Mathematics - 2015-06-07 (page 6 of 9)

Karim Mansour

16:00

good I am planning to go that route too @Semiclassical

Soham Chowdhury

@TedShifrin If $Y$ is a subset of an m.s. $X$, $x\in X$ is adherent to Y if $$\forall r>0 \;. B_r(x) \cap Y \neq \emptyset$$

Ted Shifrin

@Balarka: One has that map in the setting of manifolds and tangent bundles, as well.

@Soham, empty or nonempty?

Fargle

@TedShifrin Oh, I'm not! I'm used to the "brick-wall effect", as it were. It's part of why I'm still doing math.

Soham Chowdhury

@TedShifrin what is?

Ted Shifrin

In your definition, did you mean to say $\ne\emptyset$?

Soham Chowdhury

16:01

oho, yes.

Semiclassical

what that attitude misses is the possibility of emergent behavior, whose physical meaning can't be understood in a reductive fashion. one can certainly describe it consistently, but one wouldn't try to build it from the ground up

Mary Star

Could you take a look at my question: math.stackexchange.com/questions/1315879/… ? @robjohn @KarimMansour

Ted Shifrin

Modern language is to call $x$ a limit point of $Y$.

Semiclassical

@KarimMansour ah, neat

Fargle

There's a definite gratification you get from breaking a barrier in your knowledge or conceptualization.

Shin Kim

16:01

@Fargle Hey man, I've just created my mathblog yesterday to post what I've learnt and studied, but I need a feed back from others or basically need some friends to study with. Wanna check it out?

Soham Chowdhury

@TedShifrin ah, thanks.

Ted Shifrin

Sure, @Fargle, or in cracking a stubborn problem you've worked on for a while.

@Balarka: Were you going somewhere with that stuff?

Balarka Sen

now, my question is $\phi$ seems to be taking the group of $\mathfrak{C}^1$ functions to the group $\text{hom}(\Bbb R^n, \Bbb R)$. i haven't learned about higher derivatives, but can we extend this to something else on the right using the notion of higher directional derivatives (if that even makes sense!).

Karim Mansour

@Semiclassical but you see the more well rounded you are in something the more connections you'll make and you will make connections among first unrelated concepts. I took AI class before and our professor said that knowing stuff on broad terms is better than being specailized in something.

Fargle

@TedShifrin Oh yes. I found some of the group theory proofs I did in abstract algebra to be quite satisfying in this way.

@ShinKim Yeah, I'd love to.

Semiclassical

16:03

oh, i certainly agree with that attitude

Balarka Sen

chain complex obsession, so sincere apologies if i am raving nonsense.

Soham Chowdhury

Is it common that one doesn't feel like moving on if one knows all the steps that have to be taken to prove a theorem but one doesn't "get it"? @Ted?

Karim Mansour

@Semiclassical I think that too broad is better than knowing about one thing and one thing only.

Semiclassical

but there are limits to it.

Ted Shifrin

You want to think of these as vector spaces, not as groups, @Balarka, and, yes, there are higher-order derivative maps. They are multilinear maps or tensors.

Semiclassical

16:03

it's a balancing act, to be sure

Ted Shifrin

Sometimes one should go on, use the theorem, get a better feel for it, and then go back, @Soham.

Soham Chowdhury

yes, but I somehow have this fear of being bitten if I do that.

Balarka Sen

@TedShifrin hmm, that's interesting.

Soham Chowdhury

I have to learn to be comfortable with this "incremental" sort of learning, I guess.

Semiclassical

but i've found that, in my research at least, the surprising applications have not come from 'foundational aspects'. a lot of it has just come from realizing the sheer breadth and depth of applied mathematics

Ted Shifrin

16:05

So, for example, @Balarka, $D^2f(a)$ is a symmetric bilinear form, and $D^2\vec f(a)$ is a vector-valued symmetric bilinear form (so a tensor of type $(2,1)$, etc.).

Shin Kim

@Fargle Alright cool. Here's the link to my website- scheenija.tistory.com

Balarka Sen

cool.

Karim Mansour

@Semiclassical I agree but you never know what you might make connection amongst later in your research maybe it will require also foundational aspects

Balarka Sen

i'm going to do the exercises in the previous chapter (along with your continuity problem! :)) to get to higher derivatives.

i want to think about this thing.

ok, i have to go now.

thanks for that information, @Ted.

Semiclassical

perhaps. but i'm very much someone who prefers to take seemingly-simple models and pull something interesting out of it, rather than tackle somthing deep and obscure

Ted Shifrin

16:07

bubye, @Balarka. Sleep well :P

Karim Mansour

@Semiclassical I see

Fargle

@ShinKim I like what you have so far, even if it's not very much

Karim Mansour

@Semiclassical have you read this book before I bought it 2 weeks ago it is very nice read amazon.ca/Course-Modern-Mathematical-Physics-Differential/dp/…

altough some of the stuff at first is covered in set theory and analysis

Shin Kim

@Fargle Yeah good to hear that. Like I said, I've created this website yesterday, like 30 hours ago.

Ted Shifrin

BTW, @Fargle, have you taken a class from Ken Knox? He was a student of mine years ago.

Semiclassical

16:08

i know of it, but for very random reasons

Fargle

@TedShifrin Oh my God, yes I have! 561 Topology.

Ted Shifrin

He was at my retirement party. He was telling me about a few students.

Did you like him?

Semiclassical

(the university library computer said that i had checked it out, and i hadn't. i ended up having to order a replacement copy to avoid paying a ridiculous fee)

a rather irritating situation, all things considered

Fargle

I did. He was a fantastic lecturer--was able to make point-set and algebraic both much more intuitive to me. I ended up dropping for out-of-school reasons, unfortunately, but my general impression of him was good. (Hopefully vice versa!)

Karim Mansour

lol

Ted Shifrin

16:11

Well, since I don't know your name, I can't ask :) He told me about one student who asked for permission to take the class without the prerequisites and ended up deciding to be a serious math major.

Fargle

@TedShifrin It might have been me. I didn't take analysis before, but I studied independently through the department head.

Ted Shifrin

Ken was a music major when he ended up in my classes. He took the second semester of multivariable and diff geo together ... poor guy ... But he turned out fantastic :P

Goodnight, @MikeM.

Soham Chowdhury

This limit point / closure definition of closed sets is so much nicer.

Fargle

I wouldn't have pegged him as a music guy, but a lot of mathematicians are musicians as well.

Ted Shifrin

Well, you have to be a bit careful in general topological spaces, @Soham, but in metric spaces you can do whatever you want :P

Mike Miller

16:12

Morning, @Ted. Just landed in SJC. Captain mistakenly said it was 10:10 local time and I was terrified I just flew to the wrong San Jose.

Ted Shifrin

ROFL @MikeM

I highly doubt you were terrified.

Soham Chowdhury

@TedShifrin Yeah, haven't got to those yet.

Fargle

Vihart and Dr. Thistlethwaite come to mind.

Ted Shifrin

@Fargle: Yeah, pianist and also serious amateur actor.

Soham Chowdhury

@Fargle Ah, are you a cuber?

Fargle

16:13

@SohamChowdhury Not intensely, but Morwen is a professor at my university

Semiclassical

oh, ted. while working on math-physics research i found a reason to be very very glad i knew about wedge products already

Ted Shifrin

@Fargle: He told me he wanted to teach the undergrad diff geo (and use my notes), but that not enough students registered. Maybe you should volunteer to sign up for it next time :P

Mike Miller

@Ted: It would not have been pleasant to wake up in the wrong state

Semiclassical

namely, understanding what a grassmannian is :)

Fargle

@TedShifrin I very well may. I did try to sign up for it, actually, but it didn't make.

Soham Chowdhury

16:14

My mental picture of an open set is sort of like points tapering off at the boundary, so taking a complement of one is sort of hard to visualize, @Ted. That's what.

Ted Shifrin

Well, @Semiclassical, grassmannians have been a major portion of my math life.

Soham Chowdhury

Do you know about Thistlethwaite's algorithm, @Ted?

Ted Shifrin

Nope.

@MikeM: Is the other San Jose actually in a state? I thought it was called something else.

Karim Mansour

grassmannians

what is that @TedShifrin

Soham Chowdhury

It's an algorithm that approximates optimal (fewest moves) solutions to a scrambled Rubik's cube. What does he teach? I'd guess algebra.

Ted Shifrin

16:16

Generalization of projective space: It's a space that parametrizes all $k$-dimensional subspaces of a vector space.

Mike Miller

I don't know, @Ted, but you can be sure I opened up Google maps to see where I was...

Semiclassical

ah, neat. i finally read a discussion which allowed me to grok the equivalence of free fermions and infinite grassmanians

Ted Shifrin

LOL, it's a lot further (and souther) to the other one, @MikeM.

Karim Mansour

oh

Ted Shifrin

@Soham: I know the name from knot theory/geometric topology.

Mike Miller

16:17

Well, I don't even have the best send did direction when I'm on the ground.

Fargle

@SohamChowdhury He tends to teach stuff related to algebra or topology--his initial research was knot theory, where he helped prove the Tait conjectures.

Ted Shifrin

Good point, @MikeM. You're lucky to find yourself.

Soham Chowdhury

Yes, I just googled him. There are a lot of non-mathematicians who know a lot of group theory and work with the cube, so I never guessed. :)

Semiclassical

@TedShifrin what did your research re: grassmannians involve?

Fargle

@SohamChowdhury Yeah, he's pretty awesome. Easily the most famous mathematician at my school.

Ted Shifrin

16:19

They're a natural way to study submanifolds that aren't curves or hypersurfaces. The Gauss map of such a thing naturally maps to a Grassmannian (or generalization thereof).

Semiclassical

Gauss map?

Ted Shifrin

Gauss map assigns to each point of a submanifold its tangent space at that point.

Shin Kim

What term will be appropriate to use when you want to call the audience in this chat room? I was about to say "hey guys" but this looks quite rude, I guess.

Semiclassical

hmm

Ted Shifrin

"call"? You mean interrupt? :D

Semiclassical

16:21

i usualy just say 'hi chat'

Ted Shifrin

Saying hello is fine .... some of us do that.

Karim Mansour

I say hey guys

lol

Shin Kim

This is public chat room. Why do you think it's interruption? ;)

Ted Shifrin

I was teasing, @Shin.

user147690

I say hi, so I don't think its a problem lol

Ted Shifrin

16:22

oh, hi, @AlexC.

user147690

Hey @Ted studying for finals these days. 15th,19th,23rd

Soham Chowdhury

@AlexC: Dan is awesome

Ted Shifrin

Good luck to you, @AlexC !

user147690

@TedShifrin Thanks, going to get at worst case a 7,6,5 and at best case a 7,7,6(where 7 is the best)

Ted Shifrin

Is 7 the best?

Semiclassical

16:23

i've been trying to understand things regarding Sato's infinite grassmannian and the relation to the KP hierarchy of integrable equations

user147690

@TedShifrin Yep - 4 pass, 5 credit, 6 distinction, 7 high distinction

Ted Shifrin

One of my colleagues here did a lot of stuff on integrable systems and KP, but I don't know much of that, @Semiclassical.

Semiclassical

kk. which collegue, out of curiousity?

Ted Shifrin

Malcolm Adams. Mitch Rothstein did a bit of stuff on it, too, I think.

Semiclassical

ahh

Ted Shifrin

16:24

@AlexC: So there are 3 failing grades?

user147690

@Soham I can never do the pedals :P

user147690

@TedShifrin Yep, and a 0 for not doing anything at all

Ted Shifrin

Bizarre.

user147690

If you get a 3 you can sit a sup exam

Soham Chowdhury

@AlexClark he started at 19. six or seven years ago.

user147690

16:25

If you get 1 you get an academic warning

Ted Shifrin

And what does 2 mean?

A single course can give you an academic warning?

user147690

@TedShifrin Have to repeat but no warning, and no option for sup

user147690

@TedShifrin Yep and 3 is getting kicked out

Karim Mansour

lol

user147690

@SohamChowdhury He is a beast

Karim Mansour

16:26

what if person gets 0

Ted Shifrin

Interesting.

Karim Mansour

or 1

what will they do with him @AlexClark ?

user147690

1 is warning, and 0 is also warning(but it is usually an indicator that they are depressed or something, since it means literally 0%[or actually if you cheat you get a 0 aswell, but you only get 1 warning with cheating])

Karim Mansour

oh

Soham Chowdhury

@Ted, when Balarka writes ${\sf Gal}(\bar{\Bbb Q}/\Bbb Q)$, does $\bar{\Bbb Q}$ mean $\Bbb R$?

Ted Shifrin

16:28

No, @Soham. Algebraic closure. All the algebraic numbers in $\Bbb C$.

Soham Chowdhury

Oh.

Algebraic numbers as in not transcendental, right?

or something else?

Ted Shifrin

Yes to the former.

Agawa001

wth

Semiclassical

the linkage with fermions comes from taking a wedge product like $e_{1}\wedge e_2 \wedge e_3$ and interpreting $e_r\wedge$,$e_r\lrcorner$ as anticommuting fermionic operators

Soham Chowdhury

oh, okay. thanks.

user147690

16:29

:22059772 $\lim \limits_{x\to0}\lim \limits_{n\to\infty} \frac1x$ IS--------- $\lim \limits_{x\to0}\lim \limits_{n\to\infty} \frac1x$

Fargle

@TedShifrin Er, will I sound silly if I'm still not intuiting #8 correctly?

Ted Shifrin

Sounds like geometric algebra, @Semiclassical, which one of the SE members loves to champion.

Semiclassical

yeah, muphrid i think

Agawa001

nvm

latex is a hell

Remember me

Hello@TedShifrin

user147690

16:30

Gotta use \lim \limits_{blah}

Ted Shifrin

@Semiclassical right :)

Agawa001

its indefite anyways

Ted Shifrin

I said hello ages ago, Remember.

@Fargle: I've forgotten.

Agawa001

@TedShifrin that might got responded ages ago i assume

Remember me

@SohamChowdhury Sorry I didnt have net connection for a moment... Any probd?

Semiclassical

16:33

the main point in this context being, for example, $$(e_{-1}\wedge e_{-1}\lrcorner)\omega+(e_{-1}\lrcorner\wedge e_{-1})\omega=\omega$$ for all $\omega$

user147690

@Kaj Have you heard any of NeObliviscaris?

Agawa001

$\frac{holy}{crap}$

wtf

user147690

Don't forget \Huge{} around that

Mike Miller

Did you see the question I pinged you yesterday, @Ted?

Fargle

@TedShifrin It's the one where you prove that the straight line is the shortest path. I'm going to try at it again before I ask again for serious help, but it's boggling me, which is more frustrating than I care to admit, haha.

Agawa001

16:35

@AlexClark i cant find fittin explanation apart a nag bug

Ted Shifrin

Yes, @MikeM ... I don't know the answer.

Semiclassical

so $\beta_{-1}^\dagger \beta_{-1}+\beta_{-1} \beta_{-1}^\dagger = 1$ (and i'm rambling at this point, so i should probably stop :P)

Ted Shifrin

Oh, right, that one @Fargle.

Also, Fargle, be aware that * means that there's an answer or hint at the back. (Doesn't apply to that problem.)

Mike Miller

FYI: you can't expect to be able to do this smoothly. You definitely cannot get tubular neighborhoods of your invariant $S^n$, at least when you're finding an invariant $S^3$ in $S^4$.

Ted Shifrin

I assume your involution is smooth, @MikeM?

Fargle

16:38

@TedShifrin Thanks, that will certainly help.

baxx

how do i get the negative reciprocal of y=0x +c ? basically it's just a straight line, so do I just set x to c instead of y?

Ted Shifrin

You don't do it by that formula, @baxx. You do it by thinking.

Mike Miller

Not necessarily, @Ted.

baxx

@TedShifrin ace

Mike Miller

I meant above that in the case it is smooth, you can't nexessarily do this smoothly.

Ted Shifrin

16:40

@baxx: I assume you mean you want the perpendicular line?

I understood, @MikeM.

baxx

@TedShifrin yeah, i get that it will be x = something. but I'm not sure how to go about it

Ted Shifrin

It's a cool question. One I should know the answer to. I was thinking about making a metric invariant under the involution, but then I didn't know what to do. For arbitrary metrics, I don't know the totally geodesic hypersurfaces have to be topological spheres. But someone may know that.

@baxx: Did you try drawing pictures? They really help.

Shin Kim

Hey people, a small question: When you define a term, which way do you think is better between by using a logically equivalent statement or just using a directive equivalent relation?

For example,
We say $L$ is a language iff $L$ is a method of communication. or
A language is a method of communication.

The reason why I ask is the later one is a bit ambiguous to me since its necessary condition is not guaranteed.

Ted Shifrin

Besides, @baxx: Perpendicular line through what point?

baxx

@TedShifrin yeah I've been messing about with Desmos, it's nice for visualising things

Ted Shifrin

16:41

Just do it by hand, @baxx. No computer needed here.

Shin Kim

apparently the later one is not equivalent relation.

baxx

@TedShifrin I have points A(-7,7) and B(1,9), line between them is y = 0x + 1, and I wanted to draw the perpendicular bisector through that line

@TedShifrin desmos saves paper and stuff though, seems handy

Ted Shifrin

No, no, @baxx, that line doesn't go through those points!

Fargle

@ShinKim Yeah, generally, if you want to give necessary and sufficient conditions, it's less confusing to use the biconditional.

baxx

whoops

c(3,1) d(-7,1) :)

Ted Shifrin

16:43

OK, now the line is $y=1$.

You want the perpendicular bisector?

iluso

@Ashwin So it is indistinguishable balls to indistinguishable urns ? I'm sorry I do not see the relation. Could you propose a short answer to explain your way ?

baxx

yeah

Mike Miller

@Ted: As a corollary of this, if $M$ has $S^n$ as a double cover, then $M$ is homeomorphic to $\Bbb{RP}^n$. The reason you can't do this smoothly is that it would imply the above is a diffeomorphism, and that's just not true.

Ted Shifrin

So what point must it go through? And what direction does the line have?

baxx

I can set x to the center value, but I wasn't sure if there was a method I should use or something. It's a vertical line

it'll be x = -2

Ted Shifrin

16:44

OK, so it's $x=?$ ?

Yup, and you're done.

baxx

but sometimes i feel as though I should know a way or something?

Ted Shifrin

Thinking is never bad :)

baxx

guess that's just really obvious, thought maybe there was a method or something I was missing :P

Ted Shifrin

In general, when the line isn't horizontal or vertical, you can use your negative reciprocal slope stuff.

baxx

I'm self teaching so there are a fair few gaps :)

Ted Shifrin

16:46

Hmm, @MikeM, remind me why diffeo to $\Bbb RP^n$ fails.

Semiclassical

you could presumably work with the line $y=mx+1$, figure it out for $m\neq 0$, and then intelligently take $m\to 0$

Ted Shifrin

No, @Semiclassical, because you'll end up with infinite slope.

baxx

LaTeX isn't rendering for me - is it working for everyone else in here?

Fargle

@TedShifrin Is there some WLOG I'm allowed to do here? Or is that not really necessary?

Shin Kim

@baxx me neither

Ted Shifrin

16:47

No WLOG needed, @Fargle.

Mike Miller

@Ted: There's a fake $\Bbb{RP}^4$, constructed by Akbulut. Its double cover was long proposed as an exotic $S^4$; it was proved standard in the early 90s by Gompf.

Ted Shifrin

@Shin, @baxx, there's a LaTeX in chat link to the right.

Oh, ok, @MikeM ...

I'm not sure I knew this.

Semiclassical

eh, yes, but it'll be of the form $y=-x/m+c$ and so $my = c-x\to x=c$

Ted Shifrin

Sure, @Semiclassical, fine for people more comfortable with more algebra.

Semiclassical

right. just pointing it out

Ted Shifrin

16:49

Teaching involves knowing the level of the student :P

Semiclassical

psh

Mike Miller

I don't know whether there are examples of fake $\Bbb{RP}^n$ for higher $n$. You should be able to answer this with Wall's smoothing theory, that should all be classified.

Semiclassical

the main reason i'm thinking like that, i suppose, is that when i think of perpendiculars i think of orthogonal families of curves, i.e. $y'(x)X'(Y)=-1$.

Ted Shifrin

Sure, sure ;)

Shin Kim

@TedShifrin Thx. Now it renders

baxx

16:51

@TedShifrin I haven't got a LaTeX in chat link that I can see?

@ShinKim where is it?

Shin Kim

@baxx on the right

math.ucla.edu/~robjohn/math/mathjax.html

iluso

Does anyone know if math.stackexchange.com/questions/1315109/… is related to a "simple extension of 'balls in urns' formula for unordered partitions" ?

baxx

@ShinKim i can see it on that page but not in the chat

Shin Kim

@baxx can you see the 'start ChatJax' hyperlink on that link?

baxx

@ShinKim yeah, I selected that and came back here, no dice

Shin Kim

16:54

@baxx You have to bookmark it first. and came back, click it.

then it renders

Semiclassical

though if i really want to be pedantic i should write it in terms of a wedge product of 1-forms :P

Shin Kim

comback*

baxx

@ShinKim ah, it's working, nice one

Ted Shifrin

@Semiclassical: What you're writing there doesn't make sense to me.

Semiclassical

which?

Ted Shifrin

16:56

Seems like you want one exterior product and one interior product?

Mike Miller

Ok, @Ted, it's not true. I should have googled longer: seems you only get homotopy equicalence, so the cell structure I want to build won't work.

Ted Shifrin

@Semiclassical There.

Semiclassical

er, yes

badly typed

Ted Shifrin

@MikeM: You've lost me.

Semiclassical

$$(e_{-1}\wedge e_{-1}\lrcorner)\omega+(e_{-1}\lrcorner e_{-1}\wedge)\omega=\omega$$

Mike Miller

16:59

I'll write down the details when I sit down. Just arrived at SCU - have some time before my lunch date.

Semiclassical

though i'm not liking $\lrcorner$ for scalar product

Ted Shifrin

The interior product shouldn't be a subscript, @Semiclassical. I'm not sure I've ever used the symbol in LaTeX. I tend to write $\iota_v$.

Say hi to any of my old friends you run into, @MikeM.

Transcript for

$Mathematics$ Mathematics