Mathematics - 2015-06-27 (page 1 of 2)

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12:02 AM

@evinda hello evinda! I am fine. Just very tired. But weekend is coming so it is fine. :- ) I hope you are doing well too.

what is hott vs in programming?

12:39 AM

guys, I have question. I noticed that practically everything in any math formula are functions derived of the sum, what make sense, to the "limit" of exponential functions. What Im trying to say is that everything can be write in function of sums, mulplications, divisions, powers, roots, logs and exponential by general. In many cases you need expressions that comes from integrals or series as harmonic numbers, dilogarithms, gamma functions and so on

so I was reading these days about differential equations and the very hard task that "write" solutions in the majority of cases... so I thought that maybe a good strategy to define new functions based on integrals or series for these kind of solutions that generally you only can see by numerical methods

Soham Chowdhury

1:04 AM

@Masacroso no, isomorphismes was asking about what the word "type" means in HoTT and how that meaning differs from the meaning in programming.

1:35 AM

@isomorphismes Type theory. It is the same idea

Soham Chowdhury

@Balarka, @Ted, everyone else: thanks for suggesting that I get comfortable with metric spaces before hitting Munkres/whatever.

It's helped a bunch.

i have not seen you talk much algebra @SohamChowdhury

(lately)

Hello @AlexClark

@Masacroso You may find the following perspective on Painleve equations intriguing (link).

the first half, anyways. the second half is more a review of the historical literature arising from Painleve's work.

2:04 AM

Hello!! Is someone of you familiar with computability and especially with reduction?

1 hour later…

Soham Chowdhury

3:05 AM

@PaulPlummer Haha. I actually read half of Artin's groups chapter yesterday, because the prof's asked me to. I'm more concerned with finishing all the point-set I need once and for all right now. Algebra is a pleasure. ;)

I saw this brilliant talk yesterday.

That guy is so animated. It's a joy to watch. He talks about Euler characteristic and stuff.

So I'm going to be busy with topology for a while, @Paul.

3:24 AM

@SohamChowdhury Ah Farb, he is a geometric group theorist! (well at least he does quite a bit of work in the area)

Soham Chowdhury

Yes, I saw that. :)

I will give it a watch later today

Soham Chowdhury

Yeah.

You'll probably love it too.

3:37 AM

@SohamChowdhury lol, still with a chalk and blackboard ?

What do you mean "still"?

Should he be projecting his thoughts as some hologram?

3:51 AM

not a hologram lol, maybe a datashow ?

maybe is it government school ?

which supports itself by donations and taxes ?

4:32 AM

What is a datashow? @Agawa001

1 hour later…

5:40 AM

@SohamChowdhury A very good talk

Ughhh, It looks like tomorrow will be 106F (41C) and 109F (43C) the day after

where the hell are you

Apparently Hell... (Idaho)

I am hoping it is just google spying on me and trying to make me feel good for staying inside all day

Soham Chowdhury

5:56 AM

@Agawa001 blackboard > whiteboard / projector / etc. in many people's opinions, me included ;)

@SohamChowdhury I think @Agawa001 is a time traveller and he was just surprised that in 2015 people still used blackboards

6:25 AM

@Semiclassical hey, thank you very much @Semiclassical! This was about I was thinking. See you later.

6:45 AM

@SohamChowdhury "helped"? You mean it will help a bunch, or did you just finish metric spaces?

iwriteonbananas

hi balarka

hi

6:59 AM

hi

did you figure out the whole hopf fibration business, @Huy?

I understand it a bit better, but I'll definitely look at it again next week. I usually don't succeed if I just sit and try to understand it for hours but rather if I look at it repeatedly and get some more out of it each time.

true, that happens to me too.

Chris's sis the artist

News!!!

I found a very brilliant way of calculating $$\sum_{n=1}^{\infty} \frac{H_n^2}{n^4}$$

Did you know this integral has a very nice closed form? $$\int_0^1 \frac{\text{Li}_2(x) \text{Li}_2(1-x) \log (x)}{1-x} \, dx$$

iwriteonbananas

@BalarkaSen Let $X$ be the quotient space of $S^2$ by identifying $x\sim -x$ on the equator.

does this have the same homology groups as $S^2$?

7:14 AM

I forgot, let me check.

I forgot what a homology group is but I think the answer is no.

Why don't you just do cellular homology?

It has a single 0-cell, a single 1-cell and two 2-cells

the 2-cells are attached via degree 2 maps

seems like there should be no problem.

iwriteonbananas

ehh right

dumb question, but why does the attaching map have degree 2 again?

attaching a $2$-cell at the bd of a disk with $f(z) = z^2$ identifies $x$ with $-x$.

Chris's sis the artist

@iwriteonbananas @BalarkaSen @Huy One question: do you laugh while doing mathematics? I laugh a lot, have a huge amount of fun, because otherwise I wouldn't do mathematics. Mathematics is about having fun, never forget that. :-)

iwriteonbananas

7:18 AM

right

i am not checking whether what you say is true or not though

too lazy.

iwriteonbananas

np lol

would appreciate if Chris'ssis would stop advising everyone on how to do mathematics and what to do while doing mathematics.

Chris's sis the artist

@BalarkaSen You're too serious, really. I don't imagine you ever laughing while doing math, but I might be wrong. :-)

iwriteonbananas

@Chris'ssistheartist sighs

7:25 AM

@Chris'ssistheartist: I don't laugh too much, but do have fun.

Chris's sis the artist

@Huy That's great.

Or this one

$$\int_0^1 \frac{\text{Li}_2(1-x) \text{Li}_2(x) \log (1-x)}{x} \, dx$$

@BalarkaSen You should find this interesting, from the video Soham posted, its a sketch of a proof that if a manifold has a smooth vector field without zero vectors, then it has Euler characteristic 0

here

I'm sure Balarka already knew that?

ah? i didn't know of the result.

o.o

7:28 AM

@Huy no, why do you think I already knew that?

The proof is very cool, by Thurston apparently (at least the one given in the talk)

That's an if and only if.

I know, but he does not provide the other direction

although it is obvious for the torus

the other direction is not hard.

ps : did you not mean 2-manifolds?

No

I didn't mean a 2-manifold

7:30 AM

every 3-manifold has euler char 0. I don't think it's true that it admits a smooth vector field

at least, I'd be very surprised if it does

Sorry about that.

Sorry about what, that he doesn't think it is true?

I'm going to take that as a "you're wrong".

yes :)

@MikeMiller that's a pretty fascinating fact.

7:33 AM

I really liked his jab at people who study a bunch of machinery but end up not being able to prove anything

@BalarkaSen: Just assumed you'd know such a result considering how much you know about alg top etc.

Every orientable 3-manifold is parallelizable, which is of course much better than having a single nonvanishing smooth vector field.

The correct statement above was "an orientable manifold carries a nonvanishing smooth vector field if and only if it has trivial Euler characteristic". I don't know if this extends to non-orientable things. Probably not.

The statement that having a nonvanishing vector field gives $\chi = 0$ doesn't depend on orientability.

@Huy I don't think it's purely an algebraic topology fact. In any case, I am only familiar with basic algebraic topology, or only a portion of it.

@PaulPlummer Not everyone is Bill Thurston :P

You can use the nonvanishing vector field to build a fixed-point-free map

@BalarkaSen: Yeah, it's not and I didn't mean to offend you by any means if you think that.

7:42 AM

@MikeMiller interesting, but I dunno what parallelizable means.

no, no, @Huy. I wasn't offended at all.

Trivial tangent bundle.

Oh yah definitely. The more I learn, or hear about it it does seem like the statement "he is the most underrated mathematician" becomes more and more true. Did you watch it @BalarkaSen

oh, ok.

no, I didn't watch it. my internet is too swingy at the moment.

I will, though.

He's underrated? I think plenty of people rightfully think he's one of the most important mathematicians of the later 20th century :)

I think that is the point of the statement, even then he is underrated (I have seen this multiple times in different sources) @MikeMiller

7:48 AM

@MikeMiller ok, sure. but I am not seeing what that will accomplish.

Please be more specific, I'm on my phone so don't see where arrows are pointing.

Random Variable

Anyone familiar with the Schwarz reflection principle?

you wanted to construct fixed point free maps using that nonzero vector field.

do you have a specific reason to do so?

7 mins ago, by Mike Miller

You can use the nonvanishing vector field to build a fixed-point-free map

@PaulPlummer: Fair enough. Maybe my view of what other people think is biased by my own thoughts; I find myself influenced by much of Thurston's mathematics.

@Balarka Lefschetz...

7:53 AM

ah, I see what you mean. that should give a proof of the "if" version.

The other direction is hard.

I can believe that. Even trusting that "only if" direction seems to be hard.

I can't parse that sentence, could you rephrase?

Soham Chowdhury

@BalarkaSen I've played a bit with metric spaces in the past, so now working with Munkres, it's easy to see the motivation for why topologies are defined this way.

Save it for the video :) @BalarkaSen

@SohamChowdhury You are spoiling the plot!

Soham Chowdhury

7:56 AM

Thurston's proof almost hit me as hard as diagonalization and Euler's (underhanded :P) proof of $\zeta(2) = \frac{\pi^2}6$.

@MikeMiller I mean, I don't think someone with a meager knowledge of topology would find that "only if" statement easy to believe.

(i.e., like me).

what electrons, @Soham?

I am not sure how obviuos it is suppose to be

I don't remember which arrow goes where :) Which is the only if direction? $\chi(M)=0 \implies M$ has a nonvanishing vector field?

Soham Chowdhury

@BalarkaSen watch the video, you'll see. ;)

@MikeMiller yes.

I don't want to have electrons going in and out of the mathematics.

Soham Chowdhury

7:58 AM

@BalarkaSen I knew it!

I agree, it's surprising. (Remember I'm only willing to state it for orientable manifolds.) It's an application of obstruction theory.

Soham Chowdhury

They just help with thinking about the vector field, I guess.

in any case, Mike's hint almost makes the proof a tautology. should have figured that one out :(

interesting, @MikeMiller.

Lefschetz is far from a tautology.

Chris's sis the artist

Do you know by any chance if it's a know fact that all the series $$\sum_{n=1}^{\infty} \frac{H_n^2}{n^m}, \ m\ge 2$$ can be expressed in terms of zeta function? I can prove that, I have a way to show it, but not sure if it's a known fact. I also work on a solution to the general case - that is pretty complicated as I can see so far.

8:02 AM

true, especially for me who hasn't yet read the proof (which uses simplicial approximation)

but it's an easy corollary.

There are plenty of cases where physics has an immensely valuable impact on mathematics. It's not something to sniff one's nose at. (Not that I know any, I just respect the influence it has on my own field.)

Chris's sis the artist

@RandomVariable I mean in terms of zeta function only.

Lefschetz is pleasant. (You should be careful - how does one go from a nonvanishing vector field to a fixed point free map?)

It does seem like physics provides some interesting ways to look at some (many...) questions and problems

@MikeMiller I have to sniff my noses at something, now that I have decided to stop doing that with mathematics!

/joke

Soham Chowdhury

8:06 AM

@BalarkaSen noses? hmm, not sure I want to meet you in person :P

Is sniff's one nose at the right phrase?

Random Variable

@Chris'ssistheartist My question is about how it applies to the dilogarithm. I seem to be misunderstanding something.

I think I typo'd snift. But I'm not sure if that's a thing.

I feel like I've embarrassed the math stack, being on the front page..

I think it is turned or maybe flipped ones nose

Random Variable

8:09 AM

@Chris'ssistheartist Sorry. I thought you were talking about the question I asked earlier.

Chris's sis the artist

@RandomVariable I didn't study the matter enough to provide with a proper opinion on what you asked. I'll check some papers I might have on that.

@BalarkaSen: I might suggest pop culture; it's a popular choice for condescension, and often not unwarranted.

cool, I'll try that, haha.

Random Variable

@Chris'ssistheartist Can it just be expressed in terms of zeta values for $m=3$? I don't remember off the top of my head.

Chris's sis the artist

@RandomVariable Yeah, for all values of $m$ according to a proof of mine.

@RandomVariable Actually, the variant with $m=3$ is going to be the next question I'll propose in a math magazine (well, I also thought of making it an article, but no) after another question of mine will be published in that magazine.

@RandomVariable it's $7/2 \zeta(5)-\zeta(2)\zeta(3)$. The interesting thing is that one can get an elementary solution by only using classical tools of analysis.

Random Variable

8:23 AM

@Chris'ssistheartist Let me check if that's one I've done.

Chris's sis the artist

@RandomVariable OK

It also appears in the famous paper by Flajolet that treats these series.

Random Variable

@Chris'ssistheartist It doesn't look like I have. Strange.

Chris's sis the artist

@RandomVariable It's posted also here on MSE somewhere, I don't remember where now, but i might try to find it.

Random Variable

@Chris'ssistheartist I have $\sum_{n=1}^{\infty} \frac{H_{n}^{(2)}}{n^3}$.

Chris's sis the artist

42

Q: How to prove that $\sum_{n=1}^{\infty}\frac{(H_n)^2}{n^3}=\frac{7}{2}\zeta(5)-\zeta(2)\zeta(3)$

Prove that $$\sum_{n=1}^{\infty}\frac{(H_n)^2}{n^3}=\frac{7}{2}\zeta(5)-\zeta(2)\zeta(3)$$ $H_n$ denotes the harmonic numbers.

real-analysis sequences-and-series complex-analysis analytic-number-theory riemann-zeta

@RandomVariable And today I also derived $$\sum_{n=1}^{\infty} \frac{H_n^2}{n^4}$$ Did you try this version?

$$\sum_{n=1}^{\infty} \frac{H_n^2}{n^4}=\frac{97}{24} \zeta (6)-2 \zeta^2 (3).$$

Random Variable

8:40 AM

@Chris'ssistheartist It seems like I've only done the alternating version with $m=1$ and the regular and alternating version with $m=2$. I really thought I did the $m=3$ case. But I don't see it in my notebook.

Chris's sis the artist

@RandomVariable Well, I didn't attend the alternating series with $m=1$, but I should try that. Or better check my paper first to make sure.

Random Variable

@DanielFischer Is it too early in the morning for a question about Schwarz reflection principle?

@RandomVariable Don't know, let's try.

Chris's sis the artist

@RandomVariable I'm calculating it now.

@DanielFischer: In the proof that for every compact $m$-manifold $M$ that is Hausdorff, there is an integer $k \in \mathbb{N}$ and an embedding $f: M \to \mathbb{R}^k$, we explicitly create such an embedding, namely we cover $M$ with finitely many coordinate charts $\phi_i: U_i \to \mathbb{R}^m$ and use a partition of unity $\{\theta_\alpha\}_\alpha$ subordinate to that cover to define
$$f(p) := \begin{pmatrix} \theta_1(p)\\ \theta_1(p)\phi_1(p)\\ \vdots \\ \theta_\ell(p)\\ \theta_\ell(p) \phi_\ell(p) \end{pmatrix}.$$ I can see that $f$ is an embedding as claimed, but what exactly does $f$ …

Soham Chowdhury

8:52 AM

So this video is a nice proof of V - E + F = 2. It's the same one that Armstrong has, but animated, of course.

Euler's Formula and Graph Duality

@SohamChowdhury Just take the 1-skeleton of your polyhedra, and delete pieces of the graph systematically

It's a not-so-hard-to-do proof.

Soham Chowdhury

Ah, I know the proof.

Random Variable

@DanielFischer Knowing that the dilogarithm is real-valued on the real axis for $x \le 1$, I assumed that the principle would only apply if you reflected over that portion of the real axis. But it seems to apply to the entire real axis.

But you should ponder on it. It's a topological property, not just a combinatorial one.

Soham Chowdhury

I know.

The video I saw yesterday taught me that.

8:54 AM

No, you don't.

Soham Chowdhury

Well, $V - E + F = 2 - 2g$ seems quite topological.

It's a low-brow way to state that the sphere has cyclic homology at $0$th and $2$nd homology and vanishing homology for higher dimensions.

Soham Chowdhury

I can't prove it, yes.

@SohamChowdhury Yes, that's the corresponding statement for homology of $\Sigma_g$.

Soham Chowdhury

@BalarkaSen Haha, "kids like big words", don't you think? ;)

8:56 AM

@RandomVariable Identity theorem. If $U$ is connected, symmetric with respect to the real axis, and $U\cap \mathbb{R}\neq \varnothing$, you look at $f$ and $g\colon z \mapsto \overline{f(\overline{z})}$. If $f$ and $g$ coincide on any nonempty interval, then $f\equiv g$.

@SohamChowdhury You can prove that the sphere has $H_i = \Bbb Z$ for $i = 0, 2$ and $=0$ otherwise?!

Soham Chowdhury

Anyway, I think I understand why it's not just combinatorial. It'll take me (quite) a while to learn about homology. :P

@BalarkaSen "can't"

@SohamChowdhury It's not "big words". It's an actual translation to something more powerful.

My glasses are bent because I put them in my case for my sunglasses and now I'm getting a headache after reading for a few minutes because the whole world is bent in front of me.

:(

@Huy You have local embeddings. You want to glue them together so that they don't disturb each other. So you put each local embedding into a different space, and take the product.

8:57 AM

@SohamChowdhury oh, I misread. yes, I know you can't prove it, but I am asking you to give it a bit more thought.

Soham Chowdhury

I know, and I'm being defensive because the only thing I know about homology groups is that they talk about n-dimensional holes (in some sense, don't kill me). :P

@BalarkaSen oh, of course I will :)

Chris's sis the artist

@RandomVariable I got $$\sum_{n=1}^{\infty} (-1)^{n+1} \frac{H_n^2}{n}=\frac{1}{12} \left(9 \zeta (3)+4 \log ^3(2)-\pi ^2 \log (2)\right)$$

And now I'm leaving for a bit to buy stuff. BBL

@DanielFischer: That's actually a good explanation. Thanks.

@MikeMiller: What qualifies as pop culture?

@Soham holes are a bad way to think of homology, so don't listen to that analogy much. did the lecture you saw tell about euler characteristic of genus g surfaces?

'cause most of the places I know of talks about euler char of polyhedras, which destroys the topological gem hiding behind it.

Random Variable

9:01 AM

@DanielFischer So the fact that it's real-valued on a continuous portion of the real axis allows us to apply the principle to entire axis?

@RandomVariable Yes. (Provided the domain is connected.)

@Chris'ssistheartist Isn't that an old one ?

Chris's sis the artist

@Hippalectryon I didn't find it in my paper, so I calculated it again. @RandomVariable told me about it.

@Hippalectryon I also see the possibility of calculating easily $$\sum_{n=1}^{\infty} (-1)^{n+1} \frac{H_n^3}{n}$$

@Chris'ssistheartist You've already done it

Chris's sis the artist

@Hippalectryon loooll, OK! Thanks! :-))))))))

@Hippalectryon Then it should be somwhere in these papers.

@Hippalectryon Oh, right! :-)

@Hippalectryon btw, great news here! :D

9:07 AM

?

Chris's sis the artist

@Hippalectryon Today I found a very very nice and simple proof to $$\sum_{n=1}^{\infty} \frac{H_n^2}{n^4}=\frac{97}{24} \zeta (6)-2 \zeta^2 (3)$$ by only using elementary stuff.

:D

Chris's sis the artist

:D

@SohamChowdhury if you think a chalk is better than felt pen and you have allergic issues , then it really bad choice

someone here came up with geometric square equation

some threads in MSE are simple and interesting

Random Variable

@DanielFischer Is the notification that is sent to the OP retracted if you delete your answer? And if so, what happens if you undelete it? I asked this yesterday, but seemingly no one knew.

9:17 AM

@RandomVariable I think the notification is retracted when the answer is deleted, like comment notifications are. I don't know what happens when you undelete.

Random Variable

@DanielFischer My guess is that it's not sent again. Or maybe the OP just didn't like my answer.

@RandomVariable You can always post a comment when undeleting, "I fixed [not so good or even bad stuff], not sure whether you get notified when an answer is undeleted, therefore I ping you with a comment." or something like that.

Culture that is popular qualifies as pop culture @Huy :D

Chris's sis the artist

@Hippalectryon Basically I have in mind the solution to calculate any series of the type $$\sum_{n=1}^{\infty} \frac{H_n^2}{n^m}, \ m\ge2 $$, but there are many steps to do, I don't have an optimized solution such that I can publish it. The good thing is that I have a way, even in these conditions!

@skillpatrol: I like Taylor Swift.

9:22 AM

Me too.

<3

:P

Random Variable

@Chris'ssistheartist I used to write down everything that I thought was interesting. I stopped doing that several months ago.

Chris's sis the artist

@RandomVariable Why did you stop?

Random Variable

@Chris'ssistheartist I don't know. I might start again.

Chris's sis the artist

9:24 AM

@Hippalectryon I can also derive easily (well, it's some to write) $$\sum_{n=1}^{\infty} \frac{H_n^2}{n^5}, \ \sum_{n=1}^{\infty} \frac{H_n^2}{n^6}$$

That's great :-)

Chris's sis the artist

@Hippalectryon Even the version $$\sum_{n=1}^{\infty} \frac{H_n^2}{n^{2015}}$$ if you want, but there will be a hell to write down. :-)

xD

Chris's sis the artist

@RandomVariable Do you also like to create problems?

Sometimes I feel I like to create problems more than solving them. Creating problems is a fascinating process, at least to me.

Random Variable

@Chris'ssistheartist Every time I create a problem, M.N.C.E. can seemingly solve it.

Chris's sis the artist

9:28 AM

I might dare to say that one can even learn more from the creating process rather then solving problems at random.

rather than*

@RandomVariable yeah, he's pretty good. Do you know each other outside the site? I was inclined to say that Cleo and M.N.C.E. are the same person.

Random Variable

@Chris'ssistheartist Only on this site and the other site.

@Chris'ssistheartist He finds approaches that I would never have even thought of.

Chris's sis the artist

@RandomVariable Perhaps it's about much experience.

Chris's sis the artist

10:11 AM

@Hippalectryon the shocking thing is that we cannot find here algo.inria.fr/flajolet/Publications/FlSa98.pdf $$\sum_{n=1}^{\infty} \frac{H_n^2}{n^6}$$ and that's really curious.

hmm weird indeed

Chris's sis the artist

10:39 AM

@Hippalectryon I ask myself if I'm the first in the world that did that. If you find any paper about that, pls let me know.

@Chris'ssistheartist sure

1 hour later…

11:44 AM

I went to this roller coaster today which I found out was a mobius strip... Pretty nice!!

Soham Chowdhury

@Agawa001 yes, I have asthma. that is indeed a problem :(

@BalarkaSen aye, it did.

I actually quickly read through half of The Knot Book. Fascinating stuff.

stay away from chalk then

12:04 PM

Hello@Soham

Soham Chowdhury

Hey.

Its better to stay with chalk than whiteboards..
Whiteboard markers contain toulene!!@Soham

Hello@huy

Soham Chowdhury

@Rememberme so where do I go? :(

Pen and paper(though I prefer a pencil)@Soham

hi @Rememberme.

12:06 PM

Brown papers ... I use the cover of my notebooks when I take my own tests

hi, everyone.

Hey@Balarka

Soham Chowdhury

My god.

Braids.

Braids?

@Soham glares

Soham Chowdhury

12:09 PM

Braid groups are cool.

I don't think skimming through a bunch of names would do you any good at all.

Soham Chowdhury

I'm not skimming, Balarka.

Seriously.

Hey@Balarka Did you see the non hausdorff space I pinged you... ?

Yes, you are. Braid groups are hard, and I don't think you'd understand it in a minute.

nope, @Remember

Let me write them again then :)

Soham Chowdhury

12:10 PM

I'm actually reading a book, which happens to be sort of not a textbook. More of a popular science sort of thing.

But they are cool.

I mean, you can understand what braids are and the definition of braid groups. But they are deep stuff.

@Soham Topological quantum field theory is cool too.

Soham Chowdhury

@BalarkaSen I didn't deny that.

But I am not really "skimming definitions". :)

$(X,\tau)$ where $\tau = (\emptyset ,X)$ @Balarka

Tobias verified it

If you want to read some pop-math book, try Courant's "What is mathematics".

Soham Chowdhury

Ah, this one is great fun. You might want to try it.

12:12 PM

I like that one.. I read when I started maths..@Balarka

Soham Chowdhury

@BalarkaSen Wasn't that your first?

Oh, his too.

Yes.

It has heaps of exercises.

Pretty quiet exercises I would say :)

3

Q: Can a fractal be a manifold? if so: will its boundary (if exists) be strictly one dimension lower?

So it is weekend! and I am reading a nice book, "The Poincaré conjecture", written by a mathematician (Donal O'Shea, topologist). The book introduces step by step basic concepts of Topology, and talks about the mathematical advances since Poincaré stated the conjecture up to the moment it was dem...

general-topology manifolds fractals

@Rememberme eh, well.

Its a boring one@Balarka

Soham Chowdhury

12:13 PM

Anyway. The Knot Book is actually brilliant, and has zero prerequisites (which might not be something you look for :P)

I mean the non hausdorff space

I want to find interesting ones.....

@Soham nah, I don't want to read pop-math anymore. I'll persuade prof to let me study knot theory at some point of time.

Soham Chowdhury

And it's cool how Reidemeister moves are connected to braids.

@Remember Here's a visual example :

Soham Chowdhury

@BalarkaSen He gets into Seifert surfaces and knot polys and stuff. ;)

12:14 PM

Take two spheres.

Soham Chowdhury

Anyway, your call.

Okay...

So you have two copies of spheres $X_1$ and $X_2$. Equip $X_1-\{p\}$ and $X_1-\{q\}$ with subspace topology from R^3, for points $p$ in $X_1$ and $q$ in $X_2$.

And let open nbhds of $p$ be disjoin union of nbhds of $p$ in $X_1$ and $q$ in $X_2$ and vice versa.

I am not sure if I have been able to explain this very well : you can think of this space as two spheres brought "infinitely closer", but not touching each other.

@BalarkaSen I didnt get that one

@SohamChowdhury Not sure if there's any point learning something which I can't use in proving anything.

personal opinion.

12:20 PM

Though I understood the last statement ...

@Rememberme It's a bit complicated to explain using words.

Okay...

Essential idea is to bring the two spheres infinitely closer, so that each neighborhood of one of the "infinitely close" points would contain the other point.

@Balarka I have this question .. If you dont mind that is..

Anyway, easier and less visual example : cofinite topology on $\Bbb C$.

Chris's sis the artist

12:21 PM

What amazing results I got!!!!!!!!

Ahh......

Chris's sis the artist

I almost cry! I'm simply overwhelmed.

@Balarka (from my mobius band trip) Why is there such a weird intersection of surfaces in a Kline bottle

You can't embed the Klein bottle inside $\Bbb R^3$. It's a fact.

Tobias Kildetoft

@BalarkaSen I was trying to get @Rememberme to construct a a topology where all constant sequences but one would converge to a single point, and the last one would converge to everything.

12:24 PM

That is really difficult for me @Tobias

Anyways Hi@Tobias

Tobias Kildetoft

@Rememberme Hi, yeah, it might be a little too tricky

@TobiasKildetoft ah.

Tobias Kildetoft

(though it can be done with a set consisting of just $2$ elements)

it is of course slightly more interesting on an infinite set

That is what I was talking about in our last discussion but you didnt reply to it @Tobias

Tobias Kildetoft

@Rememberme I must have missed it

12:28 PM

No worries

Hmm. Actually, I have never thought about why the Klein bottle cannot be embedded in R^3.

Tobias Kildetoft

@BalarkaSen Well, "nice" forms of embedding are ruled out by it having torsion in its fundamental group

I know that space with torsion in H_1 cannot be embedded in a nice way into R^3.

But I am not sure if I can rule out the niceness.

Soham Chowdhury

@BalarkaSen oh, there are all sorts of interesting proofs in the book.

Tobias Kildetoft

mathoverflow.net/questions/4478/… is related

12:32 PM

@SohamChowdhury nah, I don't think anything interesting can be done with knot theory without knowing any prereqs at all.

Soham Chowdhury

but there's a chapter on applications to bio, physics and chem. you'll get a perma-wrinkle in your noses. :P

I tried doing Rolfsen, but it turned out I need cohomology.

@TobiasKildetoft thanks, I am having a look.

Soham Chowdhury

@BalarkaSen well, you get to show that the left and right trefoils aren't the same using HOMFLY

heh, whatever.

yes, but HOMFLY has deep interpretations.

Tobias Kildetoft

@BalarkaSen I also asked a similar question on MSE a while back, which is where I was directed to that MO question

12:34 PM

the combinatorial definition isn't of my taste.

@Tobias oh, ps : did I ask you how HOMFLY comes from representation theory? I think Mike referred me to you when it came up in a discussion.

I don't remember if I asked it.

Tobias Kildetoft

@BalarkaSen I don't think you did

But I am not actually that familiar with how it comes up

ah, I see.

Tobias Kildetoft

I don't even recall if it can be obtained from Khovanov homology as the Jones polynomial can

oh?

Tobias Kildetoft

(I never really studied any of this)

12:40 PM

I don't even know what Khovanov homology is :P

Tobias Kildetoft

@BalarkaSen It is a way to realize the Jones polynomial as the Euler characteristic of a complex, and then using those complexes as an alternative invariant, which is stronger

interesting. if you're willing to elaborate a bit more, I am prepared to listen.

or refer me to anything, if that's way too much work.

the sea sounds like pretty blond lady callin me to bed, and i m like no, i have another prettiest gf made from {1,0}

Tobias Kildetoft

@BalarkaSen That explanation was pretty much the extend of what I know about it

I mainly know about it because it was one of the original examples of a categorification, so it tends to be mentioned in anything related to that

@BalarkaSen There is a fairly accessible paper by Dror-Natan about it (someone asked about it recently on the main site, let me go find it)

arxiv.org/abs/math/0201043

thanks, I am having a look.

er, that looks a bit too algebraic for me.

12:52 PM

@Chris'ssistheartist I really think you should watch this :) this is very interesting : youtube.com/watch?v=oYp5XuGYqqY

hey everyone :) anyone who has time... watch this , its a very interesting TED talk

Donald Hoffman: Do we see reality as it is?

3

Tobias Kildetoft

@BalarkaSen Yeah, it will invariably be somewhat algebraic (I think the original paper by Khovanov is too)

oh well.

ps @Soham : if you want to read popmath, try Ash-Gross. it leads you through the beautiful theory of elliptic curves. i just recalled, so thought i might as well mention that.

whats popmath

(the real motivation behind this is to derail you from the algebraic topology obsession)

me?@Balarka

1

Q: Point set where each point has unity distance to all other points ($L_1$ metric)

I want to construct a point set where each point has the same (w.l.o.g., unit) distance to all other points in the $L_1$ metric. Example: The points $\left(\frac{1}{2},0\right)$, $\left(-\frac{1}{2},0\right)$, $\left(0,\frac{1}{2}\right)$ and $\left(0,-\frac{1}{2}\right)$ have unit distance betwe...

geometry metric-spaces

12:58 PM

@Tobias by the way, it turned out I need Alexander duality to prove that the klein bottle cannot be embedded in R^3.

which is not good, since I don't know cohomology yet :)

Whats $L_1$ metric?

Tobias Kildetoft

@BalarkaSen I see

@BalarkaSen Then learn cohomology :)

Yes, I plan to. I am skimming through the topology I have learned so far, as a warm-up.

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