The h Bar

Danu

8:00 PM

So I took a mini course on mathematical GR

Loong

@Obliv In China, the default water is hot.

Danu

@Loong Fo' real?

Obliv

that's so weird.

Danu

Gonna need some evidence for that

Obliv

It is healthier so it doesn't surprise me since Chinese people live so healthily :p

barrycarter

8:03 PM

@Obliv Depends on how hot it is, I think.

Loong

@Danu chinesecultural.ca/chinese_culture/chinese_culture.html

Obliv

@ACuriousMind Like given $A = \{a,b,a\}$ the set of equivalence classes of $=$ is not equal to the binary relation on $A$?

since the binary relation on $A$ is the set of ordered pairs right?

ACuriousMind

What is $\{a,b,a\}$ supposed to be? Sets do not contain the same element twice.

Danu

This should honestly probably be in the math chat at this point :P

Obliv

they can

Danu

8:09 PM

I doubt that---shouldn't any set be a union of singletons?

Obliv

frequency and order is irrelevant in a set, iirc.

ACuriousMind

@Obliv No, they cannot, at least not in the standard axiomatization.

Danu

4

Q: Can elements in a set be duplicated?

If $A = \{x \mid x \text{ is a letter of the word 'contrast'}\}$ Represent it in a Venn Diagram, and then find the $n(A)$. Do I need to write the letter 't' twice inside the venn diagram? What should be the answer for $n(A)$?

elementary-set-theory

ACuriousMind

@Danu ...aaaaaaand?

Danu

@ACuriousMind Seems pretty fucking boring

Obliv

8:10 PM

Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2,4,6}. Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics such as Venn diagrams are taught...

ACuriousMind

@Danu Could've told you that from reading this chat :P

Obliv

down by describing the set 4th paragraph

anyway this is a tangent

Danu

@ACuriousMind Sadly.

No interesting topological/geometric insights

The best they can do is using Frobenius' theorem to find some submanifolds to which certain subbundles of $TM$ are tangent

bolbteppa

Any thoughts on the statement: "Geometric Algebra has been characterized as a reflection algebra because its most primal computation, vector multiplication, can be reduced to a simple reflection"?

Danu

Geometric Algebra being...?

ACuriousMind

8:13 PM

@bolbteppa Here's a thought: "Huh?" ;)

@Danu another name for the Clifford algebra of a space.

bolbteppa

https://en.wikipedia.org/wiki/Geometric_algebra
http://faculty.luther.edu/~macdonal/GA&GC.pdf

Danu

Oh, I'm planning on going to a summer school on Clifford algebra

ACuriousMind

It has got to be about more than just "Clifford algebras", right?

Danu

Not really, I think :D

ACuriousMind

Hey @StanShunpike! How are you?

Danu

8:20 PM

cs.unitbv.ro/~acami/summerschool.htm

ACuriousMind

@Danu Hmmmm...eh, just another topic that's much deeper than I thought, then ;P

user507974

@ACuriousMind like every topic in existence?

Danu

@ACuriousMind We'll see

user507974

also guys, i got a landau/lifshitz for $9

Danu

nice, how

user507974

8:22 PM

someone undervalued it

actually ive had friends make ALOT with physics books

Danu

hyperboleandahalf.blogspot.de/2010/04/…

bolbteppa

@user507974 which one?

user507974

they got a big box for like $300 that included almost all the LL and they sold $500 worth of other books they had copies of

@bolbteppa stat mech part 1

to amazon books, so its not like they even put a lot of effort into the selling back part

ACuriousMind

@Danu The second part is "Kähler calculus", that's certainly not just Clifford algebra! Although I am already confused at the terminology because I know something called "Kähler differentials" from alg. geo. and that's not it.

user507974

this same guy got an oscilloscope, psu, signal generat for $100

ACuriousMind

8:26 PM

@Danu The Alot! I had almost forgotten about that

bolbteppa

@user507974 Sweet, if you find the early sections frustrating this might help a bit weizmann.ac.il/complex/tlusty/courses/statphys/statphys.pdf

user507974

makes me wish i had a car to do some precision yard selling

@bolbteppa wow, virial expansion and cluster methods in ch 3,

0celo7

Apparently the lab will pay for physics books

Give me allllll the books

bolbteppa

I think Geometric Algebra is the "Euclidean Space Linear Algebra" of Clifford Algebra theory, i.e. it's basically a way to visualize this theory through plane geometry and then extends these ideas to SR and stuff, I know it arose as a way to do plane geometry coordinate-free and motivated linear algebra, but I think the real secret behind this literally is that it is just a "reflection algebra",

i.e. decomposing rotations into products of reflections then representing the outcomes of rotations via reflections

From that summer school: "Lie derivatives as partial derivatives"

"Reflection on the foundations of differential geometry as argument for canonical Kaluza-Klein geometry" wtf?

Why don't I know all this and algebraic geometry already :\

Obliv

what does a set of a bunch of sets look like?

$\{\{blah\},\{blah2\}\}$?

Danu

8:37 PM

sure

0celo7

@bolbteppa See Wald for that

The Lie derivative is just a partial derivative in coordinates adapted to the integral curves of the vector field

@bolbteppa

bolbteppa

Sweet

0celo7

@bolbteppa Also note that for functions, $\mathcal L_v f=vf$

i.e. the directional derivative

Stan Shunpike

8:52 PM

@ACuriousMind I am amazing! So busy with school and music and friends and relationships, but never better. How about you? I was admiring your point total today.

You must have the largest $\frac{d\text{points}}{dt}$ ever :,-D

ACuriousMind

@StanShunpike Good to hear! I'm fine, although not as busy currently, it's still two weeks until the semester starts again

And I don't have the fastest point increase, at least JohnRennie has gained over 80k since I joined

Stan Shunpike

O.O wow

that's amazing

he must then also have the fastest second derivative too

#acceleration

@ACuriousMind Are you taking classes or doing research Or both? if classes, what have u been taking?

0celo7

ACM doesn't research

He knows everything already

Stan Shunpike

@0celo7 hahaha I beg to differ

ACuriousMind

@StanShunpike I'll hopefully start doing research, but I'm also taking some classes. Currently "Advanced String Theory" and "Topology of Singular Spaces" are planned, not sure if I'll take more, there's not much going on that interests me and that I've not already taken

Stan Shunpike

8:59 PM

I think his profile page says otherwise

explicitly

@ACuriousMind what topics do you plan to research? will you be actually running experiments or just like interpreting data?

Obliv

@StanShunpike Was "heartbeat" played or was it synthesized?

the instrumental parts that is

ACuriousMind

@StanShunpike I'm a theoretician, I don't deal in experiments or data. ;) What I'll be doing depends on which advisor I choose, which seems like something I'll have to decide at the end of this month, if my current plans work out.

But it's going to be QFT, possibly a bit stringy, possibly rather mathematical. I keep wavering on how much of a mathematician I want to be :P

Stan Shunpike

@Obliv synthesized

i need to update that

my latest is here soundcloud.com/gotlepton/daybreak

@ACuriousMind yes, that seems like the career problem someone in your position naturally faces

you are going over to the dark side

0celo7

@ACuriousMind take some more Riemannian geometry

You can never know enough theorems about sectional curvature

ACuriousMind

::yawn::

Danu

9:10 PM

@ACuriousMind Singular spaces in what sense?

ACuriousMind

@Danu That's the mystery ;) I have no idea why the lecture is named like that, it's the continuation of last semester's sheaf cohomology.

Danu

I'm so excited for the courses this semester :3

Finally, mathematical gauge theory :D

Okay. Sounds awesome! My Riemann Surfaces course will also focus a lot on sheaf cohomology, which is why I'm excited about it.

I'm almost through the book now, but I didn't really study it very well, just read everything through without worrying too much about the details

I'm struggling with developing some intuition for sheaves

And the book I'm reading is very concrete, very short with little motivation/explanation

0celo7

@ACuriousMind :(

bolbteppa

@Danu have you seen the Laurent series motivation for sheaves?

Danu

@bolbteppa If you'd be more clear about what you mean I would be able to tell ;)

bolbteppa

9:19 PM

@Danu toperkin.mysite.syr.edu/talks/sheaves_and_more_cohomology.pdf

Danu

Thanks, I'll have a look later

bolbteppa

Basically you have a collection of open sets coming from functions and you have some constraint on the functions on the overlap of the sets, i.e. some patching condition gluing the open sets together in some way, then the cohomology comes in when you mod out the functions for which you can patch the sets together and concentrate on cases it doesn't hold

ACuriousMind

Why do you call that "Laurent series motivation" when that's a completely general description of what a sheaf and its first cohomology do?

bolbteppa

"Origins: The Mittag-Leffler Problem"

Danu

@ACuriousMind That's what I was thinking

The motivation that link gives that there was historically a question of whether you can get meromorphic functions with prescribed poles

To me, that's not much of a motivation. Why would I care about that?

bolbteppa

9:26 PM

Are you serious? You mean the historical question that motivated the very concept is useless to you?

Danu

(I only read the first few paragraphs)

@bolbteppa If you can't tell me why this historical question is significant then... yeah.

What can I do with meromorphic functions with prescribed poles?

I'm not being entirely facetious here---my point is that a "motivation" should tie it to things that I know I care about

ACuriousMind

@bolbteppa I would always call that "Mittag-Leffer" or "Cousin" motivation, not Laurent series motivation, and your "basically" description didn't actually contain any reference to Laurent series.

bolbteppa

Dude this is basic complex analysis, the Mittag-Leffler question that arises when you analyze Laurent series

ACuriousMind

@Danu Do you care about invariants? I.e. things that can distinguish manifolds or other spaces from each other?

Danu

@ACuriousMind Of course I care about invariants :D

ACuriousMind

9:29 PM

Because then the cohomology of certain sheaves on a space is just another such invariant.

Danu

...but that's a-posteriori

I'm talking about motivation to understand why people wanted this to begin with---was it obvious that this would yield invariants?

bolbteppa

You guys are being ridiculous tbh, it's like you are not thinking properly, these questions come from analyzing Laurent series if you don't see that you're still just memorizing things

Danu

@bolbteppa what are you even on about?

ACuriousMind

@bolbteppa I refrain from insulting you when you say things I consider ridiculous, and I would appreciate if you did the same.

Danu

Do you feel like you can draw conclusions on our general understanding of mathematics, and say "you're still just memorizing things", from what... 3 sentences?

That comes across as arrogant, man.

ACuriousMind

9:32 PM

@Danu Do you know a good motivation for singular homology that's not "this will yield powerful invariants"?

Danu

Sadly, I never took a course that treated homology yet.

ACuriousMind

Or "this will be a generalization of deRham/cellular/simplicial".

Danu

It's coming up this semester, and I'm confident Leeb will provide a very thorough discussion; he always does :D

I'm also OK with someone just saying: This is not going to make much sense but once we've got all the definitions there are many great things we can do with it.

The problem is that my current book doesn't even do that.

It's just Lemma Lemma Theorem Lemma, once every 5-7 pages an Example or two... Rarely any text that's not a purely formal statement.

bolbteppa

@ACuriousMind It's ridiculous to try to say that my "basically description" doesn't reference Laurent series when the pdf I linked to (which I was summarizing in general terms) begins by discussing them through it's "Origins" in the Mittag-Leffler Problem, where do you think Cousin got his inspiration from? It's memorization to try to say calling it the Cousins problem somehow means referencing it's Laurent series motivation is somehow problematic, especially when the person is having trouble

Danu

@bolbteppa why don't we stop talking about this. Nobody is enjoying this.

If you think I'm being stupid, just ignore me please (and let ACM try to explain it to me). I don't think anyone is getting anyone out of this right now.

ACuriousMind

9:39 PM

@Danu :O Do you know some sort of cohomology theory? deRham, for instance? Because I always found it reassuring that the "usual" cohomologies are sheaf cohomologies for very simple sheaves (deRham is that of the constant sheaf)

Danu

I know de Rham cohomology

And yes, I saw that come along in the book, at least.

A huge bunch of major theorems seem to follow from the fact that an exact sequence of sheaves induces this longer exact sequence on cohomology groups.

I don't know if that's the standard way of proving things, by the way.

ACuriousMind

Or rather, deRham is that of the constant sheaf resolved by the differential forms - and different resolutions of the constant sheaf give different ways to comute cohomology, e.g. singular.

@Danu Oh, the "long exact sequence in (co)homology" is a very standard tool.

Danu

Yeah, that thing.

ACuriousMind

And I hope you were tortured with proving the snake lemma involved in its construction at least once ;)

Danu

Someone I recently talked to also mentioned this snake lemma---I didn't encounter it?

bolbteppa

9:42 PM

@Danu I'm not trying to be arrogant - it comes across as really, really pedantic and conflict-baiting to try to say I'm saying something incorrect by using words more familiar to you and not technical words (words you've literally said are un-motivated and causing you problems with understanding this) when I'm just trying to help you, if you don't find it helpful that's fine, standard books use this as motivation for sheaves e.g. Griffiths Harris, I don't mind help being thrown back in my face

Danu

@bolbteppa I'm not even sure whether you're trying to say I'm being conflict-baiting or not, now. In case you were, I fail to see how.

bolbteppa

It's fine, just drop it

ACuriousMind

@Danu The morphism $H^i(C) \to H^{i+1}(A)$, where do you think it comes from?

Danu

@bolbteppa :) No hard feelings

HDE 226868

@0celo7 (Sigh) Done.

Danu

9:45 PM

@ACuriousMind This connecting homomorphism?

ACuriousMind

@Danu Yes

It exists because of the snake lemma, and I don't know an easier way to prove its existence.

Danu

The book I'm reading only describes it in about 1 paragraph and then immediate proves the theorem on the long exact sequence.

lols

ACuriousMind

lol, it "describes" it? :D

Danu

I do want to note that this book uses a lot of shortcuts which are only possible in the case of Riemann surfaces (I've noticed this in the discussion of simpler topics)

ACuriousMind

Ah

Danu

9:46 PM

I don't know if that's what's going on here, too.

ACuriousMind

Maybe you can really write it down explicitly in that case

Danu

Yeah, he gives it explicitly.

It takes about half a page (Forster, Lectures on Riemann Surfaces; section 15.11)

You did sheaves on arbitrary complex manifolds, or what?

ACuriousMind

Arbitrary topological spaces

Well, I think we assumed they're...compact and connected at the end.

Danu

Ah, of course

ACuriousMind

Or maybe not compact? I forget, but most of the theory doesn't need anything besides a topology

user507974

9:49 PM

guys, what would you guys say is the most important variable you learn about in GR

Danu

Compact is life

@user507974 $x$

user507974

@Danu a little more special to GR, its for a present im designing for fellow physics folks

Danu

@ACuriousMind The theory of compact Riemann surfaces is real nice.

@user507974 I don't think the question makes much sense.

ACuriousMind

@Danu Haha, he only give it for $H^0 \to H^1$, the trickster :Ps

In general, that "long" sequence is much longer than what's written in 15.12

user507974

@Danu each of the big fields has a couple of really important symbols you identify with the topic immediately

Danu

9:51 PM

@ACuriousMind Ah... That's interesting.

@user507974 Okay, in that sense.

Of course $g_{\mu\nu}$

ACuriousMind

The $H^n$ in general only vanish when $n$ is larger than the dimension of the space ("Grothendieck vanishing theorem")

Danu

Also $R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi T_{\mu\nu}$

@ACuriousMind Grothendieck, really? Pfft...

For forms this is trivial :D

But I guess de Rham is just a sad little dot in a sea of cohomology theories

ACuriousMind

@Danu Haha, but this is for a general "Noetherian topological space"

user507974

@Danu and what about for EM

0celo7

@HDE226868 Sigh?

Danu

9:54 PM

@ACuriousMind I also find it suspicious that you say you are going to be giving me a definition of "dimension" that makes sense for (fairly) arbitrary topological spaces

Also yeah, Noetherian ones come across as pretty much super special/pathological to me

bolbteppa

@user507974 In the strict sense of a variable, the metric $g_{\mu \nu}$ is the variable in the Einstein-Hilbert action you solve for by minimizing, so discussing the construction of the EH action, e.g. how you go from the classical mechanics action to the SR action (+ flat EM) to the GR action (+ curved EM) all because of the constancy of the speed of light (and why potential energy $V(x)$ allows infinite speeds hence CM action must be wrong) would be a good talk (Landau Vol. 2 does it nicely)

0celo7

@user507974 $g$, definitely.

ACuriousMind

@Danu I think its the Lebesgue covering dimension

Danu

Ascending chains of open sets stabilize was the defining property, right?

0celo7

If I see a $g$ it's either GR or Riemannian geometry

Danu

9:56 PM

@ACuriousMind Oh, nice. Didn't hear about that concept before.

And it agrees with the usual one in the simple cases, I hope?

user507974

kind of fitting, given that back in our youth g was associated to gravity

ACuriousMind

@Danu Yep

bolbteppa

The cohomology $\mathrm{H}^n$ is just a vector space / abelian group / category of dimension $n \in \mathbb{Z}$ right?

ACuriousMind

@Danu Noetherian looks pretty horrible, but it is equivalent to "hereditarily compact", i.e. every closed subset is compact again. This is e.g. true for compact subsets of $\mathbb{R}^n$ - they are closed and bounded, so closed subsets of them are also closed and bounded, hence compact, hence the space is Noetherian.

Danu

@bolbteppa No, it's the $n$-th cohomology group. Its dimension can be e.g. 0

bolbteppa

9:59 PM

Yeah sorry

dimension $m \in \mathbb{Z}$

Danu

@ACuriousMind Every closed subset of what?

ACuriousMind

Oh god, forget what I said, it's non-sense :D

Danu

I was going to say something about relationships between closed and compact in Hausdorff spaces, but yeah okay let's forget about it ;)

ACuriousMind

Or is it?

Now I'm confused

@Danu Every closed subset of the space.

Danu

In Hausdorff spaces compact implies closed, and in compact spaces it's the other way around

barrycarter

10:02 PM

@JohnRennie is probably asleep, but lets see if this summons him.

Danu

So it's slightly weaker than compactness, @ACuriousMind?

ACuriousMind

Ahhhh

Danu

Or is compactness actually equivalent to every closed set being compact?

ACuriousMind

Hereditarily compact means every subset is compact, not just every closed one

Danu

Ewwww

ACuriousMind

10:03 PM

So no, nothing you'd usually write down is Noetherian :D

Danu

Yeah, that's pretty damn useless.

Algebraic geooooometry

Y U SO pathological

ACuriousMind

But this means I don't know any good vanishing theorem for non-alg-geo spaces, so the sheaf cohomology can potentially get even longer than the dimension

Danu

Yeah, it's probably not easy in spaces that don't have a proper notion of dimension.

What about arbitrary cell complexes, for instance?

That seems like a nice class of topological spaces to be able to cover.

ACuriousMind

Is $\mathbb{C}P^\infty$ a cell complex? I forget, but that thing has arbitrarily high cohomology.

Danu

How'd you define that (never seen it before)?

ACuriousMind

10:06 PM

Uhhhhhhhh

It's the direct limit of the $\mathbb{C}P^n$, I think

Danu

Yeah, but then again I don't know what the direct limit is ;)

I want to say that it should be a cell complex

since the finite projective spaces all arise so "canonically" from the idea of a cell complex

But I have no real idea

ACuriousMind

The direct limit in this case is just the union

So yes, it's a CW complex with one cell in every even dimension. Or should be

Danu

Then I definitely want to say it is

ACuriousMind

Well, but it's infinite-dimensional, so this doesn't tell me there's no nice vanishing theorem

Danu

Hahaha

ACuriousMind

10:11 PM

Oh well, it's not that relevant anyway

Danu

I guess in the case of cell complexes, there is a pretty obvious/natural definition of dimension

ACuriousMind

My original point was that the "long" cohomology sequence is much longer than the pithy truncated one Forster presents ;)

Danu

which in term leads me to believe there should be no irregularities with nonvanishing cohomology

@ACuriousMind Well, it is the full long cohomology sequence for a Riemann surface :D

ACuriousMind

is it?

There's no $\to 0$ at the end

so it could go on

Danu

Oh, yikes, could second cohomology classes also be nonvanishing? :P

ACuriousMind

10:14 PM

That is what I mean by "can be much longer"

Danu

No right? Or is it somehow tied to real dimensions?

@ACuriousMind My point was that the thing is one-dimensional.

So the classes should hopefully vanish for larger than $1$ (or $2$ if it's the real dimension that counts)

ACuriousMind

@Danu Yeah, but that the second cohomology of an arbitrary sheaf vanishes would be such a vanishing theorem I'm searching for

Danu

Sure, but my experience with exactly 1 cohomology theory proves to me that it's always the case :D

ACuriousMind

But that's just the constant sheaf! You have no idea how horrible these things can get ;)

Hm, best I can find are vanishing theorems for complex manifolds, but given GAGA, that's not much better than the alg. geo. case.

Danu

@ACuriousMind But Riemann surfaces are complex manifolds.

ACuriousMind

10:18 PM

Oh well, I guess I just found my first question for my prof in the course :)

Danu

:)

Yeah... Let's just forget about those singular spaces. Can we go back to sheaf cohomology? :D

I think that, if you guys cared about shit like Noetherian spaces etc in the previous course, the singularities mean things like cusps and stuff in curves---that's the kind of things algebraic geometers study, right?

ACuriousMind

@Danu No no no, we didn't care about Noetherian spaces, that's from my alg. geo. course

We didn't do any kind of vanishing in the sheaf cohomology course at all

We spent most of our time constructing weird categories :D

Danu

Ah, okay.

ACuriousMind

And I think the ultimate goal is Verdier duality and perverse sheaves

Danu

Perverse sheaves... I've heard it said many times that Grothendieck would be furious if asked about this terminology.

I really admire the guy's passion for proper names and definitions. That's some under-valued stuff, and it can be so helpful.

ACuriousMind

10:24 PM

Looking over my notes, I see we really only restricted to locally compact spaces. Which pretty much kills alg. geo. applications, as these spaces are Hausdorff :D

Danu

I approve.

If it's not Hausdorff it doesn't exist (is there any known counterexample to this?)

Exist, here, in the physical sense.

I remember when Leeb tried to argue that non-Hausdorff spaces can be natural too.

The line with two origins... So natural :3

ACuriousMind

Haha

Danu

Heading to bed for now. Bye!