Mathematics - 2019-12-22 (page 2 of 6)

Balarka Sen

02:00

@TedE Depends. I can visualize $\Bbb{RP}^2$ reasonably well. I think nothing with torsion fundamental group embeds in $\Bbb R^3$, but I wouldn't be able to give you a proof.

Akiva Weinberger

@TedE Yeah the real summary is that I'm dumb and can't do physics and that Google searches can take a liter of 30C water to about 30.25C

@skillpatrol Wow

Wait sorry five months?

skillpatrol

1 pound /week

Akiva Weinberger

How long are World Chess Championships usually

Also now I'm wondering about physical limits on the energy consumption of computing

Landauer's principle

Landauer's principle is a physical principle pertaining to the lower theoretical limit of energy consumption of computation. It holds that "any logically irreversible manipulation of information, such as the erasure of a bit or the merging of two computation paths, must be accompanied by a corresponding entropy increase in non-information-bearing degrees of freedom of the information-processing apparatus or its environment".Another way of phrasing Landauer's principle is that if an observer loses information about a physical system, the observer loses the ability to extract work from that system...

Semiclassical

bah, sniped

Akiva Weinberger

There's also this page

Limits of computation

The limits of computation are governed by a number of different factors. In particular, there are several physical and practical limits to the amount of computation or data storage that can be performed with a given amount of mass, volume, or energy. == Hardware physical limits == === Processing and memory density === The Bekenstein bound limits the amount of information that can be stored within a spherical volume to the entropy of a black hole with the same surface area. Thermodynamics limit the data storage of a system based on its energy, number of particles and particle modes. In p...

which discusses lots of similar topics

Wait it's proportional to the operating temperature of the computer?

nbro

02:10

@AkivaWeinberger What are you studying?

skillpatrol

> According to Ashley, India's first grandmaster, Viswanathan Anand, does two hours of cardio each night to tire himself out so he doesn't dream about chess

nbro

Good to know!

Akiva Weinberger

@nbro Currently on winter break wooo

Rithaniel

Akiva studies whatever captures his attention, I believe

Kind of like me

skillpatrol

@nbro espn.com/espn/story/_/id/27593253/…

Akiva Weinberger

02:12

but I'm gonna major in math

nbro

@AkivaWeinberger Yeah, but, of course, I was not wondering about that

Ted E

Do you play chess @AkivaWeinberger

Akiva Weinberger

(Don't technically have to declare a major until my sophomore year I think)

Ted E

I'm playing the bullet time ones

Akiva Weinberger

@TedE No

nbro

02:13

@Rithaniel So, according to the same Wikipedia page, ${\displaystyle P(A\mid C)=\sum _{n}P(A\mid C\cap B_{n})P(B_{n}\mid C)}$

Akiva Weinberger

I'm probably gonna take complex analysis next semester @nbro

if that's more like what you mean

nbro

No, I mean what kind of degree are you pursuing.

Akiva Weinberger

I'm a freshman undergrad I don't really need to know that yet

nbro

Yes, but what kind of bachelor's are you getting?

Rithaniel

Yeah, that's right. I haven't worked through the equations in the paper you were referencing, but whenever I see sums of conditional probabilities like that, I think "LoTP"

Akiva Weinberger

02:15

BS

nbro

BS means bachelor of science, but in what? Mathematics, computer science, etc?

Akiva Weinberger

Math

nbro

Ha, ok

What about you @Rithaniel?

Akiva Weinberger

4 mins ago, by Akiva Weinberger

but I'm gonna major in math

Balarka Sen

@TedE A visualization of torsion in fundamental group might be: Consider configuration spaces of two points in $\Bbb R^2$. This literally means all unordered pairs of points $\{p, q\}$ such that $p \neq q$. This would be $(\Bbb R^2 \setminus \Delta)^2/\Bbb Z_2$

nbro

02:17

@AkivaWeinberger Yes, I read that, but that's the major

Even though if you major in math you probably only studied math before

Balarka Sen

Then the loop traced out by switching the position of the two points from say, $(1, 0)$ and $(-1, 0)$, is non null homotopic

Rithaniel

I have just finished my undergrad. Bachelors of Science in Math. Focus in abstract math. I'm going into a Masters Degree in math next

Balarka Sen

It's double is

nbro

I had a friend who had initially tried to study math, but that didn't go so well, so he decided to study computer science

Akiva Weinberger

@BalarkaSen Isn't that the braid group on two strands?

$B_2=\Bbb Z$

nbro

02:19

@Rithaniel Abstract math is stuff like category theory?

Balarka Sen

@Akiva Well that can't be right because $\Bbb Z_2$ is acting freely on $(\Bbb R^2)^2 \setminus \Delta$

Akiva Weinberger

@BalarkaSen Can you help me with this

Rithaniel

That's probably one of the more abstact branches, yes. Though, abstract algebra has my attention, as well as model theory, knot theory, differential topology, and stuff like that

Akiva Weinberger

twitter.com/jtbrazas/status/1208057254529949698

Balarka Sen

So it's not the braid group

Akiva Weinberger

02:20

> Meet the Hawaiian mapping torus - the mapping torus of the shift map. First singular homology is infinite cyclic generated by inner loop. Can you find the non-trivial elements of H_2? If you look at the image, it might feel H_2 is trivial since there is no "enclosed space!"

Balarka Sen

Maybe braid group is fundamental group of some other configuration space but I can't remember what

Akiva Weinberger

Balarka Sen

Oh fuck

Akiva Weinberger

Oh fuck re: configuration space or Hawaiian earring

Rithaniel

That's a neat idea

Balarka Sen

02:21

Hawaiian torus

This is a good puzzle. Wow

nbro

@Rithaniel When did you decide you want to study math? At early age or only during high school?

Akiva Weinberger

I'm like 80% sure it's trivial and the person's messing

@BalarkaSen Why not

Balarka Sen

If $X$ is a simply connected Hausdorff space, $G$ acts freely on $X$, then $X/G$ has fundamental group $G$

Rithaniel

Partway through college. I was originally trying for aerospace engineering, then chemical engineering, but I kept coming back to math because I enjoy the "puzzle" aspect of it.

Akiva Weinberger

Why would this be simply connected @BalarkaSen

Rithaniel

02:25

When I was in high school, everyone was telling me to go for a math degree in college, though.

Balarka Sen

$(\Bbb R^2)^2$ minus the diagonal? That is simply connected, no? It's a 2D subspace thrown out of a 4D space, transversality, etc.

Rithaniel

Other pressures led me down other paths for about half a decade, though, before I finally came back to college

Balarka Sen

Anyway I got the correct configuration space

Here my points are indistinguishable

Akiva Weinberger

@BalarkaSen 1D thrown out of 3D isn't simply connected

Why would 2D thrown out of 4D be

nbro

What's the difference between college and high school? At what age you started college?

Balarka Sen

02:26

If you use $n$ distinguishable points, so ordered tuples, you get braid group

Secret

Last night dream, there are two functions:

an array of delta functions spaced by one unit x (y=x)

nbro

I thought that high school lasted until like 18 years old

Balarka Sen

@Akiva Oh fair. Let's do $(\Bbb R^3)^2$ minus diagonal

That's 3D in 6D, so now it's simply connected

Akiva Weinberger

That makes much more sense

Balarka Sen

Thanks

Akiva Weinberger

02:28

'Cause otherwise it's literally the definition of the braid group

nbro

Depending on your skills, then you may finish earlier or later

Rithaniel

College is usually began around age 18 or 19, and the most major difference is that college is not generally state-sponsored

nbro

But that's the usual age to end the high school

But isn't college the same thing as university?

Akiva Weinberger

No, that's pure braids

Balarka Sen

What am I saying

Maybe

Akiva Weinberger

02:29

where they end where they started

Balarka Sen

Yeah

Akiva Weinberger

$B_n\to S_n$ is a map, the kernel is pure braids

Rithaniel

Universities generally have graduate programs. It's my understanding that colleges might not

nbro

Well, where I live, I am not sure we have colleges

Are you in the US?

Rithaniel

Yeah, I'm on the East Coast

Akiva Weinberger

02:30

Where are you from?

If you're OK with saying

nbro

I live in Europe

Balarka Sen

@Akiva Right, $S_n$ acts freely on $(\Bbb R^2)^n \setminus \Delta$ and quotient is something with fundamental group $B_n$. So by the Galois exact sequence $1 \to \pi_1(X) \to \pi_1(X/G) \to G \to 1$ you get a SES $1 \to \pi_1(\Bbb R^{2n}\setminus\Delta) \to B_n \to S_n \to 1$

Ted E

@BalarkaSen Do you really want $(\Bbb R^2\backslash\Delta)^2/\Bbb Z/2\Bbb Z$? Rather than $(\Bbb R^2\times\Bbb R^2)\backslash\Delta / \Bbb Z/2\Bbb Z$? Currently it looks like you're taking unordered pairs of points in $\Bbb R^2$ remove the diagonal, and permitting them to be the same point, or maybe I'm just too tired

Balarka Sen

Where $\Delta$ here is union of all the diagonals

@TedE Yes, there were two typos

Ted E

Oh ok

Akiva Weinberger

02:32

@nbro If English is your second language, are you OK with a little grammar note

Balarka Sen

Or at least one was a typo another was a mistake

nbro

And I didn't study math, so, of course, I know nothing about measure theory, etc

Ted E

I'm just reading downwards :P

Balarka Sen

I want $(\Bbb R^3)^2\setminus \Delta/\Bbb Z_2$

Secret

function 1

nbro

02:33

@AkivaWeinberger Go ahead. I speak multiple languages and English is not my native language, but I doubt I made a mistake

Akiva Weinberger

"At what age did you start college?", not "At what age you started college?"

Balarka Sen

Unordered tuples of points in $\Bbb R^3$

nbro

Ha, that's a typo

Secret

function 2

Akiva Weinberger

Arright it's a common mistake so I wasn't sure

sorry

nbro

02:34

When I am tired, I even write "no" in place of "know"

Akiva Weinberger

@BalarkaSen This contracts to RP^2 I think

Ted E

Okay, sets of two points in $\Bbb R^3$

Secret

The question in the dream is to compute $\lim_{x \to \infty} f_1(x)f_2(x)$. Problem is, in reality, it diverges

Ted E

and what is the fundamental group here?

Akiva Weinberger

@nbro Well, that's spelling, not grammar :P

Balarka Sen

02:34

$\Bbb Z_2$, @TedE

Ted E

Ok

Balarka Sen

@Akiva Yeah

nbro

Yeah, but thanks man

Sorry if I get rude sometimes

Ted E

I think my reading comprehension has been decreased as a result of playing bullet chess lmfao

Or maybe the chat room has too many simultaneous threads active

How am I meant to visualise this configuration space?

Akiva Weinberger

@BalarkaSen I wonder about the configuration space of circles in 3-space

Ted E

02:36

I mean, I can certainly visualise points

nbro

@AkivaWeinberger Btw, I've just checked, I wrote "At what age you started college?"

What's the problem with that?

Akiva Weinberger

(Don't know if it matters if we specify Euclidean vs topological circles)

Balarka Sen

@TedE Think about moving the points around, simple, right?

nbro

You can do that

Akiva Weinberger

@nbro In a question, you replace "started" with "did start"

Balarka Sen

02:37

@Akiva You mean geometric circles right

nbro

Well, not always

Balarka Sen

Otherwise you have some massive space of knots

nbro

You can do that

It's like a half question and half statement

Balarka Sen

Hatcher wrote some paper about $\text{Emb}(S^1, \Bbb R^3)/\text{isotopies}$

Ted E

Circles are just planes plus radiuses

so $\Bbb RP^3\times \Bbb R_{>0}$

Oh mine go through the origin

Err

nbro

02:38

English is a Germanic language, but incredibly it feels a lot like a Romance one

Akiva Weinberger

I think the other option would be something like "You started college at what age?"

Balarka Sen

You can shift the center around

so a factor of $\Bbb R^3$

Ted E

$\Bbb R^3\times\Bbb RP^3\times \Bbb R_{>0}$

Seems excessive maybe

Balarka Sen

But anyway that's geometric circles

Ted E

Is there a unique point per circle in that description?

nbro

02:40

@AkivaWeinberger You're an English native?

Akiva Weinberger

Yeah

nbro

I've been listening to English films since I am like 5 (with subtitles though)

Balarka Sen

(center, plane, radius) determines the circle so yes

nbro

@AkivaWeinberger So, what's the difference between that and what I said? You just changed the position of certain words

Ted E

You position changed just of words certain the

Akiva Weinberger

02:42

English cares about word order, dunno what to tell you

"When did you start" is OK, "When you started" is not

Ted E

What age were you when you started

nbro

I asked "At what age you started"

Akiva Weinberger

Yes

Same problem

When you have "what" you have to have a dummy "do" verb

skillpatrol

at what age did you start?

nbro

Guys, I know how to use do, but that's pretty useless, very often

Ted E

02:44

I've definitely never heard "At what age you started" from a native

nbro

If I add a ? (question mark)

Akiva Weinberger

You can't just add ? to a sentence to turn it into a question (though you can in Spanish and a bunch of other languages)

Ted E

It's clear what you mean, but I've never heard those words spoken in that order (other than pseudo-spoken in my subvocalisation just now)

Akiva Weinberger

You have to do "subject–auxiliary inversion"

skillpatrol

@nbro it takes practice :-)

Akiva Weinberger

02:45

and in many cases that requires "do-support"

nbro

I don't like that sometimes. I am lazy

Akiva Weinberger

(that's what the Wikipedia article calls them)

nbro

You understand me very well. I am breaking the rules of your language, which naturally evolves over time thanks to the outlaws like me

Akiva Weinberger

In any case, just a note

nbro

Thank you

But how can I remember that?

That's quite specific

Ted E

02:47

Also, while we're being excessively pedantic, I think "You're an English native?" should be "Are you a native English speaker?"

Akiva Weinberger

I've never had to learn English as a second language so I don't have any tricks for learning it

@TedE Yes - this is the same thing

almost

nbro

Yeah, I am super lazy

Or should I have said "Is English your mother tongue?" :D

Akiva Weinberger

(the inversion, not the Do-support)

Ted E

@BalarkaSen I mean, is there a way to visualise the whole space?

Akiva Weinberger

In any case, if you just want to be understood, you don't need to change anything, you're doing fine

nbro

02:48

Well, I want to be perfect, but often I don't care

:D

Balarka Sen

@TedE It seems pretty visualizable. You know what points in the space are, you can walk from one point to another.

If you can maneuver around you're visualizing it

nbro

@AkivaWeinberger So, when is it fine to ask a question without "do" (excluding can, etc)?

Ted E

@BalarkaSen I can visualise an entire sphere though

Balarka Sen

Visualization doesn't have to be "seeing from outside" or extrinsic!

Ted E

It's like global visualisation verses local visualisation

Balarka Sen

02:50

It's intrinsic vs extrinsic, yeah

If you're interested in these sort of things I recommend Thurston's 3-manifolds book

He has a very pedagogical approach to understand what being able to visualize means

It's kinda cool

nbro

@TedE Yeah, that sounds better

Akiva Weinberger

@nbro If there's already an auxiliary verb in the sentence, you move it up. (Ex: "I can go" -> "Can I go?", "It is safe" -> "Is it safe?") If there is no auxiliary, you create one with "do" (Ex: "I went there" -> think: I did go there -> "Did I go there?")

nbro

Yes, I know that

Ted E

So a path from 1 point of the space to another, looks like a pair of paths in space, that connect the pairs up

Balarka Sen

Ya

nbro

02:51

I was just wondering about leaving out the auxiliaries

Balarka Sen

Exactly

Akiva Weinberger

"You're an English native?" is good if you're surprised

Balarka Sen

We have a pair of identical bumblebees and we're looking at their joint trajectories

nbro

Ha, ok, I see your point

Akiva Weinberger

Like you know the answer is "yes" but you want to be sure

nbro

02:52

Well, I was surprised, because of your name

Akiva Weinberger

Similarly "You started when?" or "You did what?"

Balarka Sen

@Akiva ????

Akiva Weinberger

@BalarkaSen ????

Balarka Sen

? ??

Ted E

האם לתרגם טוב טוב? האם אתה קורא עברית?

Akiva Weinberger

02:53

אני מדבר עברית אבל לא היטב

nbro

Really?

Ted E

Ман мебинам

Balarka Sen

Почему мы говорим на иврите

nbro

Blyat, you're using Google Translate, I suppose

Akiva Weinberger

Ya nyet speak Rooskie

Ted E

02:55

Mine wasn't russian

nbro

That's Cyrillic though

Akiva Weinberger

Google Translate offers Tajik

I'm guessing it's Turkic at least

Tajik would be random

Ted E

Oui

It was Tajik

I just used it because I'd never heard of it before

Balarka Sen

obscure

Akiva Weinberger

Wait Tajik isn't Turkic

nbro

02:56

Ya govoryu nemnogo po-russki

Akiva Weinberger

Woops

nbro

No Google Translate used

Ted E

mdr

Akiva Weinberger

Wikipedia says it's a variety of Persian

> The issue of whether Tajik and Persian are to be considered two dialects of a single language or two discrete languages[8] has political sides to it (see Perry 1996).[7]

nbro

I see you're fond of foreign languages

Have you ever tried to study one?

Akiva Weinberger

02:57

Yeah

nbro

I suppose that many English native speakers really don't care about that

Because English is the most spoken language

Akiva Weinberger

Learned Hebrew since I was relatively young, Spanish in high school, and currently studying Japanese

nbro

That's cool

I also like Japanese and I will study Hebrew at some point

And I understand like 80% of Spanish

Akiva Weinberger

I read somewhere that 2/3 of English speakers speak it as a second language

nbro

Hebrew?

Or Spanish?

Akiva Weinberger

02:59

English

nbro

hahaha

Akiva Weinberger

Meaning, only 1/3 of English speakers are native speakers

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$Mathematics$ Mathematics