Mathematics - 2016-07-21 (page 1 of 3)

0celo7

00:01

@BalarkaSen The theory that my prof hasn't opened the book in 10 years was spot on, he didn't know Milnor used that much algebraic topology and he didn't understand the proof either

@BalarkaSen Exercise: show that GP's definition of the Euler characteristic is equivalent to the (co)homological one.

My prof knows of a proof that uses geometric analysis, maybe you can come up with a topological proof?

free_mind

00:32

hi

i realize this may be a silly question, but i figured i'd ask anyway. Is this true or false The negation of "S is an open set" is "S is a closed set." I believe it's true. I realize this is essentially $\forall s, s$ is open, and the negation would be $\exists s, $~(is open)

Krijn

No, a set that is not open is not the same as a set that is closed.

A set can be both closed and open

free_mind

ahh yes

like the set $\mathbb R$

Krijn

Or $\emptyset$

free_mind

00:55

i have a general question for you guys. i love math, but i feel like i'm completely horrible at it. i feel as though i constantly have to look up things, or practice regularly to retain anything. I've made all A's and B's in my math classes, but I feel like I still don't "get it". when does this happen?

I'm currently taking real analysis and it's been difficult and I'm passing with a B and may end up with an A, but I still don't think this speaks to any ability I have in math. i feel like i can just pass a class.

Krijn

That is common, or at least, to me.

After a few years you get used to it.

free_mind

well real analysis is my last class for my minor and i eventually want to go back and finish up my undergrad in math, but i suppose you mean a few more years of grinding through math and not just letting it go haha

do you recommend just coming here and trying to help others with their problems as a means of getting better?

Krijn

Yes, that helps

Also the discussion in this chat helps a lot

free_mind

thanks!

Krijn

01:10

@Balarka Although I had to get used to the slow pace of the movie, it was a beautiful movie. Elegantly bleak through all these magnificent shots and good use of symbolism as well. Still confusing though, should watch it again some time

free_mind

OK i have another true/false question from my book. For what it's worth I believe this is false, but I don't have a full enough grasp on cardinality to know for sure. "If S is a subset of the real numbers and S contains an open interval, then the set S
is uncountable." i believe this is a false because of S being a subset of the real numbers, however, the definition seems to indicate that $f:S \to \mathbb N$

Krijn

If you think this is false, then you should give a subset $S$ of the real numbers which contains an open interval, which is countable

An explicit subset

Which definition seems to indicate your function $f$? I don't get what you mean by that

free_mind

From wiki: By definition a set S is countable if there exists an injective function f : S → N from S to the natural numbers N = {0, 1, 2, 3, ...}

sorry definition of a countable set

Krijn

Ah, yes, sure. So have you found a set $S$ that satisfies these qualities?

free_mind

So every subset of $\mathbb R$ is uncountable since there are infinitely many numbers between the integers?

Krijn

01:15

Well, the set $\mathbb N$ is a subset of $\mathbb R$ and certainly countable?

free_mind

Ahh... so every set containing an OPEN interval would be uncountable

which is a subset of $\mathbb R$

Krijn

You need to prove (or disprove) that

0celo7

@free_mind any open interval is diffeomorphic to $\Bbb R$...do you want a hint for that

free_mind

i don't know what diffeomorphic means

0celo7

diffeomorphic is stronger than what you want

Krijn

01:18

@0celo7 This is not gonna help for a student in real analysis

0celo7

but it works

@Krijn actually it does, because diffeos are bijections

Krijn

And Galois Theory works to prove that 2 is a prime

0celo7

and the canonical diffeo in this case is a really nice bijection

Krijn

So tell him that

0celo7

are we telling him the answer or helping him

Krijn

01:20

Tell him that any open interval of $\mathbb R$ has a nice bijection to $\mathbb R$

We're trying to help, I hope

0celo7

@free_mind There is a nice bijection of $(a,b)$ onto $\Bbb R$, using elementary functions.

I think there's one using rational functions as well

I think we used a rational function in my analysis course

free_mind

im having a hard time wrapping my mind around this

0celo7

@free_mind Let me give you a graph to help you

First, do you see that $(a,b)$ has the cardinality of $(-1,1)$?

free_mind

yes

0celo7

alright, getting an image from my analysis book

@free_mind does that help at all?

free_mind

01:27

OK @0celo7 perhaps i did not understand how $(a,b)$ has the cardinality of $(-1,1)$. can you elaborate a little more

Krijn

Could you think of a function that maps $(a,b) \to (-1,1)$

free_mind

x^3

0celo7

uhhh

We want a bijection between $(-1,1)$ and $(a,b)$

free_mind

sorry i mean $-x^3$

0celo7

@free_mind First, do you see why you may take $a=-1$?

free_mind

01:31

no i don't sorry

0celo7

you can translate $(a,b)$ so that the left endpoint is $-1$

free_mind

i mean i understand that it's mapping a to $-1$

0celo7

you can edit messages

press the up arrow

free_mind

right and $b$ is mapping to $1$, is that what you mean?

0celo7

no

basically, what I have in mind is: move $(a,b)$ so that $a=-1$. Then "stretch" it onto $(a,b)$

free_mind

01:38

perhaps it would help if i explain to you how i'm seeing what you're showing me. I see an open set $(a,b)$ where $a$ and $b$ represent some numbers not in the set, but are boundary points of the set. And between $a$ and $b$ there are infinitely many numbers.

Krijn

Yes

And we want to map that open set to the open set $(-1,1)$

free_mind

so are $-1$ and $1$ just values you picked out of a hat?

Krijn

So for example, if our set is $(0,2)$ we could do that by the map $x \mapsto x - 1$

0celo7

@free_mind No, because the above graph shows $(-1,1)\sim\Bbb R$.

I can give you the function if you want

You want to show that $(a,b)\sim\Bbb R$, right?

free_mind

right

0celo7

01:39

@Krijn you understand what I'm trying to do, right

then give it a linear stretch

free_mind

i think so

Krijn

@0celo7 Yes, although I wonder if it's the best approach

0celo7

@Krijn One can show $(a,b)\sim\Bbb R$ directly.

Krijn

It's not necessary to show that for this exercise

free_mind

@Krijn OK that last part of this confuses me just a little bit

Krijn

01:41

And may be above his level, I wouldn't know

0celo7

Although...giving a 100% rigorous proof is not easy

@Krijn What's he trying to do?

Krijn

Proving/disproving "If S is a subset of the real numbers and S contains an open interval, then the set S
is uncountable."

0celo7

Oh...the only way I know how to do that is show it's isomorphic to $(0,1)$, I know how to show that's uncountable

I think one can do that with Lebesgue measure?

free_mind

@Krijn so are you saying that in $(0,2)$ if I do that by the map $x \to x -1$ I can get a bijection?

0celo7

But there's an elementary proof

I'm no set theorist

@free_mind the map $x\mapsto x+c$ is just translation

Krijn

01:44

@0celo7 Cantor's diagonal argument

0celo7

if you slide your set around $\Bbb R$ you won't change the cardinality

Krijn

@free_mind Let $f: (0,2) \to (-1,1)$ by $x \mapsto x - 1$, do you see that this is a bijection?

0celo7

@Krijn Ah, the book I used has two proofs

We didn't cover Cantor's proof

Although I do recall reading it...but I'm no set theorist

free_mind

@Krijn I think so. Is it because I can take the point $0$ and subtract $1$ and end up in the other set?

0celo7

@free_mind Hint: you can write down the inverse map

@Krijn what's your field?

Krijn

01:49

@0celo7 I'm interested in algebra and number theory, but I'm still just a student

0celo7

@Krijn I know nothing about algebra, but I found a copy of Lang in my adviser's office today that was not Springer

thought it was interesting

What does number theory do?

We did a little in my basic algebra course, but I felt like it was just tricks and little factoids

Nothing really deep

Krijn

Yeah, elementary number theory felt the same for me

free_mind

@0celo7 by "basic algebra" I assume you mean Abstract Algebra haha?

Krijn

But algebraic number theory was much more fun, as is galois theory

It looks at field extensions and in particular number fields

0celo7

@Krijn The algebraic number theory class requires you to enroll in the modern algebra course, which I have no time for

And I think I pissed off the prof who teaches it, anyway

Krijn

01:53

What do they teach in modern algebra?

0celo7

@free_mind yes

@Krijn Hungerford and Lang

Algebra is far from my interests, I have not inquired about it any further.

Krijn

What are your interests?

0celo7

Geometry

Krijn

That's broad

0celo7

I want to learn geometric analysis but I don't know much analysis

Krijn

01:54

I like algebraic geometry

0celo7

@Krijn Number theory and algebra aren't? :P

Krijn

There are a lot of flavors to geometry

0celo7

Differential

Riemannian, even

Krijn

Symplectic, too?

0celo7

Don't know much about it besides Arnold's book

And I didn't appreciate that very much because the meat was very analytical -- I'll return to it next summer, perhaps.

The last chapter on stability is what the book is all about

@Krijn I quite like general relativity, too

But the interesting stuff there requires nonlinear hyperbolic PDE theory

Krijn

01:57

I don't know anything about general relativity, but a fellow student of mine explained it as "differential geometry with a physics flavor"

Anyways, I'm off too bed. G'night.

0celo7

Night.

free_mind

@Krijn night and thanks!

0celo7

@free_mind which sets do you know are uncountable

do you have any theorems about such sets at your disposal?

free_mind

I'm still trying to understand cardinality - it's a new subject for me

at its surface it sounds very simple - the number of elements in a set

but things get hairy when we start discussing infinite countable sets

Semiclassical

hi chat

0celo7

02:05

evening

free_mind

So I just read this... "If an uncountable set X is a subset of set Y, then Y is uncountable."

0celo7

Yup

free_mind

Since $\mathbb R$ is uncountable, that would mean my subset is also uncountable

0celo7

No

Is $\Bbb Q\subset\Bbb R$ uncountable?

free_mind

No it not uncountable

0celo7

02:08

The converse of that theorem is false

free_mind

oh i see

Semiclassical

$X\subset \mathbb{R}$ not $\mathbb{R}\subset X$

free_mind

if i have an uncountable subset then its... ugh parent set is uncountable

0celo7

Do you see how to prove it?

free_mind

i honestly don't see how to prove it

0celo7

02:12

@free_mind Proof by contradiction.

free_mind

ahh ok

"If S is a subset of the real numbers and S contains an open interval, then the set S
is uncountable." so is it the open interval that makes this true?

0celo7

Is what the open interval?

Do you know that open intervals are uncountable?

free_mind

@0celo7 thanks for your patience. I'm still very confused by this class and most of my questions are met with questions. Please know i'm not intentionally being obtuse, I'm just trying to learn and understand

0celo7

@Semiclassical Ugh, I still need to complete that circle exercise

Semiclassical

good luck

0celo7

02:23

I will procrastinate

Time to read more Morse theory...

Semiclassical

the main problem is that while the intuition is straightforward now

getting it actually organized is a pain

that's what'd be annoying for me, at any rate

0celo7

what is a "counter-image"

Semiclassical

shrug context?

0celo7

Semiclassical

i'm not parsing that either

0celo7

02:26

$Df$ is the pushforward

$M_x^*$ the cotangent space

Semiclassical

maybe it just means that there's a bijection between the set of critical points of $f$ in the vector space and the submanifold of zeros in the tangent space?

0celo7

@Semiclassical yes, roughly

Semiclassical

not sure i like that terminology---sounds more suggestive than precise

0celo7

$Df:M\to T^*M$

Semiclassical

mmkay

0celo7

02:29

so, $x\in M$ is crit point if $Df(x)=0$

what the heck is $Z^*$

Semiclassical

i'm a little confused too. why wouldn't there just be one zero?

0celo7

Well you can have many critical points

like $\sin(x)$

Semiclassical

well, sure.

but then what's Z^*?

which, of course, is what you were asking

0celo7

Yeah

The "set of zeros" is $(Df)^{-1}(0)\subset M$

that's not generally a manifold either

Semiclassical

what is $\approx$ supposed to mean?

0celo7

02:32

diffeomorphic.

Semiclassical

hm.

is there not a unique zero vector in $T^*M$?

0celo7

oh...no...don't tell me

$Z^*$ is probably the zero section of the cotangent bundle

oh I'm stupid

It's not $(Df)^{-1}(0)$

Semiclassical

smarter than me on this point, i'm afraid

0celo7

it's not $0$ you need, but the 0 bundle

er, zero section?

Yeah, zero section

Semiclassical

that's probably the better way to write it, yeah

0celo7

02:35

@Semiclassical Basically, the zero section of a vector bundle is the points on the base manifold with the zero vector attached

Semiclassical

hmm

that doesn't seem much different from the base manifold itself, which is perhaps the point

once one makes precise that notion of 'not much different'

0celo7

It's an embedding of the base manifold, yeah

Semiclassical

kk

0celo7

$M$, viewed as a subspace of $E$, is the zero section.

Semiclassical

right.

0celo7

02:38

I recall Ted's definition of a Morse function

lol

@Semiclassical Apparently "$\partial_i\partial_j f$ is a nondegenerate matrix when $\partial_if=0$" is the same as $Df\pitchfork Z^*$.

Semiclassical

i'm just going to smile and nod

0celo7

I'm about to read the proof

Lord help me

Semiclassical

what does \pitchfork mean?

0celo7

Transversal

Semiclassical

ah. no wonder i don't know it

0celo7

02:41

Let $Z\subset Y$ be a submanifold, and let $f:X\to Y$

If $df_x(T_xM)+T_{f(x)}Z=T_{f(x)}Y$ for each $x\in f^{-1}(Z)$, then $f\pitchfork Z$.

Semiclassical

ow

not sure what's meant by "space + space = space" here

0celo7

@Semiclassical At the most basic level, it gives the condition for the preimage of a manifold to be a manifold

@Semiclassical vector space addition

the two spaces on the left are subspaces of the right

their combined span equals the one on the right

Semiclassical

ah, and they generate the entire vector space

0celo7

yeah

I like how Future's latest track reminds me of Pokemon Mystery Dungeon

@Semiclassical Proof: "it's easy to see"

Time to get out the scratch paper

Semiclassical

...

there are times when I like being a physicist. at least then I actually get to compute things :P

not things anyone necessarily cares about, but...

0celo7

02:47

The other day I had to use multivariable calculus to derive an error formula

Someone in the lab said "hey math major"

Apparently this means I can compute partial derivatives

I haven't even taken Calc 3

Semiclassical

that's just a little pathetic

0celo7

"easy to see"

Semiclassical

the pathetic was re: someone needing a 'math major' to do partial derivatives, to be clear

0celo7

He didn't need it

He wanted me to check what he did

Semiclassical

ahh

hmm. whether to do computations or watch youtube

0celo7

02:52

oh, maybe I should work in a chart.

Semiclassical

this pencil is sharp, guess i'll compute

0celo7

sounds like a plan

Semiclassical

yup

0celo7

Ah, it is trivial

:P

Semiclassical

it is now that you see it, heh

0celo7

02:54

God, $L^2(\Bbb R^n)$ is horrible notation

Semiclassical

square-integrable functions on euclidean space?

0celo7

no

it's linear selfmaps of $\Bbb R^n$ here...

Semiclassical

wut

0celo7

@Semiclassical Uhhh, is End(V) the invertible selfmaps?

Semiclassical

hell if i know

is that endomorphisms?

0celo7

02:56

Yeah, does that imply invertibility?

Semiclassical

no idea. i can't remember the various prefixes

dfn seems to be 'morphism from an object to itself'

0celo7

Shit, it's Aut(V)

Semiclassical

automorphism looks to be 'invertible endomorphism', yeah

like adding 1 to any integer versus multiplying an integer by 2

0celo7

I hate LaTeX

It randomly decided \Aut is already defined

Semiclassical

$\Aut$

0celo7

02:58

AFTER I defined \Aut and used it

Semiclassical

huh

weird

you're trying to compile some Latex stuff?

0celo7

Yes, I called Aut(V) End(V) in my homework

Semiclassical

ah

so now you're trying to replace the definition?

0celo7

just need to fix it, and I want a new command for \operatorname{Aut}

Now it works

Semiclassical

kk

0celo7

03:02

¯_(ツ)_/¯

Semiclassical

ugh, load faster, 4.6 MB pdf that i only need one page out of

0celo7

is that a large pdf?

Semiclassical

so my computer would have me believe

0celo7

I wonder how Witten spends his free time

Semiclassical

doing more physics, probably

0celo7

03:08

I want to write a book

It would have the best notation and typesetting

@Semiclassical what's the bible of your field?

Semiclassical

not really sure there is one. the field of condensed matter physics is too broad as a whole to have one, and the stuff i do tends to be a bit all over the place

Rrjrjtlokrthjji

Morning!!!

0celo7

uhhh, morning?

Semiclassical

welcome. i have to confess, your name looks like something out of Cthulhu

0celo7

that $f$ is really curvy

@Semiclassical hah

MickLH

03:12

@0celo7 o ya, she curvy

0celo7

I want to read one of those books

Rrjrjtlokrthjji

I noticed that on a mall about a year ago.

@Semiclassical it's just gibberish!

MickLH

@Rrjrjtlokrthjji Damn kids graffiti everywhere... Some thugs just wanna watch the world learn.

Rrjrjtlokrthjji

@MickLH what grafitti? That's decorative paper on the wall! :D

0celo7

@Semiclassical what do you do?

Rrjrjtlokrthjji

03:14

Illuminati!

Semiclassical

i've done a few things

they could be broadly placed under the rubric of asymptotics, but eh. i just do whatseems interesting

0celo7

you have that much freedom?

Semiclassical

i wouldn't put it quite like that, but then i probably put it too broadly

it's just a bit tough to encapsulate the stuff i've done in a neat way

0celo7

One more week until MAJOR KEY is released :D

Semiclassical

that and i don't feel like trying atm :P

0celo7

03:20

Me: hey prof book A is too hard

Prof: ok, take a look at book B

Book B: JET TRANSVERSALITY

Noooooo

Semiclassical

lol

0celo7

03:50

@Semiclassical One of our professors is leaving academia and is giving away his books

My advisor got like 15 books

I'm trying to snag something

Transcript for

$Mathematics$ Mathematics