Mathematics - 2018-01-21 (page 2 of 6)

Secret

05:00

$\Bbb{R}^0$ is just a point?

Akiva Weinberger

Yeah

Daminark

Yeah

Sniped

Secret

well, I don't know how one can even define tangent vectors of isolated points

orbit-stabilizer

360

@AkivaWeinberger thanks for the paper. it's a fun read

Daminark

@Secret it's 0 dimensional stuff so gonna be one of those things where you define it by convention so that your theorems work nicely

(e.g. degree of 0 polynomial being negative infinity)

Akiva Weinberger

05:04

I guess it's a vector in a 0-dimensional vector space

Secret

yeah, 0 dimensions are weird..
(The polynomial case kinda make sense though, since there is no well defined powers of x in the zero polynomial and defining its degree that way will ensure its degree stays unchanged when you differentiate it)

Akiva Weinberger

$\{0\}$

Oh, my convention was always that $\deg(0)=\pi$

Daminark

Lmao

Akiva Weinberger

Heh, apparently this sort of thing is only possible for 1, 2, 4, or 8 squares

(product of the sum of N squares being the sum of N squares)

Secret

(Actually might play with this idea later in mathworks: Define an alternate polynomial like algebra such that it has the property that $p \neq 0 : deg (\frac{d p}{dx}) = deg (p)$)

Akiva Weinberger

05:07

It's moot anyway since every number is the sum of four squares, and thus the sum of N squares for all $N\ge4$ (fill it up with copies of $0^2$)

but I guess they still work in more general settings like rings and whatnot

Daminark

Well, one example use of the negative infinity convention is that it is now true that $deg(fg) = deg(f) + deg(g)$ without having to specify that it's non-zero

etc etc

Also hello @Ted!

Akiva Weinberger

There's a similar thing with 16 squares but you need division; it expresses it as the sum of 16 squares of rationals

Hey, Ted! We were talking about how $\sin$ was analytic (someone was unsure), and then discussed the radius of convergence of $\dfrac1{e^x+1}$ which then managed to get segued into a discussion about the quaternions and normed division algebras in general

Secret

OEIS don't have finite sequence, thought I can just lookup how many times 1,2,4,8 appeared in mathematics

Akiva Weinberger

There's that one which goes 1,2,4,8,16,31

about cutting up a circle

Balarka Sen

Alerting the drama here

What are we doing?

Akiva Weinberger

05:12

Found it

Secret

Balarka: Akiva give a nice summary on what happened in the past hour

Semiclassical

And then Secret summed up for Balarka how Akiva summed up the last hour to Ted

(I won't continue)

Secret

And then repeat the above $\omega$ times :P

Akiva Weinberger

Re the 1,2,4,8 thing, that's a nice puzzle. I can't remember why it does that, though

Balarka Sen

@Akiva @Daminark Small subtlety: $TS^2$ is not isomorphic to $S^2 \times \Bbb R^2$ as a vector bundle over $S^2$ because of hairy ball. There's a little bit of trickery (We Are Number One tune starts) involved in proving that they are not homeomorphic as manifolds.

Secret

05:15

There's a proof for the normed finite division algebra case (which I never get to look in detail), but for the sum of squares identity, I have no idea

Akiva Weinberger

@BalarkaSen I mentioned that

'round here

Or do you mean to say that that's not the only reason?

Balarka Sen

Well that's the reasoning to prove they are not isomorphic as vector bundles.

Akiva Weinberger

@Secret Oh, no, I meant the image

with the circle divisions

Daminark

@Balarka is it easier to prove that they're not diffeomorphic?

Akiva Weinberger

You have $n$ points on a circle, join them together with lines such that no three are concurrent, as in the image, and ask how many regions you get

Balarka Sen

05:17

Yes, mildly.

It's hairy ball theorem, truly, but a little fuckery is required. $TS^2$ has a submanifold with self-intersection number 2

$S^2 \times \Bbb R^2$ has no such thing

Akiva Weinberger

@BalarkaSen Hm. I guess it's not too hard to triangulate $TS^2$

Do the homology groups agree?

Balarka Sen

They do :)

I mean both are homotopy equivalent to S^2

Contract the plane fibers

Akiva Weinberger

Oh, right

Balarka Sen

What does not agree is the cup product structure

Akiva Weinberger

Doesn't that only depend on homotopy type?

Balarka Sen

05:19

And not even that. It's the compactly supported cup product structure

@AkivaWeinberger Not if you think compactly supported. Consider the Moebius strip and the actual strip

If $\alpha$ is the center circle,

Akiva Weinberger

Minus boundary (so they're not compact)?

Balarka Sen

Right

$\alpha \cup \alpha$ (compactly supported cup product) is nonzero in the first and zero in the latter

Because perturbing the center circle a little bit makes it intersect back with itself

Akiva Weinberger

Makes sense

Would this be different for the compact versions (with the boundaries)?

Balarka Sen

I gotta go

Sorry

Akiva Weinberger

Give me a yes or no, at least!

Well then.

But yeah I guess in $S^2\times\Bbb R^2$, you can shift your canonical copy of $S^2$ in there "up" a bit so it doesn't intersect itself anymore

but for the canonical copy of $S^2$ in $TS^2$, doing so would solve the hairy ball problem.

@BalarkaSen I'm gon' ping you so you can answer my question when you get back and/or wake up

Ted Shifrin

05:33

Howdy, DogAteMy

Thanks for the hello, Demonark.

I just figured out I can make a video with my doc cam or even live stream. Amazing, technology. Still, pretty boring to just write on a piece of paper and drone on — I preferred having live students to interact with.

Ah, I see the difficult question came up about whether non-isomorphic vector bundles can be homeomorphic (diffeomorphic) as manifolds.

And on cue MikeM shows up.

Akiva Weinberger

Doc cam?

Ted Shifrin

Document camera ... your teachers probably use those in their classrooms occasionally.

Adeek

hi @TedShifrin

Akiva Weinberger

They don't

Ted Shifrin

hi Karim

Akiva Weinberger

05:37

but suddenly I want one

Ted Shifrin

ROFL

Adeek

I think my master thesis will be like 150 pages @TedShifrin

:D

I am happy

Secret

How interactive are your classes?, most math lectures I been to often have professors using doc cams alot and wrote stuff on, and some even just decided to play youtube videos of their prerecorded lectures

Ted Shifrin

Even I used one occasionally in college teaching, DogAteMy. Projecting a book up on the screen or something from someone's piece of paper.

Akiva Weinberger

Because, you know, how else will I post videos to YouTube of me drawing on a piece of paper?

Other than making something out of cardboard to prop my phone on, I guess

Adeek

05:39

@TedShifrin if everyone had your teaching abilities then I would like attending classes

Ted Shifrin

Secret: I had a lot of interaction with students. And in certain classes I tried to get them to tell me how to do proofs, make up examples, etc. Particularly did that in algebra and point-set topology.

Adeek

With advanced math I noticed that people can't really teach it

unless they are talented in teaching like you @TedShifrin :)

Akiva Weinberger

I imagine it's a practice-makes perfect sort of thing

Ted Shifrin

I have known a surprising number of excellent teachers, Karim, but also some horrid ones.

Akiva Weinberger

but if you don't get a lot of feedback it's hard to improve

Ted Shifrin

05:40

The geometry kids in the AoPS class I substituted for were extremely interactive (a bunch of 12-13 yr olds, I think).

Akiva Weinberger

Maybe it's easier to, like, "train" yourself by tutoring one person about something, before trying to do it in front of a class

Adeek

yeah

Also, with advanced math you need to introduce all ideas

Ted Shifrin

DogAteMy, our IT fellow back at UGA tried to talk me into using a regular webcam with a whiteboard on an easel or something, but I think I made the right decision.

Adeek

because missing 1 key idea can ruin people focus in class.

Secret

Ted: That's good, cause often the students have to just do it in order to understand some of the proofs, kinda like in fine arts where one just have to do it before they will get what will happen and why it is the case.

Abstract agebraic proofs are often quite abstract for most students thus guiding them should help them to learn better on what kind of thinking process to be used and proof strategies, I think...

Adeek

05:41

especially with like advanced things

Ted Shifrin

One-on-one tutoring is surprisingly different from teaching, DogAteMy. But it gives you some insights, if you do enough, so that you know what students will find difficult or misunderstand.

Akiva Weinberger

What you want to do is have everything relevant on the board so that they need to use their memory as little as possible

Ted Shifrin

Our set-up in the UGA classrooms was terrible. When I used Mathematica, displaying it on the screen, the screen covered almost all the blackboard. Terrible design. I bitched.

Akiva Weinberger

Same reason I think there's no point in trying to do hard multiplication problems in your head if you have paper available

Adeek

@TedShifrin did you know that there is connections between algebraic curves and differential equations ?

that is kinda very weird !!

Ted Shifrin

05:43

So, DogAteMy, we gonna have our own exploring differential topology YouTube channel? :P

Akiva Weinberger

At one point I wanted to make a thing on Brouwer's fixed point theorem, which never happened

Ted Shifrin

Semiclassic can give you a lecture on that, Karim. Also the whole notion of monodromy ties together both algebraic varieties and differential equations with singularities.

Akiva Weinberger

I mean, it still can, it just hasn't happened yet

Secret

Akiva: Some problems are simply way too hard to do mentally, for me, if a problem have more than three inequalities, then I have to do it on paper to even get my bearings

Adeek

@TedShifrin very cool

Ted Shifrin

05:44

DogAteMy: Here's a problem I'm going to give my AoPS kids tomorrow. You might like it.

Adeek

@Semiclassical maybe you can tell me something about this ?

Akiva Weinberger

I think part of the reason 3Blue1Brown so good is that he generally leaves important stuff on screen

Ted Shifrin

Take a regular $n$-gon inscribed in the unit circle. What is the product of the lengths of the segments from one vertex to the $n-1$ remaining vertices?

Akiva Weinberger

and draws your attention to the parts of the screen that's relevant at a particular moment

Ted Shifrin

I miss blackboard teaching.

Akiva Weinberger

05:45

@TedShifrin That's a classic, isn't it?

Ted Shifrin

Oh, is it?

I put it in my algebra book, but I don't remember where I found it (or if I found it at all).

Akiva Weinberger

It's $\prod|1-\omega|$ over the roots of unity.

Ted Shifrin

Well, of course. :)

We're doing roots of unity tomorrow.

Akiva Weinberger

And $\prod(x-\omega)$ is $x^{n-1}+\dotsb+1$.

Ted Shifrin

And no, I'm not teaching them Galois theory.

Akiva Weinberger

05:46

So then you get $n$?

Ted Shifrin

Aha. That's the easiest way to do it, yes. There are other ways :)

Akiva Weinberger

And you can check for an equilateral triangle ($\sqrt3\times\sqrt3$) and for a square ($\sqrt2\times2\times\sqrt2$) to be sure

Ted Shifrin

Another way is to let $u=1-\omega$, so $u$ satisfies a polynomial and you want the product of its roots.

Akiva Weinberger

What's that end up being, $(1-x)^n-1$?

Ted Shifrin

Nope.

orbit-stabilizer

05:48

nasa.gov/astronauts/biographies/jonny-kim/biography

this is crazy

Akiva Weinberger

Oh, that divided by $x$?

Ted Shifrin

OK.

Akiva Weinberger

@orbit-stabilizer What about it?

Ted Shifrin

@orbit: Crazy cuz he's been through medical school? Interesting he went to UCSD and majored in math. I wonder if @PVAL knew him.

orbit-stabilizer

seal team 3, math undergrad, MD from harvard, now he's becoming an astronaut

he's only 33

Akiva Weinberger

05:50

Honestly I thought there was only one SEAL team

and we just called it 6 so that the enemy would think there's more of them

orbit-stabilizer

Dunno, not american.

Ted Shifrin

Oh, I missed that he enlisted right out of high school before he went to college 6 years later.

orbit-stabilizer

But, damn, he is incredibly driven. And he has a wife and kid.

Akiva Weinberger

> He will report for duty in August 2017.

I see that updating this website was not anyone's priority

orbit-stabilizer

and he's a time traveler!?!

Akiva Weinberger

05:53

Lol

orbit-stabilizer

stop the simulation.

now it's just too obvious

Daminark

@TedShifrin Drat

Adeek

what is the best way of putting an arrow with on the top of an arrow name of a function

like f denoting function from X to Y ?

Daminark

This is complete overkill for just that task but...

Ted Shifrin

So, it came across my Facebook feed (and I've shared it) that if you call the White House feedback # to speak with someone, they give a biased diatribe about how the Democrats shut down the government and so no one can take your call.

\vec, Karim

although for things that are bigger you need \overrightarrow ... can always write a macro.

Akiva Weinberger

05:55

$A\xrightarrow fB$

\xrightarrow

Adeek

nice thanks

orbit-stabilizer

Akiva Weinberger

@TedShifrin Little letter on an arrow, not the other way around

Ted Shifrin

Oh, I misread.

No, you misread.

He wants something like $\vec f\colon X\to Y$.

Or did he mean $X\overset{f}\to Y$?

Akiva Weinberger

ha

$\xrightarrow{\vec\to}$ONWARD

Ted Shifrin

05:58

His sentence was garbled. I quit.

Akiva Weinberger

FedE$\to$x

Ted Shifrin

Karim: You should probably learn about using some CD (commutative diagram) stuff, too, although it doesn't do non-horizontal/vertical lines.

OK, I'm going back to Australian Open tennis.

Bye.

Akiva Weinberger

Watching? Participating? Insulting?

Ted Shifrin

Yes.

Akiva Weinberger

Have fun.

Ted Shifrin

06:00

Night :)

PVAL-inactive

@Ted I don't think he was on the research track.

He looks vaguely familiar.

Adeek

oh yeah @TedShifrin

I use XY pic

nights btw @TedShifrin

Faust

Morning] @TedShifrin

Akiva Weinberger

Let's all ping him as many times in a row as we can

Secret

Diagram of an ordinal sardine:

Faust

06:09

i think thats the wierdest thing ive seen so far today...

congrats @Secret

Secret

perfect fusion of biology and set theory

Yashas

06:23

Does power set exist for an uncountable set?

Is the power set of an uncountable set a sigma algebra?

Secret

power sets is defined for any set

and the resulting set is strictly of larger cardinality than the set itself

Akiva Weinberger

@Yashas Sigma algebra means that it's closed under countable unions and intersections, as well as complements, right?

And the empty set and full set are in there

The power set of any set is a sigma algebra, yes.

It's closed under arbitrary unions and intersections, which means it's definitely closed under countable unions and intersections.

Balarka Sen

I am back

Akiva Weinberger

YAY

1 hour ago, by Akiva Weinberger

Would this be different for the compact versions (with the boundaries)?

Balarka Sen

Well, you need to use an awkward notion of cup product again. The self-intersection number of the center circle does not change (it's 2 and 0 respectively)

Akiva Weinberger

06:34

I suppose at some point I should sit down with a Hatcher, a pencil, and a piece of paper and just work it out by hand

A Hatcher, a pencil, a piece of paper, and a hand

Balarka Sen

I need to work out a calculation explicitly

:(

Differential forms are too hard

Akiva Weinberger

Unrelated

Balarka Sen

06:49

@Akiva Btw, I think just one-point compactifying and thinking about cup product on that compact space is enough

Akiva Weinberger

The one-point compactifiction of $X\times\Bbb R$ for any $X$ sounds annoying

Balarka Sen

Nah, it's $X \times [0, 1]/X \times \{0, 1\}$

Akiva Weinberger

It's like $SX$ but you glue the two cone points together

Balarka Sen

For compact $X$

anon

the smash product of the one-point compactification of X with the one-point compactification of R?

Balarka Sen

06:50

I am pretty sure

Akiva Weinberger

Yeah, sure, I just don't like the idea of a double cone point.

Although, actually, this is $S^2\times\Bbb R^2$, not $S^2\times\Bbb R$.

Random side note: Apparently $\rm\Bbb OP^2$ exists but $\rm\Bbb OP^3$ doesn't?

Balarka Sen

@AkivaWeinberger Should just be homotopy equivalent to $\Sigma X \vee S^1$ by "thinning away" at the cone point

Akiva Weinberger

And $\rm\Bbb OP^1$ as well, though I think that's just $S^8$

@BalarkaSen Hm, yeah

PVAL-inactive

S^7

Akiva Weinberger

Why? $\rm\Bbb CP^1$ is $S^2$, it seems to be the same dimension as the field it's on

Balarka Sen

06:55

@anon Would that be "smash product of X with the one point compactification of R" if X is compact? That cannot be right; consider X = S^1

PVAL-inactive

wait nvm

Akiva Weinberger

What's the one-point compactification of $S^2\times\Bbb R^2$? It sounds like you get a higher-dimensional cone point

PVAL-inactive

Do actual topology please.

Akiva Weinberger

$S^2\times S^2/S^2\times\{0\}$?

Balarka Sen

That's it.

Akiva Weinberger

06:57

Which I can't visualize well enough to homotope it to something nicer

Balarka Sen

Whereas the one point compactification of $TS^2$ is $S^2 \times S^2$ mod the diagonal copy of $S^2$

I think

Akiva Weinberger

(like you can for the other thing being the wedge of a thing and a circle)

And I guess the diagonal of $(S^2)^2$ sits differently in the space than a horizontal section does?

Balarka Sen

Very differently.

Akiva Weinberger

In a way that's not true for $(S^1)^2$ (torus) or $(S^3)^2$

Balarka Sen

The diagonal has self-intersection number 2

@Akiva Right

Transcript for

$Mathematics$ Mathematics