Mathematics - 2015-11-29 (page 2 of 3)

@Krijn Because, geometrically, you're straightline homotoping in $\Bbb R^{n+1}$ and then projecting to $S^n$ by rays through the origin. If the straightlines pass through the origin, you cannot do that in a well-defined way.

Krijn

@BalarkaSen Nice interpretation, thanks!

Balarka Sen

No problem.

@Krijn This is something more than the hairy ball, but ok. The deal is that degree is an invariant for maps $S^n \to S^n$.

The standard proof tries to break that, assuming a tangent vector field on $S^2$.

By the way, @Krijn, are you aware of the converse of the statement "if two maps of $S^n$ are homotopic, then they have equal degree"?

L33ter

5:25 PM

@BalarkaSen can you do the Borsuk Ulam theorem without knowing the covering spaces and fundmental group of a circle ?

in munkres

Mary Star

Does it stand that $e^{x}-1 \mid e^{kx}-1$ if and only if $k \in \mathbb{Z}$ ?

Mike Miller

You could prove it without ever using algebraic topology, if you wanted.

L33ter

well, I read the the section of homotopy and fundmental group and just a little bit about covering spaces.

but as I was reading the proof yesterday about the question I asked yesterday I think I am missing some background for covering spaces.

and fundmental group of a circle.

I still didn't figure out why qh induces a continous map from $S^1$ to $S^1$

@MikeMiller

Mike Miller

What is the universal property of a quotient map $f: X \to Y$?

evinda

@DanielFischer Could I also ask you something else?

Let $A$ be a $n \times n$ matrix ($A \in \mathbb{R}^{n \times n}$). Then for each $x \in \mathbb{R}^n$ the vector $Ax$ is defined and so we can see the matrix $A$ as a function $A: \mathbb{R}^n \to \mathbb{R}^n$ and this function is linear.

I want to show that if $x_m \overset{||\cdot||_2}{\to} x$ then $A x_m \overset{||\cdot||_2}{\to} Ax$.

There is a hint that we could use the Cauchy-Schwarz inequality.

But how can we use that $|\langle x,y \rangle| \leq ||x|| ||y||, \forall x,y \in X$ ?

Mike Miller

5:38 PM

It's that if $g: X \to Z$ is constant on every level set $f^{-1}(y)$, then $g$ descends to a map $g': Y \to Z$. (That is, there exists a $g'$ with $g'f = g$.) Now apply this to $f = q$, and $g = qh$.

L33ter

oh I see

I didn't know of this universal property

Mike Miller

It is the point, and the only point, of quotient maps and quotient spaces.

Daniel Fischer

@evinda Look at the components of $A(x - x_m)$. Every such component, you can view as the inner product between a row of $A$ and the vector $x-x_m$.

L33ter

I see

evinda

@DanielFischer So you mean that it is like that?

$$||Ax_m- Ax||_2=||A(x_m-x)|_2| \leq ||A|| ||x_m-x||_2 \to 0 \text{ since } x_m \overset{||\cdot||_2}{\to} x$$

Mary Star

5:47 PM

Does it stand that $e^{x}-1 \mid e^{kx}-1$ if and only if $k \in \mathbb{Z}$ ? @robjohn

Daniel Fischer

@evinda That's when you use the operator norm. But to do that, you need to know that $\lVert A\rVert < +\infty$. If you already know that, that's more natural than using Cauchy-Schwarz. But if you don't know that, you can use Cauchy-Schwarz to obtain the result (and a bound for $\lVert A\rVert$).

evinda

It is not given that $\lVert A\rVert < +\infty$. You mean that we use the components of x_m and x to obtain the result? @DanielFischer

Daniel Fischer

@evinda No, we use the rows of $A$ (and the vector(s) $x-x_m$) to obtain the result.

evinda

@DanielFischer How do we use them?

Daniel Fischer

13 mins ago, by Daniel Fischer

@evinda Look at the components of $A(x - x_m)$. Every such component, you can view as the inner product between a row of $A$ and the vector $x-x_m$.

Dominic Michaelis

5:58 PM

hello everyone

evinda

So do we use indeces, $A_1 \cdot (x - x_m)_1, A_2 \cdot (x - x_m)_2 $ and so on... ? @DominicMichaelis

Hello @DominicMichaelis

How are you? @DominicMichaelis

Dominic Michaelis

I am fine but a bit confused

Someone told me without axiom of choice, one isn't able to prove that counatable spaces are lindelöf. But she uses that a set is countable if a bijection into $\mathbb{N}$ exists

Mike Miller

I'm not sure what else countable means.

Dominic Michaelis

And I thought about proving it in the following way: at first I show that $\mathbb{N}$ is countable which I can do via the well ordering of $\mathbb{N}$ so at first I pick some set that covers $1$ and afterwards one which covers $2$ and so on

because of the definition of $\mathbb{N}$ in this way I get a countable subcover

Now it is clear that continuous images of lindelöf spaces are lindelöf and that every function from the natural numbers is continuous

Balarka Sen

Sorry, was away

Dominic Michaelis

6:05 PM

As I have a bijection I am done

Balarka Sen

@L33ter Yes, using homology theory.

Dominic Michaelis

@MikeMiller countable means for me that there is an injection into $\mathbb{N}$

L33ter

I was asking mike a question yesterday @BalarkaSen in some proof of some corollary

but now I get it

I was asking him how come the map qh induces a continuous map k

but that is since q is a quotient map

Balarka Sen

The universal property.

L33ter

yeah

Balarka Sen

6:07 PM

It's in Munkres's chapter on quotient spaces.

Have you not studied it?

L33ter

maybe I did but I don't remember it

oh yeah

I skipped it

Balarka Sen

It's a useful tool.

L33ter

its theorem 22.2 I skipped reading i

it*

yeah I will read it quickly

Mike Miller

@DominicMichaelis: I see, thanks.

robjohn

6:23 PM

@MaryStar usually $a\mid b$ is defined for integers $a$ and $b$ or possibly for polynomials. How are you defining it here?

Mary Star

When we consider the ring $\mathbb{C}[e^{\lambda x} \mid \lambda \in \mathbb{C}]$ is the divisibility defined? @robjohn

robjohn

6:42 PM

@MaryStar How would you define divisibility there so that it is not useless? e.g. divisibility on $\mathbb{R}$ is useless since everything is divisible by everything else except $0$.

Mary Star

Aren't the elements of the ring polynomials in $e^{\lambda x}$ ? So is the divisibility defined as for polynomials? @robjohn

robjohn

@MaryStar Why are you writing them as $e^{\lambda x}-1$ if you are simply thinking of the polynomial $x^\lambda-1$?

You seem to be making things more difficult

Mary Star

But in the exponential ring aren't the elemens of the form $e^x$ ? I got stuck right now... @robjohn

Julian Rachman

Can I ask someone a simplicial set question?

robjohn

@MaryStar without more clarification, I don't know what ring you are actually talking about. Are you talking about a ring of polynomials in $e^{\lambda x}$ with coefficients in $\mathbb{C}$. That is $c_0+c_1e^{\lambda x}+c_2e^{2\lambda x}+c_3e^{3\lambda x}+\dots+c_ne^{n\lambda x}$ where $c_k\in\mathbb{C}$?

Julian Rachman

6:53 PM

Let alone anyone that knows category theory?

Mary Star

Yes @robjohn

robjohn

@MaryStar Then how does the divisibility differ from polynomials in $x$ with coefficients in $\mathbb{C}$? That is, $c_0+c_1x^1+c_2x^2+c_3x^3+\dots+c_nx^n$?

Why make things more difficult?

Balarka Sen

@JulianRachman "just ask, don't ask to ask"

Julian Rachman

I am not asking to ask @BalarkaSen I don't want to waste my time if there is no one to speak to about my problem

That can actually answer it

Balarka Sen

There might be people in here who knows about simplicial sets. Anyway, a place where you might get definitive answers is the homotopy theory chat (but I usually hesitate asking questions there unless it's very nontrivial).

Mary Star

6:59 PM

So do you mean the following?
$f(e^{\lambda x}) \mid g(e^{\lambda x}) \Leftrightarrow \exists r(e^{\lambda x}) : g(e^{\lambda x})=f(e^{\lambda x})r(e^{\lambda x})$
@robjohn

Balarka Sen

Is the question about a technical aspect, or just some theoretical question? I know what a simplicial set is, and I know why it's there but any technical question I defer to homotopy theorists.

robjohn

@MaryStar How is that different than $f(x)\mid g(x)\iff \exists r(x):g(x)=f(x)r(x)$ where all are polynomials?

Balarka Sen

How's things, by the way, @Julian?

Julian Rachman

Things are moving alright for now. And you, @BalarkaSen

Balarka Sen

I'm fine. What kind of math are you studying?

Mary Star

7:08 PM

@robjohn It is the same, just the variable is different, isn't it?

robjohn

@MaryStar I guess. This is completely divorced from any context, I don't know if this is actually applicable to how it needs to be applied.

Balarka Sen

@JulianRachman re: your question on homotopy theory - why do you want to think about categories where your hom sets are not sets?

Julian Rachman

@BalarkaSen I am currently studying Algebra and category theory from Aluffi and trying super hard to get algebraic topology into the mixed because the project I am working on is taking up a lot of time

Balarka Sen

that's a very odd thing to do, because a lot of important categorical theorems do not work there.

Julian Rachman

@BalarkaSen ? Could you elaborate?

Mary Star

7:14 PM

So the implication $e^{x}-1 \mid e^{kx}-1 \Leftrightarrow k \in \mathbb{Z}$ stands, right? @robjohn

Balarka Sen

@JulianRachman I see. A word of caution: simplicial sets are motivated by a very concrete theorem in classical homotopy theory (the nerve theorem: essentially, it says to do homotopy theory you only need to take care of the combinatorial data of the space, so homotopy theory is just a form of combinatorics), so you'd better off learning classical alg top first before doing them.

@JulianRachman Elaborate on what?

robjohn

@MaryStar if you are only dealing with polynomials, it would seem so. Since I have no grasp on the context, I can't make a definitive statement.

Paradox 101

Can I ask a question regarding revised simplex method?

Julian Rachman

On your re: to my question on homotopy theory @BalarkaSen

Balarka Sen

Yeah, I don't know what to elaborate.

Small categories need not have hom-sets as sets. They're just proper classes. So set theory is not going to work properly.

Why think about such pathological categories?

Tobias Kildetoft

7:21 PM

@BalarkaSen You mean non-small

Balarka Sen

Sorry, yeah, I mean large categories.

:P

L33ter

balarka

Balarka Sen

Yeah, @L33ter?

L33ter

did you see some topology videos by some prof

on youtube

they are good

Tobias Kildetoft

usually locally small suffices for most purposes (mainly because most locally small non-small categories one might be interested in are equivalent to some small category)

Balarka Sen

7:23 PM

By what prof?

L33ter

AlgTop26: Covering spaces

►

I think he comes to mse aswell

Balarka Sen

Aren't Wildberger's videos bad places to learn stuff?

L33ter

I saw couple of them I didn't mind it

Balarka Sen

Probably I'm wrong, but no, I did not see them. I attended algebraic topology classes.

L33ter

but I prefer munkres

I actually always prefer books rather than learning from people

Tobias Kildetoft

7:25 PM

@L33ter I don't think I would want to learn algtop from a guy that doesn't like infinite sets

Balarka Sen

Talking to people helps you acquire intuition and perspective, which books do not have.

L33ter

yeah

yeah I agree @BalarkaSen

actually last summer I had a reading class in advanced mechanics what we did is that I learned subject on my own then I meet with prof once a week to discuss what I learned

I really liked this type of learning

Tobias Kildetoft

@L33ter Have you read this question and its answers related to that guy? math.stackexchange.com/questions/356264/…

L33ter

no I didn't @TobiasKildetoft

Tobias Kildetoft

Apparently, he does not believe in infinite sets, which must make algtop somewhat tricky (or very easy of course)

Balarka Sen

7:28 PM

That was why I hypothesized that learning from his lectures must be a bad idea.

L33ter

lol

people you know think math is like physical world thing

they don't realize it is a logical system that depends on axioms

like giving analogy of math to real physical world is very bad thing.

ofcourse there is alot of uses of math to the physical world

for example Lagrangian mechanics etc

Balarka Sen

And a lot of uses of physics in math...

L33ter

yeah

but yeah I agree with Asaf's answer

Krijn

7:56 PM

@BalarkaSen I was aware of the statement, not the converse of it

Balarka Sen

The converse says any two maps $f, g : S^n \to S^n$ of equal degree are homotopic :)

I felt pretty awed by this fact when I first heard it.

L33ter

hey so in the universal property rpoof

proof *

For each $y \in Y$, the set $g(p^{-1}(\{y\}))$ is a one point set in Z.

if we let f(y) denote this point then we have defined a map f(p(x)) = g(x)

why is that ?

wouldn't that be always constant ?

@BalarkaSen ?

Balarka Sen

was away, I'm sorry.

L33ter

no its ok

Balarka Sen

@L33ter I don't have Munkres with me, no idea about the notations.

L33ter

8:10 PM

1 sec I will take a pic

postimg.org/image/svg7ozn61

let is say that $g(p^{-1}(\{y\})) = c$, where c is some constant.

so do they mean here that f(y) = c for all y in Y ?

or how did they define f ?

Balarka Sen

No, you define $f(y) = c$.

L33ter

I don't understand how did we get f(p(x)) = g(x) then

Balarka Sen

Also, $c$ is not constant. It depends on $y$.

For each $y \in Y$, $g(p^{-1}(y))$ is a point in $Z$. So you define the map $f : Y \to Z$ as $y \mapsto g(p^{-1}(y))$.

L33ter

I see

Balarka Sen

@L33ter $f(p(x)) = g(p^{-1}(p(x))) = g(x)$.

L33ter

8:18 PM

I see ok yes that makes sense

thank you

user174558

@I'mmostlyjustanidiot Are you Alex in disguise?

user174558

@BalarkaSen I am going to take a look at Munkres again. Maybe it isn't so bad after all.

Balarka Sen

@L33ter Note that the second equality comes from the fact that $g$ is constant on each preimage. So regardless of any element of $p^{-1}(p(x))$ you choose, $g$ of that is going to be $g(x)$.

L33ter

yeah

I figured that out because I was just about to ask

Balarka Sen

@JasperLoy Munkres is good. Has nice exercises.

L33ter

8:26 PM

how come g(p{-1}(p(x))) = g(x) we don't know if p(x) is injective

but I realized yeah it is constant on the pre-image

Balarka Sen

$p^{-1}(p(x))$ is just a notation for the whole fiber over $p(x)$.

L33ter

yeah

Balarka Sen

PS, Karim, I wouldn't recommend Wildberger's lectures. He's sometimes sloppy with notations and the intuitions he convey aren't quite accurate either.

Well, to be honest, intuitions are never accurate but the pictures he draws are not quite standard, or so I gather from seeing a few of his lectures.

Plus, I don't like his notations.

L33ter

alright

I will just stay with books prof and you guys :D

next semester I will be taking algebraic topology and will meet with prof weekly

so hopefully will be able to cover alot next semester

ambigram_maker

Anyone an expert on binomial-theorem here?

I have a question.

Balarka Sen

8:36 PM

By the way, can you see why the universal property implies $\pi_1(X, x_0)$ can be analogously defined as homotopy classes of maps from $(S^1, s_0)$ to $(X, x_0)$, relative to $s_0$?

hi @SanathK.Devalapurkar

user105491

Good day.

Balarka Sen

How's life?

user105491

It's pretty good, you?

user105491

I'm trying to learn theoretical computer science now, so less algtop haha

Balarka Sen

By life, I mean mathematical life. (What else kind of life is there?)

@SanathK.Devalapurkar I'm cool, went to a topology & condensed matter physics conference a few days ago.

user105491

8:39 PM

Nice, what did they talk about?

L33ter

hm thinking 1 moment

Balarka Sen

@SanathK.Devalapurkar Ah, then I believe Sanath will turn out to be a homotopy type theorist :P

user105491

@BalarkaSen No comment.

Balarka Sen

lol.

user105491

lol I'm just kidding, I don't know what I'm going to become yet.

user105491

8:40 PM

Evolutionary biology is also really appealing right now.

Balarka Sen

@SanathK.Devalapurkar Um, bunch of stuff. I didn't really understand much. There was a lecture on K-theory and physics where the word K-theory was never really mentioned.

user105491

As is theoretical physics.

user105491

Oh wow

user105491

I only know a little about general relativity and QFT

user105491

(from a physics perspective, that is)

Balarka Sen

8:42 PM

Pretty interesting way to lure mathematical people into lectures, right?

user105491

lol yup, for sure

Balarka Sen

I don't know any QFT, unfortunately.

user105491

I only know the basics

Balarka Sen

I know what a topological QFT is, but I never understood where the field theory jumps in.

user105491

at least I should say I only know whatever is established in Feynman-Hibbs

user105491

8:43 PM

(Atiyah's original paper is great btw for physics and TQFTs)

user105491

so how's (mathematical) life for you?

Balarka Sen

@SanathK.Devalapurkar I'll note that down. I learnt TQFTs from Lurie.

user105491

Oh man Lurie's cobordism hypothesis paper is dense

user105491

at least physics-wise it doesn't give you intuition

Balarka Sen

I didn't really went through everything. The first 2 chapters were exciting and very easy to understand, iirc.

user105491

8:44 PM

although it's great mathematically, of course; Lurie's exposition is simply amazing.

Balarka Sen

I learnt about them a year ago.

user2154420

If a continuous function from R^2 to R^2 has |f(x)-x|<epsilon for every epsilon does this guarantee that f(R^2)=R^2?

Balarka Sen

@SanathK.Devalapurkar ah, indeed

@SanathK.Devalapurkar I'm great. Do you know any geometric topology?

user105491

No, I don't think I do.

Balarka Sen

Out of curiosity. I have been getting slowly lur(i)ed into that stuff lately.

user105491

8:46 PM

hahaha your puns really are becoming better

Balarka Sen

:P

user105491

I think I'm getting pulled more into number theory now.

Balarka Sen

Ah, welcome to the dark side. What kind of NT?

I don't know much, but I was once interested in it.

(still am)

user105491

I do recall that :-)

user105491

Elliptic curves

user105491

8:48 PM

I'm currently working under Noam Elkies, who I presume you've heard about

Balarka Sen

So, 2-dimensional representations of Gal(Q). Good stuff.

Yeah, I know, you told me that you were working under him.

user105491

mhm

user105491

Anyway, what are you learning right now?

Balarka Sen

Do you understand how 2-dim reps of Gal(Q) arise from modforms yet? I still don't grok it.

@SanathK.Devalapurkar I am trying to get some multicalc straight because I want to get to differential topology. I did some basic stuff from Guillemin-Pollack during my stay on the topology & condensed matter physics conference.

Want to learn some Morse theory.

user105491

@BalarkaSen Unfortunately, I don't understand it.

Balarka Sen

8:54 PM

@SanathK.Devalapurkar A question I had: Serre (?) posed the question of whether there's an easy way to see if two algebraic varities X/\bar Q and X'/\bar Q belong to the same Gal(Q)-class (Gal(Q) acts on algebraic varieties over \bar Q naturally). Do we know the answer for elliptic curves?

All I know is that Grothendieck did some work on this for algebraic curves, using the theory of dessins. I think it's an open-ended problem, though.

user105491

@BalarkaSen I didn't know about that. Do you have a link to a reference?

Balarka Sen

No, I don't think I have a reference :)

user105491

Ah, ok :-)

Balarka Sen

what nonsense at 15:30 here.

up to that point, he was being sloppy. now he's making downright false claims.

user105491

lol wut

user105491

9:02 PM

R/Z is isomorphic to the circle group, its torsion subgroup is isomorphic to Q/Z haha

Balarka Sen

i feel sorry for the students attending that lecture, and the ones watching this.

user105491

So do I

Krijn

Is he not also the same guy that says infinte sets do not exist or something like that

Balarka Sen

yeah

user105491

that's incredible.

Frank Science

9:08 PM

@BalarkaSen Is that related?

Krijn

I do agree with his claim that we are often too nonchalant with infinity

Frank Science

He sticks to rational numbers.

Balarka Sen

oh lord

sparebytes

9:20 PM

I just watched this video on e^(pi*i) = -1:
https://www.youtube.com/watch?v=F_0yfvm0UoU

He says that numbers can be thought of as 3 things, a point on a number line, an adder, and a multiplier. He says combining 2 adders results in another adder, combining 2 multipliers results in another multiplier, and that e^x takes an adder (x) and turns it into a multiplier. Does this actually make sense to anyone? ... What do you get when you combine an Adder and a Multiplier?

Krijn

I do not understand what he is trying to say either. Maybe he is trying to explain in laymens terms that $a \mapsto e^a $ is a group homomorphism from an additive group to a multiplicative group?

Balarka Sen

9:42 PM

ok, after some research, I am now totally awed by Wildberger's view of mathematics. He's the greatest math crank I have ever known.

L33ter

@BalarkaSen web.maths.unsw.edu.au/~norman

Balarka Sen

Yeah, it beats me how he is an associate professor of a university.

L33ter

he even got his phd from yale ...

Danu

@BalarkaSen Is this the guy with the algebraic topology course on youtube? :P

Balarka Sen

Yes.

Huy

9:50 PM

is that like the JD of maths?

Danu

^lol

Balarka Sen

what's the JD?

Danu

Well this guy actually has a degree

@BalarkaSen "that guy" in the h bar that always spouts nonsense

Balarka Sen

ah.

L33ter

ok good @BalarkaSen I am starting to understand now the proof of h isn't nulhomotopic

good

Balarka Sen

9:52 PM

What is $h$?

Danu

In physics?

Huy

^lol

Danu

$\hbar$ is one of the fundamental constants of nature

L33ter

that is

Danu

It is derived from $h$ by dividing it by $2\pi$

L33ter

9:53 PM

If $h : S^1 \rightarrow S^1$ is continuous and antipode preserving then h is nulhomotopic.

Danu

Planck constant

The Planck constant (denoted h, also called Planck's constant) is a physical constant that is the quantum of action, central in quantum mechanics. First recognized in 1900 by Max Planck, it was originally the proportionality constant between the minimal increment of energy, E, of a hypothetical electrically charged oscillator in a cavity that contained black body radiation, and the frequency, f, of its associated electromagnetic wave. In 1905 the value E, the minimal energy increment of a hypothetical oscillator, was theoretically associated by Einstein with a "quantum" or minimal element of the...

Huy

@Danu: I don't think $h$ or $2 \pi$ exist, really.

Much like $\infty$.

Danu

^lol?

L33ter

I have to finish understanding it today as Ulam Borsuk theorem follows as corollarly

I will also do the bisection theorem today

then I will rewrite my own notes for the thing tomorrow

because I will present it next week

the bisection theorem seems very interesting

is there also other interesting application of ulam borsuk theorem ?