Mathematics - 2017-06-14 (page 3 of 9)

s.harp

1:13 PM

Are weakly contractible compact spaces contractible?

Balarka Sen

The closed topologists' sine curve is weakly contractible, right? But not contractible at all.

s.harp

Ah thanks! I had just asked this as a question I'll delete it :)

Its funny that if every compact subspace of a space is contractible the space is weakly contractible

something I realised today, I was wondering if the converse also held: the above would have been equivalent to the converse

Balarka Sen

Ah I see

s.harp

is it easy that all higher homotopy groups of the sign curve are zero?

Semiclassical

hi chat

@s.harp sign curve? :)

s.harp

1:24 PM

(the above statement should have been "is contractible in the space", not that hte compact subspaces themselves should be contractible D: )

haha

Simply Beautiful Art

Mmm.... anyone have any idea what $\omega^{CK}_{\omega_1+1}$ is? Where $\omega^{CK}_1$ is the Church-Kleen ordinal

Semiclassical

Can any 1D curve have nontrivial higher homotopy group? I imagine the answer is yes but it seems bizarre.

Balarka Sen

@s.harp I don't think it's as obvious as it looks but I have a proof-sketch. Give me a second.

Daminark

Hai everyone

Balarka Sen

@Semiclassical Well, 2D surfaces can have nontrivial 3D homotopy groups, eg pi_3(S^2) :D

s.harp

1:26 PM

its a continuous curve together with an interval @semiclassical , but you have continuous curves that surject onto $S^2$ sooo

Semiclassical

hmm

Fair enough.

But it seems like you'd still eventually become trivial.

oh, wait.

s.harp

its certainly something I would believe!

Balarka Sen

pi_*(S^2) is eventually Z/2 I think

not trivial

s.harp

but in mathematics you can do more than belief

Semiclassical

yeah.

nah, I'm being silly. the higher homotopy groups of S^2 don't become nice, for instance.

Balarka Sen

1:29 PM

@s.harp Say $W$ is the curve (W for Warsaw circle, also a name for the thing). Let $f : S^n \to W$ be a map; if image of $f$ is disjoint from the bit of the y-axis in $W$ on which nearby things accumulate - aka the bit of $W$ where $W$ is not locally path connected - then $f$ is nullhomotopic, because complement of the y-axis in $W$ is an arc $(-1, 1)$.

Semiclassical

I mean, one has $\pi_1=0$ but that's about the only trivial case.

Balarka Sen

So let $f(0) = w$, where $w \in W$ is on the bad bit, the y-axis.

$0$ some point I marked on $S^n$.

s.harp

oh yeah, if it leaves that bit then you can walk out with a path in $S^n$ right?

Semiclassical

and more generally $\pi_k(S^n)=0$ if $k<n$ and $\pi_n(S^n)=\mathbb{Z}$ but is arbitrarily complicated for $n>k$.

Faust7

Good morning

Balarka Sen

1:32 PM

If $p, q$ are points on $S^n$ in a neighborhood of $0$ such that $f(p)$ is on "the left" of $w$ and $f(q)$ is on "the right of $w$, then $f$ can't possibly be continuous at $0$; choose a path from $p$ to $q$ which goes through $0$ and you have a path $[0, 1] " =[p, q]" \to S^n$ which lies inside a neighborhood of the bad y-axis, and is on both "half" of it. That just breaks path-connectedness of the topologists's sine curve.

So $f$ has to miss some small topologists' sine curve subspace $T \subset W$, containing the $y$-axis. But $W - T$ is just... homeomorphic to $[0, 1)$. Contract. $f$ is nullhomotopic.

Modulo details this is how the proof should be like.

I meant $[0, 1] "= [p, q]" \to W$ above, typo.

s.harp

it appears you were thinking about a different space than me, i was just looking at $\{0\}\times[-1,1]\cup\{(x,\sin(1/x)\mid x>0\}$

Balarka Sen

That is contractible.

s.harp

really?

yep

nope

hm

Balarka Sen

The closed topologists's sine curve is that "closed up", so you add an arc from $0 \times [-1, 1]$ to the end of the sine curve

Secret

[Random] What maths is the most abstract, is category theory already the supremum of abstractness?

Semiclassical

1:37 PM

heh. yep -> nope -> hm

Balarka Sen

Semiclassical

a nice encapsulation of what research feels like at times :)

s.harp

@Balarka yeah I know that guy too, sometimes they also have a sin on the other side

Faust7

some kind of messed up homeoclinical loop?

Balarka Sen

Yeah I don't know what that one is called, @s.harp

It's easier to prove weak contractibility if you have the sine curve on both sides I guess. I am pretty sure that one is not contractible either.

s.harp

1:41 PM

@Secret I think things I havent understood are super abstract, like algebraic QFT or anything where people do QFT in a mathematically sensible framework suddenly involves all knowledge about algebraic geometry, d-modules, infinite dimensional super-lie-algebras etc that has ever been compiled

@Balarka its not path connected so it cant be contractible

Semiclassical

QFT is in general just plain weird

People talk about the unreasonable effectiveness of math, but for me I focus on the unreasonable effectiveness of QFT.

Balarka Sen

@s.harp Oh right, that closed double topologist's sine curve is not even path-connected.

Semiclassical

How can something that confusing work so well? :P

Faust7

wth is QFT

Semiclassical

quantum field theory

Faust7

1:42 PM

oh god

Semiclassical

classical field theory would be stuff like electrodynamics

Faust7

Non-sense qith a side of Quarky

Semiclassical

quantum field theory is stuff like quantum electrodynamics

Faust7

yeah i was a physics student

s.harp

there is a book "Towards the Mathematics of QFT" by Paugam, there is a chapter called "Functorial Analysis" and in the introduction to n-categories he says "the student interested in learning regular category theory may simply set $n=1$ in the following"

Semiclassical

1:43 PM

Eh, if it didn't work so damn well it'd be nonsense.

Faust7

before i was spirited away by the math dept

s.harp

almost everything in the earlier chapters of that book is so incomprehensible to me

Faust7

lol

Semiclassical

That abstract nonsense approach to QFT is hard for me to appreciate at all.

s.harp

some parts of the later chapters degenerate into the usual physics blah blah though unfortunately

Semiclassical

1:45 PM

eh, I can appreciate the usual physics blah blah blah

if only because it gives you actual calculations to do.

Faust7

love the physics blah blah blah abrviation

Semiclassical

Better a somewhat sloppy thing which I can understand enough to test than a 'mathematically respectable' formulation which is so abstract as to be inaccessible.

s.harp

I prefer a theory where I know what the things I'm looking at are

Faust7

i dunno physics at least sort of makes sense to me sometimes

s.harp

which is impossible in the usual physics way of doing it

Faust7

1:47 PM

except dif geo that is pure unadulterated non-sense

Semiclassical

Nah, diff-geo is good stuff. (I don't know it well, but I can recognize that it's well-founded and respectable.)

s.harp

I understand that doing physics in the way physicists do is real science in a way mathematics is not, but my personal perspective is that doing it that way is horribly unfufilling

Semiclassical

n-category stuff, on the other hand...idfk

It's a bit of a counterpoint, but when I see stuff like that I tend to remember a certain article I read

Faust7

Your a smarter man than i if you can understand dif geo lol

Semiclassical

Basically, there's a certain author out there who wrote a book about topos theory applied to music theory.

Which is pretty absurd sounding already.

There's a counterpoint article out there, and a response to the counterpoint.

Faust7

1:50 PM

lol

Semiclassical

which you can find here: encyclospace.org/special/answer_to_tymoczko.pdf

Take a look, if you would, at the first page.

The comparison they make: prob/stats is to topos theory as counting the members of a city is to understanding Mozart's music

Secret

I do slightly better on concepts that looks like linear algebra, and more recently abstract algebra as well.

Interestingly, things I don't understand well, such as real analysis, is actually less abstract than abstract algebra, because well, how often you can draw an algebraic structure on paper...?

is it satire?

Noooope

hmm

1:53 PM

because the first page so unpleasantly pretentious

Faust7

what is algerbraic geometry

Semiclassical

lolyes

I mean, seriously

Steve DL

Hi everyone. Graph theory question. I know there is a paper out there that discusses metric equivalence between temporal and non-temporal graphs and states explicitly there is no equivalence between the reachability of non-temporal graphs and the reachability of their temporal counterparts (i.e. non-temporal reachabiltiy isn't a viable proxy / estimation of temporal reachability). But I can't find the paper any more and Google is not helpful. Does anyone happen to know/remember it?

Semiclassical

One can make a defense of topos theory in music, to the effect of: "While it's hardly the obvious thing to do, maybe one really needs these mathematical tools to appreciate a structure as complex as Mozart's music."

But the comparison to the statistical example is so absurd as to undermine the very argument.

s.harp

I'd rather say: because of the abstraction of topos theory it is not an unnatural position that using it can give you non-trivial results not accessible before

Semiclassical

1:56 PM

Right.

My point of contrast, though, is that statistics is (to some extent) quite literally the subject of counting large populations and coming to conclusions about those populations.

Faust7

i dunno stats never seem to be all that correct

(no offence)

look at any election polls

Semiclassical

Well

s.harp

offence taken

you are mistaking the mathematical theory of statstics with how people apply it

Semiclassical

Right.

s.harp

in election polls you are not sampling in an unbiased way

Faust7

1:59 PM

ic

i dunno i worked for a large company for quite awhile

Semiclassical

Plus in election polls there's a hell of a lot of demographic extrapolation one has to make.

Steve DL

(Found what I was looking for: cl.cam.ac.uk/~cm542/phds/johntang.pdf)

s.harp

when you make a poll you are sampling people from some distribution $P$ and then getting the means for some set of random variables

Semiclassical

Not just "how many voters were in favor of X" but how representative the sample is of the general population, and how likely voters in various demographics are to actually vote.

Faust7

the company would conveniently manipulate stats to make the numbers they want to improve better and numbers they didnt want to improve worse etc constantly i guess you could say it let a really bad taste in my mouth about statistical data

Semiclassical

2:01 PM

I think it's fair to say, though, that there's a lot of bad use of stats.

s.harp

the election will ask everybody in some set $X$ for their choice, the distribution $P$ from which you have sampled can be very far away from an approximator to what people in $X$ say

Semiclassical

All the same, for all the flak election polls have gotten lately

Faust7

i understand the difficulties of trying to reach a correct conclusion based on a finite sampling of a total population

Semiclassical

While there have been some high-profile failures, those have all tended to be polling where the outcome was within the margin of error

Faust7

in theory stats is something one could apply usefully humans are alittle too fickle

s.harp

2:02 PM

@Faust7 its worse than finite sampling, its BIASED sampling

where you dont really know what your bias is

Faust7

hence the bad taste ^^

Semiclassical

That's really a modeling problem, though.

Anyways, my point in bringing this up

was that I'm never sure where stuff like category theory falls in relation to QFT

With some people, it seems like they really want to make the argument that it's indeed 'obvious' that tools like that are needed to understand QFT.

Secret

2

Q: Can a finite state machine contain an unreachable state?

If a finite state machine is formally defined with a set of states $\{q_0, q_1, q_2\}$, where $q_0$ is the start state, is it a valid machine if there is no path from $q_0$ to $q_1$ and no path from $q_2$ to $q_1$? In other words, there is no way to reach $q_1$ at all, but it is still considered ...

finite-automata

Semiclassical

But I find that really hard to believe, given how abstract that is in relation to what a practicing physicist actually does.

I've basically given up on trying to climb Mount QFT, though.

Faust7

do you know what a c*-algerbras is?

Semiclassical

2:07 PM

Something something Banach space?

I've tried to read on that stuff in the past but never got far with it

Faust7

sums up myself

found an article i wanted to read about it

but a couple things r alittle beyond my understanding

Semiclassical

John Baez has some discussion of c*-algebras in physics here: math.ucr.edu/home/baez/cstar.html

Faust7

hmm thanks ill check it out

hmm very good explanation of why one should learn what it is

Sometimes i feel like i understand so little O.o

s.harp

the book by Bratteli & Robinson: Operator Algebras and Quantum Statistical Mechanics is nice

so is Murphy's C^* algebras

Semiclassical

This seems like a good article for c*-algebras in relation to numerical analysis, which is where I've brushed up against them myself

maths.tcd.ie/pub/ims/bull45/S4502.pdf

Faust7

2:19 PM

i think its pre requisite knowledge that was myproblem

s.harp

if you want C^* algebras for QFT then Local Quantum Physics by Haag, maybe also this funky little paper: arxiv.org/pdf/1307.4458.pdf

what do you want to learn about C* algebras

Faust7

the problem with coming into this chat is i collect pdf's of intrestings things at a rate much faster then i can read them all

(thanks though) :p

Semiclassical

hah, yeah

Faust7

math.uvic.ca/faculty/putnam/ln/Smale_spaces.pdf

this article was my most recent attempt at C* algebras

w8 no wrong one

math.uvic.ca/faculty/putnam/ln/C*-algebras.pdf

that one lol

math.uvic.ca/faculty/putnam/ln/C*-algebras.pdf

hmm doesnt want to link

Semiclassical

yeah, it's breaking it for some reason

s.harp

2:24 PM

[dfa sdf ] (http:ddd)

without spaces

Faust7

[math.uvic.ca/faculty/putnam/ln/C*-algebras.pdf]

think itstoo long of a url

http://www.math.uvic.ca/faculty/

putnam/ln/C*-algebras.pdf

s.harp

this link

Faust7

lol

s.harp

did it the wrong way around

Faust7

how'd u do that?

s.harp

2:25 PM

[text] (http:link)

without the spaces

Faust7

gotcha thank you ^^

link

yeah i got it =)

Semiclassical

hi @baymax

Was it you who was doing something with a matrix of the form $A_{jk}=\int_0^1 x^{j+k}\,dx$?

BAYMAX

yups

@semi

Semiclassical

How'd that go?

BAYMAX

not yet though1

Semiclassical

2:35 PM

mmkay

I remembered enough of the orthogonal polynomials stuff to explain how that works, if you're interested.

BAYMAX

yeah,sure can we do it exactly an hour later...sorry to me if you will not be available then?

Semiclassical

sure, that's fine.

BAYMAX

thanks I appreciate it :)

Semiclassical

For convenience, though, let's create a room in advance

BAYMAX

ha ha

sure

Simply Beautiful Art

3:11 PM

@AkivaWeinberger You are familiar with the Church-Kleene ordinal?

Akiva Weinberger

@SimplyBeautifulArt I am.

Secret

This is the shape I saw in last night dream (roughly)

Akiva Weinberger

There's no way to give every ordinal smaller than it a name such that a) it's decidable whether a given word is the name of an ordinal and b) it's decidable whether one word represents a smaller ordinal than another word.

And it's the smallest ordinal with that property.

Secret

13 hours ago, by Secret

Last night dream: I was at a desktop computer in the middle of the night running some kind of 3D/4D modelling. Akiva was nearby as a looped shape with a spherical envelop is displayed on the screen. He mention how he observed there are 3 looped geometries within. Meanwhile as I take a closer look I realised that that's no loop, those are (simple) knots. The geometry shape is also indexed by the ordinal $\omega^{\omega^{\omega^{\omega^{\omega}}}}$, visible faintly rotating with the geometric obje

Simply Beautiful Art

@AkivaWeinberger What would you think of $\omega_\alpha^{\mathrm{CK}}$?

Akiva Weinberger

3:23 PM

????

@Secret

Lol

Your dreams are weird

@SimplyBeautifulArt Oh, I don't actually know how those are defined

Remind me?

Simply Beautiful Art

Admissible ordinal

In set theory, an ordinal number α is an admissible ordinal if Lα is an admissible set (that is, a transitive model of Kripke–Platek set theory); in other words, α is admissible when α is a limit ordinal and Lα⊧Σ0-collection. The first two admissible ordinals are ω and ω 1 C K {\displaystyle \omega _{1}^{\mathrm {CK} }} (the least non-recursive ordinal, also called the Church–Kleene ordinal). Any regular uncountable cardinal...

Paweł Kusz

Hi :).I am looking for an example a non abelian group which have infinitely many abelian subgroup :)

Akiva Weinberger

@PawełKusz How about the symmetric group on infinitely many elements? $S_\infty$

The subgroups $\langle(1~2~\dotsb~n)\rangle$ are abelian

Simply Beautiful Art

@AkivaWeinberger I'm trying to make sense of $\omega^{\mathrm{CK}}_{\omega_1 +1}$

Paweł Kusz

@AkivaWeinberger You mean permutation group ?

Ted Shifrin

3:31 PM

Hi, DogAteMy.

Hi @Ted

Hi @Alessandro

@PawełKusz Yeah

Where are you know?

The permutation group on the positive integers

Semiclassical

3:33 PM

hi-yo

Ted Shifrin

La Spezia ... Cinque Terre tomorrow.

Hi, Semiclassic.

Alessandro Codenotti

It's a beautiful place, my grandfather was born there

Ted Shifrin

Wow ...

Akiva Weinberger

I suppose "$S_\infty$" is ambiguous (it could mean two things). It could mean the set of all permutations on those, or it could mean only the set of finite permutations (the ones made of finitely many cycles of finite size).

Paweł Kusz

@AkivaWeinberger I understand, thank you :)

Akiva Weinberger

3:34 PM

It doesn't really matter, though. They both work.

You're welcome

Dodsy

Not Ted again

arghhhhhh

Alessandro Codenotti

Where else will you go in Italy? @Ted

SteamyRoot

@AkivaWeinberger For me, $S_\infty$ means a group has isogredience spectrum $\{\infty\}$ :P

@PawełKusz The free group of countably many generators works rather trivially, too.

Akiva Weinberger

@SteamyRoot The what???

@SteamyRoot Same for the free group on two generators, really.

$\langle ab^n\rangle$, for example

$\langle\rm anything\rangle$, really

SteamyRoot

Well, of course. It's just a nice example where both properties are incredibly trivial to see.

@AkivaWeinberger Ummm, well... For $[\varphi] \in \operatorname{Out}(G)$, the isogredience number of $[\varphi]$ is the number of equivalence classes given by the relation $\forall \varphi_1,\varphi_2 \in [\varphi]: \varphi_1 \sim \varphi_2 \iff \exists \iota \in \operatorname{Inn}(G): \iota \circ \varphi_1 = \varphi_2 \circ \iota$

Jeff

3:48 PM

Hi everyone. Anyone know is there a common name for this identity? $\sum_{k=0}^n C(n,k) =2^n$ (I know it comes from the binomial theorem, but I wonder if there's a specific name for it).

SteamyRoot

And the spectrum is the set of all those numbers.

Akiva Weinberger

Remind me what ${\rm Out}(G)$ is?

SteamyRoot

$\operatorname{Aut(G)} / \operatorname{Inn}(G)$

Semiclassical

@Jeff I'm not sure either. But another way in which that comes about is as the row sums of Pascal's triangle.

Akiva Weinberger

@SteamyRoot Where does this come up?

Jeff

3:51 PM

@Semiclassical Yes. In fact, the class notes I'm writing develop that identity with another method: summing up all the possible ways of flipping $n$ coins and comparing it to the number of $n$-character binary strings.

Semiclassical

(that each row is twice the previous one is evident from how each row is constructed from the last one)

Yeah, that's the common counting proof

Jeff

@Semiclassical Oh yeah... that one, too.

@Semiclassical Isn't Pascal's triangle fascinating?

Semiclassical

it's pretty neat.

TheGreatDuck

@Jeff Can you inscribe it within a finite space?

Ted Shifrin

@Alessandro: Moltrasio, Trieste briefly, then on to Croatia.

TheGreatDuck

3:54 PM

@TedShifrin vacation?

Semiclassical

I'm sorta bothered by the fact that I can't remember a specific name for that identity, though

Ted Shifrin

Yup

Jeff

@TheGreatDuck I'm not sure what that is. I'm teaching a 200 level course (juniors).

Semiclassical

I mean, as a consequence of the binomial theorem it's obvious

but hmm

TheGreatDuck

@Jeff can you put the entire pascal triangle in a finite space?

Ted Shifrin

3:55 PM

No name that I've ever seen, @Jeff.

Jeff

@TheGreatDuck topology? I haven't taken topology.

Semiclassical

As a name, I guess I'd call it the "row-sum property of Pascal's triangle" (stole that from somewhere else)

That's awkward, but it's what I got.

TheGreatDuck

@Jeff lol no silly. I'm just asking whether or not you write all of the rows of pascal's triangle on a finite piece of paper.

you're overthinking it. XD

Akiva Weinberger

It also comes from $(1+1)^n$

SteamyRoot

@AkivaWeinberger In covering theory and fixed-point theory. It connects self-maps and lifts of self-maps on spaces with the induced automorphisms on the fundamental group.

Semiclassical

3:57 PM

yes, that's what we mean by binomial :P

Akiva Weinberger

@TheGreatDuck No. The answer is no.

Jeff

@TheGreatDuck Yes, I can.

Semiclassical

Given that Pascal's triangle is infinite...

TheGreatDuck

@Jeff really? How many rows are there in pascal's triangle

Semiclassical