Mathematics - 2020-10-05 (page 1 of 2)

Unknown x

00:45

This has $D_4$ symmetry. right?

Am I correct?

anakhro

01:07

@TedShifrin any ideas for a cool theorem to work towards in a lecture on either hyperbolic or spherical geometry? The Euclidean one I did was to do the Three Reflections Theorem.

I am less experienced with hyperbolic and spherical geometry and so I can't think of any theorems with a special "wow" or "interesting" factor.

I know there is a TRT on spheres.

Finding areas on a sphere is another somewhat interesting thing to do.

I suppose hyperbolic area would be the analogous thing.

Ted Shifrin

01:27

You could aim for AAA congruence theorems.

anakhro

That's an idea.

Also there are inversions.

Ted Shifrin

Then do the inversive triangle inequality and the Fermat point of a triangle.

Unknown x

Is the given figure any other operation? other than 4 rotations, 4 reflections?

anakhro

@TedShifrin I have never heard of this Fermat point. What is special about it?

Unknown x

sorry for the grammatic error. does the given figure have any other operations other than 4 rotations, 4 reflections?

Ted Shifrin

02:01

@anakhro The point that minimizes the sum of the distances to the vertices. Look at Pedoe's book on circles.

Unknown x

02:40

It has $D_4$. symmetry. right?

Ted Shifrin

02:55

@Unknownx can you draw an obvious square?

Unknown x

yes

Ted Shifrin

OK, then.

Unknown x

This has symmetry similar to that of square.right?

Ted Shifrin

Identical.

Unknown x

My friend told me that my operations are incomplete. So, I tried to find more operations

Thank you@Ted Shifrin

Ted Shifrin

02:58

Sure.

Oh, your friend is right. There is also a subgroup of $\pi/4$ rotations.

Offhand, looks like a group of order 16.

No, that’s wrong, never mind.

Unknown x

@TedShifrin But when we rotate $pi/4$ angle. it doesn't preserve symmetry

I am not getting the same figure

Ted Shifrin

Right. This is the problem with scrolling.

Unknown x

okay. Thank you very much. I am happy that I am right :P

sorry. I don't undertsand the meaning of 'This is the problem with scrolling.'

I am again confused. only eight operations possible.right?@TedShifrin

William Torkington

06:54

Sup team, I'm new here. Doing Computer science in New Zealand, but I think I want to start a degree in math next year.

What are some of y'alls recommended online resources for learning?

Sayan Chattopadhyay

07:24

Barring Milnor, what are some good texts/lecture notes on Morse Theory?

Balarka Sen

07:34

Morse theory is too broad

do you want Morse homology

Sayan Chattopadhyay

Yeah like the contents of milnors first few chapters. But I do not like the typesetting, its very hard to read

Balarka Sen

08:06

@Sayan Milnor does vastly different things in his book

I dunno you could try Nicolaescu, I haven't read it but my ex-roommate has

he says its good

Sayan Chattopadhyay

I see, here's my reason on why I want to study some Morse stuff. I see them physicists using a lot of Picard-Lefshetz and I want to know what exactly are they doing. Plus in the mechanics course, some dude was talking about Floer Homology, so the instructor went on about Fukuya categories and stuff. Some googling tells me that they were first introduced in Morse Homology.
And to my naive brain, the idea of studying critical points and getting info about topology seems like a nice mishmash of topology and analysis that I seem to be liking since the past few months.

So maybe what I should be looking for is Morse Homology??

Balarka Sen

for Morse homology a good reference is a chapter from "Monopoles and 3-Manifolds" by Kronheimer-Mrowka

idk picard-lefschetz or whatever but come read Arnold's classic singularity theory text with me

Arnold-Goryunov-Lyashko-Vasil'ev, "Singularity Theory I"

chapter 2 is picard-lefschetz

Sayan Chattopadhyay

Oh these russians... How much acquainted do I need to be with Morse Homology to read Arnold's text? Mind you I do not know anything about Morse stuff

Balarka Sen

Arnold's text doesn't do Morse homology, it does singularity theory; some kind of generalization of Morse lemma

it tries to classify germs of smooth/holomorphic maps (R^m, 0) -> (R^n, 0)/(C^m, 0) -> (C^n, 0)

we can take a dig at it if its too much for you feel free to chicken out lol

i might chicken out too

Sayan Chattopadhyay

08:21

Oh lol, thats cool, maybe I can give this a shot

Balarka Sen

let me know when you'd like to start reading i can set up a room and send you the text

Sayan Chattopadhyay

Is from next week fine with you, I have exams this week and then I am free

Balarka Sen

ya

thats good w me

Axoren

09:38

Is there a proper way to reference the 3-dimensional Euclidean space that doesn't have an assigned coordinate system?

Stupid question inc

Can somebody help

Balarka Sen

Ask

(Finally a book worth reading!)

Stupid question inc

so in this proof how does it proves S is countable

and he directly concluded

Balarka Sen

He listed all the elements of $S$

Stupid question inc

there exist subset T of the set of all positive integer such that S~T

Balarka Sen

09:48

Indeed.

Stupid question inc

this means he s and t has bijective mapping but how

how you prove it is bijection?

yes he listed

kinda didn't understand how listing makes it countable

you first need to prove a bijection mapping to positive integer

this is from rudin

so waht is intuitive idea behind this proof

Balarka Sen

Do you agree what he does shows $S$ is in bijection with a subset of $\Bbb N$

Stupid question inc

I don't

He hasn't showed it yet

Crap just got 10 min ;_;

but may be due to that x_{nk}

may be n and k shows bijection

Balarka Sen

First consider this map $S \to \Bbb N$: Send $x_{11}$ to $1$, $x_{12}$ to $2$, $x_{21}$ to $3$, ...

A list is a bunch of objects labelled by $\Bbb N$. First object gets sent to $1$, second object gets sent to $2$, .... 100 millionth object gets sent to $100$ million.

Stupid question inc

I thought he was gonna do x_11 to 1 x_21 and x_12 to 2 ...

Balarka Sen

09:53

No.

$i$-th object maps to $i$.

Is this a bijection?

Stupid question inc

this is bijection clearly

Balarka Sen

No

This is not even a well-defined map.

Stupid question inc

one to one and onto

Balarka Sen

No

Think again

What if $x_{11} = x_{23}$? That is possible, no?

Nobody has told you $x_{ij}$'s are distinct

But you're sending $x_{11}$ to $1$, $x_{23}$ to $9$.

Not well-defined

Stupid question inc

I thought you already clear those duplicate

Balarka Sen

09:56

3 mins ago, by Balarka Sen

A list is a bunch of objects labelled by $\Bbb N$. First object gets sent to $1$, second object gets sent to $2$, .... 100 millionth object gets sent to $100$ million.

I defined what I meant.

Axoren

In Balarka Sen's example, the first and second object could both be 7.

Stupid question inc

ah so it's just Label

Balarka Sen

@Stupid How do you fix this map?

Axoren

The fact that they're the first object or second object has no restriction on their intrinsic value, just their extrinsic label

Stupid question inc

2 min left ;_;

to fix this i think clear all the duplicate

Balarka Sen

09:58

Come back when you have more time.

Also I read your answers to the questions I gave, they were either ununderstandable or simply wrong.

You should write them precisely and spend more time on them

Stupid question inc

ok

kinda in hurry I will memorize your words

offline ;_;

Axoren

10:13

@Stupidquestioninc Consider that same list outlined in (17) but have it assign indexes as normal but skipping duplicate elements (this is the subset T). Now, why the list without duplicates is bijective with the set S is almost a dirty trick.

Stupid question inc

11:39

@Axoren I was trying to saying the same thing lol

@BalarkaSen axoren answered this question i guess

you got any better idea?

Also I have question about arranging

why arranging like this though?

Balarka Sen

@Stupidquestioninc Write down the map $f : S \to \Bbb N$. $f(x_{ij}) = ??$

Stupid question inc

11:57

@BalarkaSen $f(x_{ij})=x, x\in\Bbb{N}}$ but only if x_{ij}=blabla then f(x_ij)=f(blabla)

Balarka Sen

That is the opposite of "writing down" lol

Astyx

waves around

Stupid question inc

so I don't understand what to write down hehe

$f(x_{ij})=n_{ij}$

Balarka Sen

What is $n_{ij}$? it's some expression involving $i$ and $j$

Can you write it down

Stupid question inc

@BalarkaSen n_{11}=1, n_{12}=2,..., n{1j}=j and n_{21}=j+1,...

ok it's just a new definition

awkwardly defined

ok I will try to do this when I get out from bike

Stupid question inc

12:53

I don't know but the intution about that arrangement is counting by hand

but now we get on rigorous approach

rudin didn't bother about bijection

I don't think we need to be that rigorous

ok so I will do a summary

if you remove duplicate of the arranged sequence the first term maps to 1 second to 2 ... last maps to last that is bijection

Sayan Chattopadhyay

@Stupidquestioninc Proof writing in mathematics involves you writing down expressions and properly giving arguments (in this case why something is a bijection.) So you have to check that its injective and surjective, Balarka is asking you for an expression so that you can prove it is a bijection

Intuition is very good, and I personally strongly advocate for that. But you need to clean your slate once you have developed the proper intuition

Stupid question inc

13:10

so I need to find an expression to map those arrangements to natural number in bijective way

ok I will do that tomorrow today I need to finish topology in complex plane so I am bookmarking this one

user2103480

@SayanChattopadhyay wild mechanics course

Stupid question inc

@SayanChattopadhyay Do you know how to express this

I really kinda have no idea what to do for now

I am kinda in hurry got only 8 min

Sayan Chattopadhyay

@user2103480 Maybe lol, but I am enjoying it. He plans to introduce sheafs and maybe infinity categories/higher categories when he talks about Classical Field theories. All in all, I like this kind of stuff

user2103480

I'm envious. I'm still looking for a topic bordering on probability that is more a mixture of analytic and algebraic methods, not just nightmarish analysis. Unfortunately, I'm not interested in quantum stuff, and things such as random simplicial complexes seem too combinatorial to me

Sayan Chattopadhyay

@Stupidquestioninc Not as of now, but even if I did, telling you would defeat the purpose of the entire effort.

Alessandro Codenotti

13:26

@user2103480 Time to become a geometric group theorist working on random groups

Sayan Chattopadhyay

@user2103480 Oh interesting. I have never done any probability theory except what the minimal requirement that was in my measure theory course. I should do it sometime, since I am interested in all this quantum stuff

user2103480

@AlessandroCodenotti grrr

Sayan Chattopadhyay

But I need super crazy motivation for that, and as of now I do not have much except I don't know maybe Bell's Inequalities or some information theory stuff. Though don't know how much of that can be even called probability

user2103480

There's a big intersection

There's quantum probability and correspondingly quantum stochastic analysis

Alessandro Codenotti

For any field A of math there's quantum A or so it feels lately

4

user2103480

13:30

A prof recently described statistical mechanics as "basically applied probability theory"

Sayan Chattopadhyay

Mathematics is quantum mechanical

user2103480

@AlessandroCodenotti indeed

Random matrices seem to be quantum motivated as well

Ergodic theory ofc

Alessandro Codenotti

quantum algebraic quantum field theory

user2103480

@AlessandroCodenotti SPDE are already applied to quantum field theories and I'm waiting for the day that quantum stochastic partial differential equations are a thing

QSPDE

Sayan Chattopadhyay

To be honest, I fail to see the quantum aspect of such fields. For instance take Quantum groups, what is so quantum about them? All I got was that you wanted to do something similar to Pontryagin Duality with nonabelian comapct groups

Alessandro Codenotti

13:34

Are quantum groups the funny ones that don't even have a precise definition?

Sayan Chattopadhyay

Yup those dudes, fun fact they aren't even groups

user2103480

The basic idea is to encode a would-be probability space dually in its algebra of functions A, typically regarded as a star algebra, and encode the probability measure as a state on this star algebra

To quote the nlab

Alessandro Codenotti

This is starting to feel like noncommutative geometry

Sayan Chattopadhyay

I think it has a lot to do with non commutative geometry

user2103480

So in a normal probability space you get a commutative algebra of random variables and your probability measure is linear functional that maps the identity to 1

Representing the integral against a measure

And yes

We just make it noncommutative

And boom theres the quantum

Alessandro Codenotti

13:40

Yes it does definitely feel very similar to noncommutative geometry as approach

There there is a duality between unital commutative C*-algebras and compact Hausdorff spaces (Gelfand-Naimark), and then you generalize to noncommutative algebras, there is no more topological space on the other side, but somehow you can still do some kind of geometry with those things

user2103480

Apparently it is actually useful in quantum thingies

@AlessandroCodenotti yup its exactly that kind of thing. The duality for probability spaces doesnt even have a name because it's a bit easier to see

Sayan Chattopadhyay

@AlessandroCodenotti Oh interesting. That is NC geometry?

Manolis Lyviakis

im taking a graduate course in PDEs what calculus must i remind myself?

Alessandro Codenotti

@SayanChattopadhyay That's the starting point

Sayan Chattopadhyay

Damn that sounds fascinating.

user2103480

13:45

@ManolisLyviakis is it a first course?

Manolis Lyviakis

yes i ve seen very basic things on pdes as undergrad

and havent seen multi-calculus in a while . Greens theorems and all that

user2103480

I mean

Where does the course start?

Manolis Lyviakis

ohh

user2103480

At what topics

Manolis Lyviakis

Laplace equation and following on evans book

user2103480

13:47

Is it expected that you know some functional or harmonic analysis?

If not then you shouldn't worry about prerequisites I guess

Manolis Lyviakis

oh ok.

i wanted to take a course on representation of groups which is more of my style

but they said i needed to take pdes xD dont get it

user2103480

It may be useful to go over the wiki articles of the divergence and the laplace operator

Just for the physical intuition

Many standard PDEs arise out of conservation laws and corresponding integral equations via the variational approach

Manolis Lyviakis

yeah i was curious about that

where can i see that

user2103480

Gimme a minute

http://numerik.mi.fu-berlin.de/wiki/SS_2012/Vorlesungen/NumerikIII_Dokumente/Numerik3_Skript.pdf

I'm ashamed to admit but the best intro I found yet is from a numerics lecture

oh shit

its german I forgot

Alessandro Codenotti

@user2103480 We officially lost you to the dark side

user2103480

13:54

blame the system

blabbers frantically about 7 types of measurability conditions for stochastic processes

Stupid question inc

@SayanChattopadhyay ok I will try then

user2103480

@AlessandroCodenotti what is that about btw?

is geometric group theory more group theory or more geometry?

Alessandro Codenotti

Hm good question

Probably more geometry

user2103480

and what are random groups used for?

Alessandro Codenotti

I think @Balarka actually knows about those, I just know that some people study them

Balarka Sen

14:08

I don't know anything please stop accusing me

Astyx

burn the witch

Alessandro Codenotti

lol

user2103480

groups shant be random

Alessandro Codenotti

Weren't you doing some stuff about random groups just a month or two ago?

Balarka Sen

Nah just first passage percolations

Random geometry on a fixed ambient

user2103480

14:12

and I thought about taking an ergodic theory course but the focus is on applying it to ramsey theory and number theory

Balarka Sen

yuck

user2103480

yeah

Alessandro Codenotti

@user2103480 Beautiful

user2103480

"We will rather omit smooth dynamics during
this course."

>:I

Balarka Sen

@user2103480 Teach me stochastic differential equations

Alessandro Codenotti

14:15

Are there notes for that course?

user2103480

@AlessandroCodenotti I can ask but I think not yet. the literature is the following:

Alessandro Codenotti

I've read the most overkill application of Ramsey theory that I've ever seen earlier: Let $f\colon \Bbb Q\to \Bbb Q$ be any function. Then there are arbitrarily large finite sets on which $f$ is either constant, increasing or decreasing

user2103480

Literature

Obligatory:
1. M. Einsiedler, T. Ward, Ergodic Theory with a View Towards Number Theory. Springer 2010.
Optional:
1. P. Walters, An Introduction to Ergodic Theory. Springer 1982.
2. C. Silva, Invitation to Ergodic Theory. AMS 2008.
3. T. Tao, Topics in Ergodic Theory, http://terrytao.wordpress.com/category/teaching/254a-ergodic-theory/

@BalarkaSen we can start with the stochastic integral

Balarka Sen

1 sec

@user2103480 chat.stackexchange.com/rooms/78363/…

join

user2103480

done

Alessandro Codenotti

14:19

Proof: Look at the colouring $c\colon[\Bbb Q]^2\to 3$ were a pair is coloured with different colours depending on what $f$ does to it, and use Ramsey theorem that $\omega\to(\omega)^n_m$ for all $n,m\in\Bbb N$

user2103480

You'd like the applications of nonstandard methods to ramsey theory

I've seen a two line proof of the debruijn erdos theorem

something something saturation

Alessandro Codenotti

Uhm actually this should give monotonicity of $f$ on an infinite set rather than arbitrarily large finite ones

@user2103480 The model theory proof is also quite short, it's an application of the compactness theorem

user2103480

@AlessandroCodenotti the nonstandard one is probably a reformulation of that

Alessandro Codenotti

Compactness arguments can be translated into ultraproducts argument, which should get us into nonstandard territory

user2103480

yup yup

Sayan Chattopadhyay

15:33

The proof given in Steinberg that the degree of an irreducible representation divides the order of the group, is the most ridiculous proof of any statement that I have ever seen.

It's like people were running around, stumbled upon algebraic integers, and thought, why don't I fit it in representation theory

Mike Miller

What is it lol

Balarka Sen

theres no better proof i think lmao

Sayan Chattopadhyay

They use the fact that if the degree of a representation is $d$, the size of the conjugacy class of $g \in G$ is $h$, then $h \chi_{\phi}(g) /d$ is an algebraic integer

Balarka Sen

they use algebraic integers which are also rationals are integers

fucking brilliant

Sayan Chattopadhyay

I don't care about algebraic integers, I was supposed to be studying representation theory. What the hell man

Astyx

15:37

lol

Do they give a proof that rational algebraic integers are integers ?

Sayan Chattopadhyay

https://arxiv.org/abs/1102.4353
I found this, I don't know how much sense this makes. I will have to check it

@Astyx Yup

I get it if you connect various fields, but do so in a proper manner. They never talk about algebraic integers anymore in the text, never have I seen them anywhere else in representation theory but here we are!

Mike

Is there some better algorithm than Gale Shapely which gives better complexity compared to O(n*n)

Tobias Kildetoft

15:57

I think that is pretty much the standard proof of the fact that character degrees divide the order of the group

Once you have that, most similar results do use more representation theoretic techniques to improve on it.

Which gives stuff like Ito's theorem or even Ito-Michler

Sayan Chattopadhyay

Yeah what I don't get is why such a relatively self contained result about irreducible representations uses something random like Algebraic Integers

People who do representation theory must be gods, I do not see any consistency.

Astyx

Lie groups and stuff ?

Balarka Sen

@TobiasKildetoft Ito's theorem lol @user2103480

Here is what you have to do @Sayan

Just avoid finite groups

Sayan Chattopadhyay

That should be like a general principle in life

user2103480

@BalarkaSen there's actually itos representation theorem lol

Balarka Sen

16:09

lolol

Tobias Kildetoft

Just study locally finite but not finite dimensional Lie algebras. All irreducible reps are infinite dimensional, so no need to worry about divisibility of dimensions

Balarka Sen

4

Tobias Kildetoft

(locally finite and simple I meant)

user2103480

@BalarkaSen any square integrable integrable random variable which is measurable with respect to a Brownian motion, can be expressed as a stochastic integral with respect to this Brownian motion.

but I know this is something completely different lol

Balarka Sen

Ohhh that's a good theorem

Tobias Kildetoft

16:12

The Ito's theorem I was referring to extends the divisibility of the character degree to the index of any normal abelian subgroup

Ito-Michler then sorta reverses that for Sylow p-subgroups

Alessandro Codenotti

They're all points

Balarka Sen

crunch that group

Tobias Kildetoft

i.e. if no character has degree divisible by some prime $p$, then the Sylow p-subgroup is normal and abelian

Mike Miller

@BalarkaSen Really? The proof identifying dim with size of a conjugacy class isn't better? (Too roundabout?)

Tobias Kildetoft

@MikeMiller I don't think I have ever seen a proof that identifies the dimension with the size of some conjugacy class

Balarka Sen

16:14

I don't think there's any other alternative proof yeah

Mike Miller

Am I assuming we are working over alg closed char 0?

and that's my error?

Balarka Sen

Yeah sure over C

Sayan Chattopadhyay

Yeah

Mike Miller

Ok got it

Balarka Sen

No what are you proving

Mike Miller

16:15

Is what I said false

Tobias Kildetoft

It does not sound true to me (or I think I would have heard about it)

Balarka Sen

I don't understand the claim; dimension of what is size of a conjugacy class?

I don't know rep theory

Sayan Chattopadhyay

I am guessing Mike is trying to prove that the dimension of the representation is that of the size of a conjugacy class

Leaky Nun

@MikeMiller for $S_3$ the dimensions are $1^2 + 1^2 + 2^2 = 6$ and the conjugacy classes are $1 + 3 + 2 = 6$

Tobias Kildetoft

@LeakyNun So the claim does hold there

Leaky Nun

16:22

what claim?

Balarka Sen

"Epitaph" - King Crimson plays in the background

Mike Miller

Meh I was thinking about things wrong

Astyx

Something something $\log(1)+\log(2) + \log (3) = \log(1+2+3)$

Mike Miller

Matrices are a spook

Balarka Sen

Truth

Astyx

16:24

Matrictober

Balarka Sen

What you need to know is number of irreps is number of conjugacy classes, sum of squares of degrees of representations is group order and uhh

And nothing else

And the class equation like Leaky wrote it

Sayan Chattopadhyay

Yeah with that info you can solve most problems

Maybe some Schur here and there

Balarka Sen

In the cheat sheaf of character theory you have to write row Hermitian orthogonality of characters and column Euclidean orthogonality of characters

Cheat sheaf lmao

Sayan Chattopadhyay

lol

Balarka Sen

rep theory of finite groups is a machine to crank out numbers

crank the shit out of them

Rithaniel

16:27

Hello folks

Balarka Sen

@MikeMiller Wachowskies announced Matrix 4

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$Mathematics$ Mathematics