Mathematics - 2018-08-02 (page 2 of 4)

Ted Shifrin

18:00

@Mathein: Look at all the stuff Balarka was reading whilst in high school ...

MatheinBoulomenos

yeah true

Rudi_Birnbaum

:" Apes don't read philosophy. Wanda: Yes they do, they just don't understand it"

Ted Shifrin

LOL @Rudi

MatheinBoulomenos

@Rudi_Birnbaum lol

I think Scholze understood quite a bit

he gave talks in reserach seminars at 16

Rudi_Birnbaum

@MatheinBoulomenos Sure!

Mike Miller

18:01

just a funny quote

Rudi_Birnbaum

@MatheinBoulomenos I actually like (PS) his approach in learning.

@MatheinBoulomenos For the one just to takle the hard stuff and try to make sense of it. And for the other that he has to completely twist everything around in his mind until it suits (so to say, dunno how to explain it differently)

Its also what I normally do, though I'm sure on much much smaller scale and with smaller capacities.

Alessandro Codenotti

I tried to read Schoenfield during my last year of high school and that didn't go very well :P

Adeek

hey

MatheinBoulomenos

I tried to read Lang and Matsumura as a freshman. In the first semester, I was mostly lost, but in the second semester it was okay

Adeek

@MatheinBoulomenos do you want closed subschemes for a sec ?

MatheinBoulomenos

18:07

I always want closed subschemes

though usually I keep them a little bit longer

Symposium

@Rudi_Birnbaum I love that film!

Rudi_Birnbaum

@Symposium which one?

Mike Miller

in high school i learned calculus lol

Symposium

A fish called Wanda.

Rudi_Birnbaum

@Symposium Oh yeah me too.

@MikeMiller I am happy that I have learned Analysis instead :P

Adeek

18:08

hi @TedShifrin

Alessandro Codenotti

@MatheinBoulomenos oh yeah, when I was an exchange student in China (fourth year of high schook) I was curious about stuff so I asked a professor there what's a group and he lent me his copy of Lang's algebra, I think I managed 10 pages at most before deciding there are better approaches to learn algebra from scratch

loch

i also learnt calculus in highschool lol

Adeek

You guys are lucky

MatheinBoulomenos

everyone learns some calculus in high school here

Adeek

I didn't learn science in high-school I learned things by myself back at home

but I guess my country is a special case

MatheinBoulomenos

18:10

(at least if you're in a "gymnasium" which is basically like a university-track high school)

Adeek

glad I am in the west now.

Alessandro Codenotti

@MikeMiller When did you decide you wanted to study maths?

Rudi_Birnbaum

@AlessandroCodenotti I think with that approach of learning it doesnt work out to just try it at some instances and do it usually a more common way. I imagine you'd have to be quite consequent in it.

Mike Miller

i was going to do political science and economics in my first year of undergrad, then applied math in my second year of undergrad, and then i stopped wanting to do applied by the end of that year

user131753

Let $(X,\tau_{X})$ and $(Y,\tau_Y)$ be topological spaces. Let $f,g:X\to Y$ be two continuous functions. If $\overline{f(A)}=\overline{g(A)}$ (respectively the closures of $f(A)$ and $g(A)$ in $Y$) for all $A\subseteq X$, then is it true that $g=f$?

MatheinBoulomenos

18:12

fun fact: I only need to take a few more courses and write a bachelor thesis, then I'd have a bachelor in Ancient Greek

Rudi_Birnbaum

@MatheinBoulomenos cool!

Mike Miller

@user170039 let $Y$ be equipped with the indiscrete topology. Then the only nonempty closed subset of $Y$ is the whole space, and every map to $Y$ is continuous. So every pair of maps to $Y$ satisfy your properties.

Rudi_Birnbaum

@MatheinBoulomenos Do you speak modern Greek?

MatheinBoulomenos

no

I only know the Ancient stuff

Mike Miller

If $Y$ is $T_1$ then the answer is yes, because you may choose $A = \{x\}$ and see that $f(x) = g(x)$.

So it seems either trivial or hopeless

Rudi_Birnbaum

18:14

@MatheinBoulomenos (OK, just thought because of you user name ...)

MatheinBoulomenos

my name is Ancient Greek

Alessandro Codenotti

I always said your abstract nonsense stuff looks like ancient greek to me!

Rudi_Birnbaum

@MatheinBoulomenos what would be the modern version?

MatheinBoulomenos

@Rudi_Birnbaum I have no idea

user131753

@MikeMiller For $T_1$ I figured it out just like you did.

user131753

18:15

I didn't think about taking $Y$ to be indiscreet!!

Adeek

@MatheinBoulomenos My mum is greek

lol

user131753

My bad.

Mike Miller

Nobody's bad at all.

user131753

Anyway thanks @MikeMiller.

MatheinBoulomenos

@Adeek cool

Mike Miller

18:16

Perhaps I should not have used the word 'trivial'.

user131753

@MikeMiller No. It really was trivial.

MatheinBoulomenos

@AlessandroCodenotti so you mean that it looks beautiful with an intricate, but really interesting structure?

user131753

@MikeMiller But I think it is not me who is bad, it's you for pointing out its triviality ;).

Alessandro Codenotti

@MatheinBoulomenos nevermind...

Symposium

Oh Scholze won the fields medal (just saw the starred section).

Alessandro Codenotti

18:18

That wasn't very surprising to be fair

Rudi_Birnbaum

Strange thought: What is more basic the concept of a group or the concept of integers? One would think of a group. Though to describe groups you need integers. Like the theorems for orders of normal subgroups and the like.

Symposium

Learning mathematics from top down is crazy to me!

Leaky Nun

integers

MatheinBoulomenos

categories

oh wait

user131753

@Rudi_Birnbaum To define an arbitrary group do you really need the concept of integers?

Alessandro Codenotti

18:22

Groups have very simple first-order characterizations, the integers are too hard

Rudi_Birnbaum

@user170039 no, of course not for the definition.

But to understand their structure, I would say.

Leaky Nun

@MatheinBoulomenos "lernen willend"?

@AlessandroCodenotti you said "first"

MatheinBoulomenos

to understand the structure of groups, depending on how far you go, you also need vector spaces, ring theory, homological algebra etc.

Leaky Nun

true ^

Mike Miller

To understand the structure of groups you of course need the notion of unitary matrices ;)

MatheinBoulomenos

18:24

@LeakyNun wollend is the correct form

Leaky Nun

oh right

MatheinBoulomenos

but yeah, that's the literal translation

Leaky Nun

discerevolens?

Rudi_Birnbaum

@MatheinBoulomenos is there then any circularity in our understanding?

Mike Miller

No

Ur crazy m8

MatheinBoulomenos

18:26

@LeakyNun yeah

Leaky Nun

apprendere volente

is that all the languages you know?

MatheinBoulomenos

my languages are really bad

I know German, English, Latin, Ancient Greek and $\varepsilon$ Italian

mercio

I know English, French, and $\varepsilon$ spanish, dutch, korean, japanese, and $\varepsilon^2$ ancient greek

Leaky Nun

so basically ich habe deinen Name ubersetzen in die Sprache dass du redest

ubersetzt?

MatheinBoulomenos

ich habe deinen Namen in die Sprache übersetzt, die du redest

mercio

18:31

matheinBoulomenos your name sounds very greek

Leaky Nun

it is ancient greek

@mercio I thought you know Spanish

mercio

I tricked you

Leaky Nun

je kunt Nederlands spreken :o

MatheinBoulomenos

@LeakyNun just "learning" in the sense of studying is too narrow for mathein, it can also mean to perceive or understand

mercio

un poco

kun* ?

I already forgot if t was only for3rd person singular

dammit

now I think you are right

MatheinBoulomenos

18:33

so you might also tranlate it with apprendere depending on the context

Leaky Nun

@mercio ik ben Nederlands leren aan het probieren

MatheinBoulomenos

Dutch is just a parody of German

Leaky Nun

except that it's got continuous tense

mercio

graag

Leaky Nun

Germans, where is your continuous tense

@MatheinBoulomenos I'm prone to believing that Dutch is more Germanic than German because it, like Low German, wasn't affected by the High German consonant shift

MatheinBoulomenos

18:35

Während du am Lamentieren bist, dass es keine solche Zeitform im Deutschen gibt, studiere diesen Satz

mercio

and somehow learning a bit of dutch made me learn a bit of german too

mysteriously

Leaky Nun

is the continuous tense used a lot in German?

MatheinBoulomenos

not that much

and we don't consider it an own tense

Tobias Kildetoft

What is the continuous tense for? Something that is ongoing, as opposed to present tense?

MatheinBoulomenos

I just wish this tense in English would have been going to be being used a bit more

Tobias Kildetoft

18:40

Now we are getting into grammar right out of the Hitchhiker's Guide

MatheinBoulomenos

@TobiasKildetoft but is it technically correct?

Leaky Nun

I'm not sure what that tense is called anymore

conditional + perfect + near-future + continuous

MatheinBoulomenos

yeah

Leaky Nun

and also passive

how would you say it in German

MatheinBoulomenos

@LeakyNun why do you consider continuous tense that important? you can just use an adverb to convey the same meaning

Leaky Nun

18:43

I see

$$H^1(\operatorname{Gal}(L/K), L^\times) = 0$$

MatheinBoulomenos

@LeakyNun you might enjoy this funny page: verben.texttheater.net/Startseite

Daminark

A+ theorem tbh

No idea how it's proven

MatheinBoulomenos

they have a lot of suggestions to improve German

oh the proof is just a calculation, mabye 3 lines

using the construction of group cohomology via cocycles mod coboundaries

not too bad

you need Dirichlet's theorem on linear independence of characters as an input

(if you allow $L/K$ infinite, then you would reduce to the finite case first)

Leaky Nun

matemaaachica

@EricSilva what do you think about the theft?

Daminark

Huh, I may check out the proof soon then. Somehow I was under the impression that theorem 90 was supposed to be quite tough. Might check it out then

Eric Silva

18:50

im mad cause it looks bad for Rio but also because the coverage is gonna be dripping with condescension over how crime ridden the city is

Alessandro Codenotti

The proof I saw of the independence of characters had more calculations than insights

Leaky Nun

@MatheinBoulomenos wuerde ich den Bewijs verstehen?

MatheinBoulomenos

@LeakyNun ja

Eric Silva

which is true, it is a crime ridden city, but the narrative is told in a kind of condescending way that isnt very productive

MatheinBoulomenos

if you look up the definition for group cohomology, sure

Eric Silva

18:51

but all in all im really mad some ass did this bc Birkar seems cool af

Leaky Nun

@MatheinBoulomenos wuerdest thu den Bewijs mir leren?

MatheinBoulomenos

@AlessandroCodenotti but it's a really simple calculation, idt you can do it more conceptually

Alessandro Codenotti

Fair

MatheinBoulomenos

@LeakyNun see page 65-66 here: math.ucla.edu/~sharifi/groupcoh.pdf

it's the same proof everywhere basically

Leaky Nun

donke

MatheinBoulomenos

18:53

so I don't think it would help if I wrote it down here in chat

Daminark

I remember in Galois theory there was the proof of the primitive element theorem using independence of characters and my prof was just like "I can never remember how that goes, I'll just prove enough of the main theorem of Galois theory so that primitive element theorem falls out"

Leaky Nun

@Daminark can man derive primitive element theorem from fundamental theorem?

MatheinBoulomenos

@LeakyNun yeah

Leaky Nun

really

MatheinBoulomenos

for infinite fields, at least

if $V$ is a vector space over an infinite field, then $V$ is not the finite union of proper subspaces

Leaky Nun

18:55

@Daminark who cares about proofs

MatheinBoulomenos

that implies the primitive element theorem if you use the Galois theory

@LeakyNun yeah nobody needs proofs

Daminark

Once you know one half of the proof, I think the one that says that you can generate a subgroup for any intermediate field, then you're just able to say that you have only finitely many subgroups there

Yeah good points proofs are nerd shit

Semiclassical

neeeeerd

Leaky Nun

@MatheinBoulomenos what's that cool letter in het pdf?

MatheinBoulomenos

\coolletter

Eric Silva

18:57

i definitely know research mathematicians who unironically think proofs are overrated

MatheinBoulomenos

$\mathcal E$? not sure

hmm

that's not it

Daminark

I think I know who you're talking about @Eric

Eric Silva

not just him!

not overrated perhaps but that their role in mathematics is not primary

Semiclassical

as is often the case, it depends so much on the discipline

Daminark

Hmm, who else do you have in mind?

Mike Miller

18:58

<-----

Leaky Nun

@MikeMiller $\nabla^2 f = 0$

Daminark

Lel

MatheinBoulomenos

I think in topology, proofs are largly supplemented by pictures

Eric Silva

@Semiclassical ima become a proof theorist who doesnt give a fuck about proofs

watch me go

Leaky Nun

and the waving of a hand

often both hands

Semiclassical

18:59

oh snaaaap

Abcd

@LeakyNun Are you there?

Leaky Nun

@Abcd no i'm not

oh wait

that means i'm there doesn't it

oh i'm there then

i'm here and there i'm everywhere

2

Mike Miller

@MatheinBoulomenos As I like to blithely respond, a student with a little competence can always turn a picture into a machine-readable proof ;)

Semiclassical

It's a rather different sort of 'picture', but I'm fond of 'proofs without words' for that reason

MatheinBoulomenos

@MikeMiller the problem with my topology courses was that our prof expected that everyone can do this

Semiclassical

19:01

namely, that a picture can convey the concept of a proof without actually stating it

Leaky Nun

May 21 at 10:46, by Leaky Nun

May 21 at 10:47, by Leaky Nun

Proof via König's Lemma that closed and bounded implies compact

Semiclassical

(the concept of a proof and the realization of it aren't the same, of course)

Abcd

Find the number of points of inflection of $(x-2)^6(x-3)^9$

Leaky Nun

differentiate it twice

Abcd

To simplify it, we know that if f(x) has n inflection points then even f(x+a) will have n inflection points for continuous f

so the problem simplifies to:

Leaky Nun

19:03

you don't need continuous f for that to hold

Abcd

$x^6(x-1)^9$

@LeakyNun Okay, I just invented it myself using problem solving didnt know that continuity wasnt necessary

Semiclassical

i'd say you're doing a bit more than that, namely you're making a linear transformation to send x-2 to x and x-3 to x-1

I agree that it's fine either way, though

Abcd

@LeakyNun -_-

Eric Silva

@Semiclassical ya the difference is the concept matters and the realization is the nerd-work

Semiclassical

oh, wait. I guess all you need is the shift here

ignore meeee

Abcd

19:04

@Semiclassical whom are you replying to?

Semiclassical

i was saying that your transformation wouldn't be as simple as just f(x)->f(x+a)

but it totally is, I was just being silly

Abcd

Okay.

@Semiclassical Now how to find NUMBER of inflection points?

Semiclassical

3 mins ago, by Leaky Nun

differentiate it twice

Abcd

we just know that the curve changes sign at x=1

@Semiclassical Thats unnecessarily complicated and long way

mercio

o..o

Mike Miller

19:07

@MatheinBoulomenos you can <3

I believe in u

Eric Silva

differentiating a polynomial twice doesnt take very long

Semiclassical

then figure it out yourself.

Abcd

@EricSilva I know but here we have 9th power , it wont be easy to see if the double derivative is 0 or not

Leaky Nun

@MatheinBoulomenos donke, ik heb het gelesen

Abcd

and for how many points it will be 0.

Leaky Nun

19:09

@Abcd three people have been telling you to differentiate it twice.

Eric Silva

@Abcd it is, in fact, in this case

i just did it

in under 2 minutes

so i think u can too

mercio

you just have to differentiate it twice

Abcd

@LeakyNun obeys

Alessandro Codenotti

@MikeMiller Formal verification of badly drawn pictures will be the next hot topic when people get bored with homotopy type theory

Mike Miller

@AlessandroCodenotti Does that mean I can submit my thesis in that form?

Abcd

19:19

@LeakyNun I get the answer to be 4, what about you?

Leaky Nun

I didn't do it

Abcd

Answer given is 1.

Someone please see.

Double derivative is:

Alessandro Codenotti

@MikeMiller only if you need a couple of decades to finish it I'm afraid

Abcd

$6(x-1)^7x^4(35x^2-28x +5)$

Leaky Nun

I guess double derivative being 0 doesn't necessarily mean that it is a point of inflection

Mike Miller

19:21

@AlessandroCodenotti I bet I can delay that long

Leaky Nun

do you even know the relevant test

cite the theorem you know

Eric Silva

@LeakyNun not a point of inflection if sign doesnt change

Alessandro Codenotti

Don't let procrastination get in the way of your dreams, but what if my dream is to procrastinate

Abcd

@LeakyNun where neither maxima nor minima occurs but derivative is 0 or where curve changes curvature

so x = 0 is not an inflection point then

but x =1 is surely

similarly the roots of the quadratic are

Then thats 3

But answer is 1 @LeakyNun

Mason

Looking for a word like "permutation"

Leaky Nun

19:29

I think it's 3

desmos.com/calculator/ljo2xrhliu

Mason

(0,0,6,3) is one solution to the question 4 squares that sum to 45.

there are 623 other solutions... and 48 of these solutions are essentially copies of (0,0,6,3).

they are (-3,0,6,0) and so on.

Anyone have a decent word for that. "copies" doesn't quite fit.

permutation doesn't quite incorporate that we are counting sign flips as different solutions.

mercio

equivalent

Leaky Nun

why does bounded mean continuous in infinite-dimensional Banach space?

Alessandro Codenotti

There exist $C$ such that $||Tx||_Y\leq C||x||_X$

Where $T:X\to Y$ is a linear operator

Did you edit your message?

Leaky Nun

I didn't

Alessandro Codenotti

19:38

I just can't read today lol

Well anyway showing continuity in $0$ is enough

Mason

@mercio. Thanks!

Leaky Nun

feels like Lipschitz

Mason

How's this:

After asking him to write down how many solutions of each type we have NAME told me that we could count
192 solutions which were essentially copies of (2,3,4,4) and
192 solutions which were essentially copies of (1,2,2,6) and
192 solutions which were essentially copies of (0,2,4,5) and
48 solutions which were essentially copies of (0,0,3,6) giving us a total of 624 solutions. Keeping himself organized while counting up these solutions is a good challenge for NAME because he is usually comfortable holding all of these numbers in his head but I have managed to find a problem whic…

mercio

o..o

who is NAME ?

Tobias Kildetoft

@Mason Hmm, do you happen to know an easy way to see how many "types" of solutions such an equation has?

Since the total number is easy, but it seems tricky to figure out how to split into types

Mason

19:43

@TobiasKildetoft. Yeah. This is Lagrange.

My students are not to take Lagrange's word for anything though. We will count and categorize types and what not. And then once we've seen it hold for many small numbers we can trust Lagrange... But also it's quite a nice combinatorics puzzle because we have to think through factorials and how zero is a seperate case and repeats complicate it and all that.

NAME is my student.

Name is my student who's name the internet doesn't need to know.

Tobias Kildetoft

@Mason So I don't assume you will be proving the formula for the total number?

Mike Miller

its our boi

Mason

3rd graders.

No. We aren't proving stuff. We are exploring numbers.

Tobias Kildetoft

So that is a no then :) (unless you know a way not involving modular forms of course)

cool

Leaky Nun

why is the set of nowhere-differentiable functions dense in $C^0[0,1]$?

Tobias Kildetoft

19:46

Have you shown them the formula to check against?

Mason

But the write up seems fine? "Essentially copies" is good.

Tobias Kildetoft

Yeah, "essentially copies" sounds good

Mason

Yes. Yes. They had the formula to check against after the first or second day.

Alessandro Codenotti

@LeakyNun sorry has to leave for a moment, bounded implies Lipschitz is easy indeed

$||Tx_1-Tx_2||_Y=||T(x_1-x_2)||_Y\leq C||x_1-x_2||$

mercio

maybe he is the kind of 3rd grader whose teacher says stuff like "there are 5 fundamental arithmetic operations, addition, substraction, multiplication, division, and modular forms"

Isa

19:49

'the pencil of parallel straight lines' means like 'a family or set of parallel straight lines' ?

Tobias Kildetoft

@mercio modular forms is not an arithmetic operation. That would be the action of the modular group.

Isa

how is pencil interpreted?

mercio

I think it was a quote from someone from a talk

Leaky Nun

@AlessandroCodenotti lol I might have misinterpreted bounded

mercio

google told me this

math.berkeley.edu/~ribet/trinity.pdf

Alessandro Codenotti

19:52

@LeakyNun How?

Leaky Nun

@AlessandroCodenotti $\|T(x)\| \le C$ lol

Alessandro Codenotti

For the other direction we use continuity in $0$. Fix $\varepsilon=1$ and we get a $\delta$ such that $||x||_X<\delta\implies ||Tx||_Y<1$. Some calculations then show that $||T||=\delta^{-1}$

@LeakyNun only the zero operator satisfies this :P

Mason

We talk about modular arithmetic but I found that the most prodigious 3rd grader can find proof writing tedious. Let's just see that it works for a few cases and leave it as a conjecture to be disproven. We can always count to double check.
@mercio. To support the spirit of the talk you posted: it's true that diophantine equations (and as a result modular forms) are pretty nice playground for gifted* young students.
*Whatever that means.

Leaky Nun

@AlessandroCodenotti right

user193319

Problem: If $K$ is closed, bounded, and equisummable in $\ell^p$, where $1 \le p < \infty$, show that $K$ is compact....Okay. Being closed implies $K$ is a complete metric space, so all I need to do is show that $K$ is totally bounded. Given $\epsilon > 0$, equisummability implies there exists $N \in \Bbb{N}$ s.t. $\sum_{n=N} |x_n|^p < \epsilon$ for every $\{x_n\} \in K$. To my mind this suggests $K$ might be contained in some nbhd of a point like $y = (y_1,...,y_N,0,...)$, showing total bdness

But I don't know how to find such a point.

Alessandro Codenotti

19:59

What does equisummable mean?

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$Mathematics$ Mathematics