex K, homeomorphic to X, together with a homeomorphism h : K → X. Triangulation is useful in determining the properties of a topological space. For example, one can compute homology and cohomology groups of a triangulated space using simplicial homology and cohomology theories instead of more complicated homology and cohomology theories. == Piecewise linear structures == For topological manifolds, there is a slightly stronger notion o...

obviously I do not know anything about that...just pointing out

0celo7

12:21

@yuggib What exactly do you know about, then?

yuggib

nothing...

apart from the fact of knowing nothing

therefore I know something

damn Russell's paradox

:-D

ACuriousMind

12:36

That's more Socrates than Russell :P

@yuggib You're not surprised about that, are you?

yuggib

@ACuriousMind No, actually not so surprised

@ACuriousMind Saying that I know of being ignorant is socrates; but saying I know nothing is paradoxical...

ACuriousMind

@yuggib The usual phrase attributed to Socrates is "I know that I know nothing", though.

0celo7

@skullpetrol No clue, he hasn't been on Skype or anything :(

We were supposed to mod Skyrim :(

ACuriousMind

His addiction to insomnia cookies finally got him

RIP

0celo7

@ACuriousMind Hey, that's me!

@ACuriousMind What are some esoteric applications of DiffGeo?

ACuriousMind

12:51

@0celo7 Huh?

0celo7

Arnold got my geometric juices flowing.

Is "Geoemetrothermodynamics" a serious thing?

ACuriousMind

I seem to recall it is

0celo7

link.springer.com/article/10.1007/s13538-011-0015-4

Is there a book on it?

Huy

wat do u mean by esoteric

0celo7

arxiv.org/abs/1111.5056

@Huy Does not...

yuggib

12:53

@ACuriousMind No, I think he was better in logic than that...with a web search I found the english form "I know nothing except the fact of my ignorance"

Huy

does not wat

yuggib

and in italian is quite similar

0celo7

not allowed to say

Huy

wat

yuggib

;-)

0celo7

12:53

@yuggib For some reason I think you're from Yugoslavia

Maybe the username?

ACuriousMind

@yuggib Well, the statement is apocryphal anyways (it does not appear in Plato's works)

yuggib

@0celo7 Don't know...but I have no Yugoslavian relatives

@ACuriousMind :-D agreed

0celo7

@Huy don't want to anger him

yuggib

@0celo7 and btw Yugoslavia do not exist since 199- something

Huy

anger who

0celo7

12:55

@yuggib really? never woulda guessed

yuggib

at the very best you have a former yugoslavian republic

0celo7

@Huy the one with long hair

ACuriousMind

@Huy I told him I found his repeated talk about the GDP annoying, so now he's trying not to talk about it.

0celo7

esoteric = useless but interesting

Huy

I'm confused

you guys are weird

3

0celo7

12:57

no, ACM is weird

seriously, what are some books on applications of geometry where it's unexpected?

that continuum mechanics book is intriguing

@ACuriousMind Do you know anything about ergodic theory?

ACuriousMind

Nope

0celo7

Seems interesting if one is into dynamical systems

which would make sense for me to be into

I think

I dunno, apparently algebraic geometry has engineering applications

wonder if number theory has engineering applications

0celo7

13:28

www2.latech.edu/~schroder/slides/number_theory/…

apparently one needs measure theory and functional analysis for ergodic theory

yuggib

@0celo7 Cryptography is always used as a practical justification for number theory. But I suspect that it may be considered a separate branch...pure number theorists have higher problems to worry about

Huy

13:48

my prof is boz in ergodic theory

0celo7

I need to learn functional anylsis

Huy

told you son

dude

0celo7

What

Huy

told you dude

son

0celo7

But I also need measure theory

Huy

13:56

then study some

easy peasy

0celo7

Why is useful math so boring

Huy

ur boring

(wasn't me)

0celo7

I'll shi your pie

Huy

nap

l2p

0celo7

I'll napalm your pie then

Works for me

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