and since it switches those two elements

If I put it back in

I would get the original back

And hence since I am conjugating on the whole of $T$

I conjugated both of them, and hence they satisfy eachother

Hey @Incurrence @DiscipleofBarney etc

@KajHansen Hello

@KajHansen Hey, how are you?

Not too bad @Incurrence. I have one fairly challenging topology set remaining, and then it's final exams for me.

Sweet, and then you are off to REU then 4th year right?

Indeed @DiscipleofBarney

@KajHansen Oh, that's good - I still have many weeks left, but I am enjoying them

I'm excited!

Yeah @Incurrence. It seems the semesters/quarters are set up much differently in other countries.

@KajHansen If you want to get really excited scroll up a little ways for some hot pictures. ;P

LOL

lmao

What pictures?

@KajHansen Have you thought much about what you are looking for in a grad school? (field wise especially )

What're you studying now @Incurrence ?

@KajHansen Group theory - trying to show that diagonal matrices are a subset of the normaliser of diagonal matrices (under the group of invertible matrices)

@DiscipleofBarney, so far I've greatly enjoyed my algebra courses, so I might look into going down that path. From what I've read (and mostly not understood) Lie algebras and the sort look enticing to me. I've also had a blast in this semester's point-set course, so I might try out algebraic topology when I can.

@KajHansen Well point-set/general topology is totally different from algebraic, its really cool though.

Have you done a similar problem to mine @kaj?

@Incurrence I've shown in the past that diagonal matrices commute with all other matrices.

Wait I screwed up

I meant to say that I am trying to show that permutation matrices are a subset of the normaliser of diagonal matrices(under the invertible matrices)

Unfortunately, my GPA is not that great, so I'm afraid I might not have that much of a choice as far as grad schools. Or at the very least, I will need to send lots of applications.

@KajHansen ^That is an open problem book filled mostly with point-set topology stuff

Though I've enjoyed my point-set course, strictly point-set topology doesn't much interest me as far as a career path. I appreciate the link nevertheless!

@Incurrence, I think it was you I've been sharing music with. I found a new band recently. Try this out: youtube.com/watch?v=tQPNredrDO0

Oh I figured that, just I think its sort of out of fashion, even though there is still tons of cool stuff (I especially like the (infinite)combinatorial and games in topological spaces). Although I guess it was always out of fashion, the fields that study the more exotic/messy stuff almost always seems to be out of fashion. @KajHansen

@KajHansen It was indeed, listening now

It just seems a bit unmotivated @DiscipleofBarney. I'm all for doing math for the sake of doing math, but something about finding a topological space with obscure properties $X, Y$ and $Z$ but not property $W$ just doesn't get my blood pumping.

@KajHansen Sounds terrible.

:P

@Incurrence, maybe try this one if you're on the fence: youtube.com/watch?v=ySjXFjLTagQ

Oh yes That song I have heard many times :)

that was the only song I had heard from them though

Ah, cool cool

I am trying to find a counter example to injectivity of the exponential map(for real matrices)

He said take real multiples of this, but it isn't working

@Incurrence, by exponential map, you mean $e^A$?

Yep

On real $A$

Are there still any open problems in calculus??

And why are all the millennium problems in number theory (except Poincare's conjecture)

What about PvNP and Navier-Stokes

Yang-Mills

Yes they are but most of them involve strict number theory......

Hodge conjecture (I am sure it is related to NT)

So that leaves two that are NT

Is there any problem related to combinatorics

Out of 7

algebraic topology

yes

web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf @DiscipleofBarney look at this article the authors says infinite sets are nonsense and dont exist

@Rememberme, such people are called finitists / ultrafinitists

See here: en.wikipedia.org/wiki/Ultrafinitism

Finitism and ultrafinitism are sort of interesting, I think it would be cool to have a coherent ultrafinitist theory of mathematics

Why are there such people i mean whats wrong with infinite set of numbers

@Rememberme Because there is no real word example

They don't exist in the real world

The universe is made up of a finite number of components

It is finite in length(depending on theory)

Then give me real word example of transcendental numbers

then even they dont exist

@Rememberme Although I probably won't read it... its written by an Australian ( ;P @Incurrence)

lol

@DiscipleofBarney It's actually a pretty good read so far

Well I skipped to page 6

Hey @Incurrence i just thought of something what is the smallest distance that you can form out of some points in a plane

@Rememberme What does that mean?

What is the minimal distance moving from the first point to the last?

Minimal distance between any two points

Take the norm

Wait i can improve on this

The minimal number of distinct distances between points in a plane@Incurrence

What does that mean?

Hi guys

@AndreiMihai Hello

Ultrafinitism on MO

If i take n points in a plane what will be the minimal number of distances between points which are different @In

You mean the minimal distance to go to each point?

between two points

That doesn't make sense to me sorry

What does minimal number of distances mean

I have a question about MVT: I saw some cases when the point c is included in the closed interval, and I know that it should be included in the open interval

therefore, is matter?

what will be the number of smallest distance between n points in a plane @Incurrence

Take n points

there will be the distance between points and the distance might be smaller or larger

so i am taking the smallest distance

now fix another point,there will be a distance which is smaller

Hi @WillHunting

what's up?

so i have to count the number of smallest distances between points by fixing a certain point @Incurrence

@Rememberme Not sure what you are talking about, but you can make the distance arbitrarily small

@AndreiMihai Hi, I am surprised you talk to me. Have we talked before?

yes

@AndreiMihai Oh, what did we talk about?

I forgot, but I remember your name

"Hunting"

I am not considering the points to coincide they will be distinct points @Disc

Yah so

I think was about a graph

@Rememberme There is no smallest distance. You can make it as small as you want. End of story.

Consider the set of $1/n$, that gets arbitrarily close to $0$

Its not there has to be something out of it

I am not familiar with Apostol Calculus, but I am pretty sure it covers these ideas, and I think you said you did every problem in Apostol

Or you are trying to say something completely different

Completely diffferent

dosent involve calculus at all

And who stars these messages always

Your talking about distance between points, and finding the smallest distance, but in calculus and analysis there is an "idea" of making distances as small as you want and it is important for the idea of continuity, integration, differentiation, etc

@DiscipleofBarney Look at this i want to find the minimal number of distinct distances between n points in a plane

And i think this is not calculus

this is geometry

Do you want to find the configurations of n points that has the minimal number of distinct distances?

Gotcha!!!!!!!!!yes

So you want them to all have the same distance?

Not same but the least distance

So for example with 3 points you can get all the distances the same, so there would be one distinct distance

Right?

yes you got my problem

And for 4 points, an upperbound would be two distinct distances (arrange points in a square). I am pretty sure that would be the least

What about n points because according to me there is nothing which you can generalize

I would guess it would be to arrange them into a regular n-gon

And the distance will be

then it would be something close to n/2 distances

@DiscipleofBarney I am not sure whether that is what he is asking.

Wait @DiscipleofBarney i think i found something very close to what i state

@Chris'ssis weekends are busy around here. I will probably be somewhat scarce tomorrow as well.

@Rememberme Can you give an example of what you mean, explicitly, with say 3 or 4 points?

@robjohn Hello rob, I am still alive. =)

Wait i a, going to show you a paper which is similar to what i say but i dont understand even a bit of maths in it

@WillHunting That is a very nice thing.

cs.umd.edu/~gasarch/erdos_dist/erdoshs.pdf look at this @DiscipleofBarney

@DiscipleofBarney Now is this related to calculus

@Rememberme Is this what you are asking about then, or it is something else?

It is related to this

It is important to phrase your question properly.

@Rememberme This is mostly some sort of combinatorial geometry, I am sure that you could use calculus to study it though

So how do you solve what i said

The thing is, we are still not sure what it is you asked. =)

Yah, still not sure, since you said it was not what the Erdos distance problem is

If you want the smallest distance between two points given n points in the plane, there is no smallest distance.

why?

Because you can make the distance as small as you want.

If this is not what you are asking, then try to think what it is you are asking, and write down the question properly first.

Consider the distance between $0$ and $1/n$

you can make $1/n$ as close to $0$ as you want by varying $n$

@DiscipleofBarney i think i am clear know, if i have points in a plane which are not collinear how do i prove that if i pass a line through n points there will always be a points opposite to the given line

@Rememberme If they are not colinear how does a line pass through all points?

And I don't know what you mean a point opposite to the line

through some points by mistake i wrote n points

Or there will always be a point in front of it

by it i mean the line

I don't know what you mean by in front of a line or opposite to a line

@DiscipleofBarney Hey.

@BalarkaSen Hey, what's up?

I'm pondering on whether knowing that every map between finite-dimensional projectivized complex vector spaces has fixed points is interesting or not.

@DiscipleofBarney its the pic

It's certainly not something nontrivial, but I "feel" it can be generalized to Grassmannians.

@Rememberme By definition of not being colinear, they can't all be in the same line

Sounds interesting @BalarkaSen

Some of them will be

@mixedmath halloa.

@Rememberme But not all of them, hence there will always be such a point

@BalarkaSen hihi

Okay....

hmmm

i thought you were writing up a blogpost on some lattice-counting problems, @mixemath.

Hello @Balarka

@Rememberme It sounds like you are asking why given a set of colinear points there will always be a point not on a line through some of the points. But that is the definition of colinear (as far as I know)

@DiscipleofBarney It's not hard to prove : every linear operator on a finite dimensional complex vector space has an eigenvalue. :P

I wonder if it's even true for Grassmannians.

@Disciple of Barney Is there any way to prove that that point is perpendicular to the line

How is a point perpendicular to a line? @Rememberme

the line passing through the point

their is always a line passing through some point which is perpendicular to any other line, @Remember

How can the line pass through the point if you are assuming it doesn't

Proof

@Rememberme look at your 8th grade geometry book.

No proof

proofs were not introduced at that point

Do you know how to make a perpendicular line given some line? @Rememberme

By making the angle 90

it's a standard construction. take your point, draw an arc. the arc cuts the line in two points. pick the two points, draw arcs again. the two arcs intersects in a point. join the original point with this point. this is your desired perpendicular.

Okay so constructions

it can be proved that this is a perpendicular. take it as an exercise.

@Balarka you like geometry right

i have hundreds of questions in it

spare me.

Good ones

not atrocious ones

Why don't you figure them out yourself

@DiscipleofBarney So, what have you been studying lately? Anything interesting?

@DiscipleofBarney this one is good

@BalarkaSen Been studying (sort of) small cancellation theory

@Balarka this is a good question please consider it

@DiscipleofBarney If you're prepared to talk a bit about it, I'm listening.

@Rememberme There is a theorem that any cutting of a rectangle into triangles has to be an even number of triangles, wish I remembered the name

I checked it it will be triangles only when n is even

'night, @Mike (and i really mean it).

@BalarkaSen Sort of, the sort of it is that small cancellation conditions on a group give a lot of control over what elements in a group can be the identity. One of the reasons certain problems in free groups are easy is that the cancellation between the relations is very small (there are no relations!), so in a sense it is a generalization of that.

It gives a geometric way to thing about the defining relations in a group

interesting. what is the geometric perspective of relations you mention?

@BalarkaSen Its sort of like Cayley graphs, but a little different. Relations can be seen as labeled circles where the boundary has the relation written on it, and conjugation can be an element would be a line sticking out from the circle. Consider $uru^{-1}$, if you think of a "balloon" where the string is labeled $u$ and going arround the balloon is labeled $r$, then as you travel up the string you pick up "$u$", then arround the balloon you pick up $r$ and then back down the string you pick

up $u^{-1}$

If you have a bunch of relations you have a bunch of balloons tied together at a common vertex basically

And you can travel around the whole outside

and come back to the start and you would have the product of all of them

ahh, ok.

Anyway relations, when multiplied can end up having some terms cancel

@DiscipleofBarney Monsky's theorem?

morning

I am home now

@MikeMiller I think so, knew it started with an M

Now to do more work

@Balarka Maybe you can give me some advice on one of my few problems

I can't tell certainly unless you give me the problems.

First of all, is $P$ the group of all permutation matrices, a subgroup of the normaliser of the diagonal matrices in the group of invertible matrices

@BalarkaSen The cancellation ends up corresponding to being able to identify "strings and edges of balloons" So that is how you can end up with the "wiener" picture I have a little ways up, because there was some cancellation (in the picture not everything that can be identified is identified)

So is $P$ a subgroup of $N_G(T)$ where $G=GL_n(\mathbb R)$ and $T$ is the subgroup of diagonal matrices, $T\subset G$

@Incurrence Yes.

Okay that makes things more difficult

Ok

I need to prove that then I guess

Small cancllation conditions gives a geometric way to restrict how much identification is possible which ends up giving algorithms for things like the word problem and conjugation problem (I have only been describing things on a sphere, but you can do these diagrammatic stuff on toruses, etc)

@BalarkaSen For example remember that group the iwrite showed us and the word that we were trying to figure out if it is the identity, I am pretty sure that group is $C'(1/6)$ so when I get to the point of studying the word problem for that class I will see if I can apply it to that group.

@BalarkaSen Also, one of the main reasons I am studying it now is I am pretty sure it will give me some tools to finally settle a problem I have been working on and off for around 2 years (maybe a year and a half)

sounds fun.

@DiscipleofBarney what's the problem?

Okay, first I will describe the "game" theoretic problem, then the "selection principles" variation, and I am pretty sure there is a (infinite) Ramsey theory variant @BalarkaSen

Say we have a group $G$, then we will play a game, I will present sets that normally generate the group $G$ and you get to choose one element from each set I present @BalarkaSen

okay

You win if you are able to collect elements so that your set will normally generate the group (this game is at most countably infinite length) and I win otherwise

So for example for finite groups you will always win

if you don't play like an idiot

My original idea for this is that you choose relations to add to the group and you win if you get the group trivial, this is obviously equivalent

i will always win if the group is finitely generated, not?

@BalarkaSen What properties should I consider to get this started? I have never worked with normalisers

@Incurrence take a diagonal matrix. conjugate with a permutation matrix.

show that you have a diagonal matrix.

@BalarkaSen So there are infinitely presented finitely generated groups, so unless you have seen something I haven't seen for 2 years then it shouldnt be obvious

Infact I think it is false

But am I not conjugating with elements $g\in GL_n(\Bbb R)$?

i'm sorry, yes.

@BalarkaSen Have you seen something, I am a little scared now...

But I want $P\subset N_G(T)$

@DiscipleofBarney wait a sec, i have to digest what i read above.

@DiscipleofBarney He was talking to me I think

@Incurrence Ugh. I am doing too many things at once. You do conjugate with a permutation matrix.

You have to show that conjugating a diagonal matrix with a perm. matrix gives you a diagonal matrix.

That shows that $P \subset N_G(T)$, doesn't it?

Hmmmmmmm

Yes

It does

Right. @DiscipleofBarney Let me read what you have written above properly.

So then I have a subset, then I need to show closure, inverse, identity?

Well, of course the set of all perm. matrices forms a group (isomorphic to S_n)

@Incurrence I don't think the problem was asking you to show it was a group

Oh yes, I already proved that

Okay yes, sorry and thank you lol

I'll be back when I am done

@DiscipleofBarney I obviously being silly, but why is it "hard" for me to win for finitely generated groups?

@BalarkaSen It may be possible for me to keep on presenting you with sets (maybe with crazy long words in them) so that in the end you are forced into an infinitely presented group, which the trivial group is not infinitely presented

Or maybe a group where you can't even decided if it is the trivial group :D

(not sure how possible that is)

wait, so you're showing me set of words on the group? i thought you were giving me set of generators.

I am showing you a set that normally generates the group, but you only get to choose one element, then I can show you a different set that normally generates the group.

and you get to choose one from that one

etc

so it is a set of words that I show you

ah, i see.

that makes things harder. for finitely presented groups, i can win, right?

Not necessarily

I don't think so

well, any set you hand me over has finitely many small words in it. i can keep picking unless i can generate the whole group, right?

or is it possible for me to end up with something else while i'm at it?

You only get to pick one word from a set I present you

But maybe every word in the set has length longer than googol plex

and every new set I present you each word will be at least a googol plex times longer than the previous words in the previous sets

So every time you pick a word, you may only "thin" out the group a small amount

but if the group i choose has "small enough" presentations, then the words won't be so long.

@BalarkaSen The free group on two generators has no relations and I can present words as long as I want

blergh. you're totally right.

hey, this is a fun problem.

off topic: to show that a series is holomorphic, why does it suffice to show it converges uniformly on compact sets?

Yah, this came out of a different game that general topologist/set theoretic topologist "play"

I present an open cover and you get to choose one element from each cover I present, can you end up with an open cover?

and there are variations where you get to choose finitely many things (but any number), etc

Interesting. I haven't heard of that one.

One can similarly win for compact spaces for this one, I guess.

sorry i meant "meromorphic" not holomorphic

@BalarkaSen For the game where you get to choose only one set from each cover not necessarily

even for compact topologies

i am gonna think about it while having a cup of coffee.

@BalarkaSen The "selection principles" variant is I present you with a sequence of sets that normally generate the group upfront and you get to choose one element from each one. If for any sequence I present you can choose elements so that when those elements are added to the relations of the group makes the group trivial then the group is "$S_1$". This is different from the game version because in the game version I can "change" which set I present depending on what elements you end up choosing

But in this one I show the whole sequence upfront

I haven't come up with a reasonable Ramsey theoretic statement that is similar to these, but I think there should be because the topology variants of these types of games have closely related Ramsey theoretic statements. So there tend to be connections between these properties and infinite combintorics on whatever structure you are studying.

@BalarkaSen So I don't think that the free group on two generators, $F_2$, has the property $S_1$, and I think player one has a winning strategy (the one that presents the sets) and I think small cancellation theory will help me show this

Basically my strategy (that I will have to figure out the details on) is this:

I will have to construct a sequence of sets so that any set will normally generate $F_2$ (so when added to the relations makes the group trivial) but any one element choice from each set will in the end give you a group that satisfies the small cancellation condition $C'(1/6)$, which means I will have to be able to construct a bunch of sets that have very little cancellation between each other. In the end I don't think it will be difficult, but to actually prove it I think I will need the

indeed, i think it is a safe bet to say that F_2 is likely an example where one may loose.

knowledge of small cancellation theory

@BalarkaSen Anyways that is the gist of it

For the integers the second player can win though (one of the first things I proved, after finite groups)

sounds good to me, although i know nothing about small cancellations.

@DiscipleofBarney hmm.

@BalarkaSen I know next to nothing about small cancellation, that is why I am studying it now ;P

well, any set you give me will contain words made out of x. if i pick some element, i will get the group nZ where n is the length of the word i chose.

the next one will give me either nZ or mZ for some m < n.

I actually came up with the idea that I need small amounts of cancellation about a year ago, but was at a loss of how to construct such a sequence and proved it had the properties I wanted, and I had heard about small cancellation theory before, but I didn't make the connection till recently! @BalarkaSen

and after countably sensible plays, one will win.

so yeah.

So for finitely generated group, if player two has a winning strategy they can always win in finitely many turns

@DiscipleofBarney cool. let me know when you prove the problem. you should write up a paper, in fact :)

(I haven't really done much with the project for about a year though, so ...)

I actually do have like 8 pages of notes typed up, so will definitely add a proof of this, if it works out, to those notes

In the end I want to see if there is an interesting "organization" of groups using these combinatorial/game theoretic properties

So for example are there any nontrivial things all $S_1$ groups have in common

sure thing.

it'd be a nice "programme"

can you refer me to some material(s) on the topologist's variant of the problem, btw?

Yes I can! Let me find of the papers

Some of the keywords though will be "selection principles" "selection principles of open covers" "combinatorics of open covers" etc

Can I have a hint for proving that $PDP^{-1}$ is still diagonal? I can see that it is true, with many of the diagonal elements being permuted, but I can't find a way to prove it

Doesn't help that I am bloody tired, but I have to do a few more questions(this is my last assignment that was handed in as garbage) to make sure I don't screw up the next one - and I have complex due tomorrow

Take pity

I'll go have a quick coffee

@DiscipleofBarney OK, thanks, let me see what I can find.

@BalarkaSen I am putting together a folder right now for you

ah, thanks a bunch!

@Incurrence try determining what $P^{-1}$ might be.

@BalarkaSen So I have only read a small fraction of what I am going to share, but I would probably recommend looking at the BoazSametRamsey files and the Math580Notes (although those are notes a friend took in class so they are not complete or polished, but they are a good spot to start)

@BalarkaSen dropbox.com/sh/v1wyzzi91eqr5d3/AAAp6qba0dx9WV3ZGwMGNErja?dl=0

i'm gonna have a look, thanks again.

ugh, my internet connection is going off and on.

The Topological Games 50th anniversary also looks nice, but I havnt read it

so true. mike and i were just talking about cannon and conner's papers on fundamental group of pathological spaces.

mike said that the small fraction of people who still thinks about those stuff are vanishing, which is a bit sad, the truth be said.

I personally think it is sad too

:21306256 many doesn't. i don't think i do too, but i think that kind of math is definitely worthed considering.

I think it is one of the reasons I like group theory, is that I can zoom in easily and look at all these super nice, interesting concrete groups or I can look around at the insane mess and complexity of general groups, but it still has some relevance (normally) to the rest of group theory and relations to other fields.

@BalarkaSen Sorry was AFK, $P^{-1}=P^T$ I proved recently

there was this guy named Daniel Rust who used to come to the chat. he's a shape theorist, and we used to discuss loads of stuff about it since at that time i was interested in p-adic solenoids (for completely different reasons : solenoid spaces provides insanely nice geometric representation of absolute galois groups).

Awesome, I always thought shape theory looked fascinating, don't know to much though :(

i had the impression that those were not-so-bad stuff.

@DiscipleofBarney me neither. i just think that some shape theory would provide a concrete connection between solenoid spaces and profinite groups. in particular, i think absolute galois group of function spaces acts geometrically on solenoid riemann surfaces.

all of it were just some pipe-dream, but i kinda keep it at the back of my head. it was a sound idea.

Its definetly fun to see if your sort of vague intuition about how something, you know next to nothing about, could interact with something you now a little bit about, I think its one of my driving forces in learning a lot of math

definitely.

(although I guess that approach can get a little out of control...)

(thinking of Rememberme :P)

haha

Its probably my favorite thing about mathematics: the connections

@BalarkaSen Do you work on projects?

depends on what projects mean. :P

Eh, I guess you sort of treat something sort of like its research, instead of working through a book or whatever

sure, i think about small, interesting, hard problems.

one of them is what i just talked about above : finding some space on which Gal(\bar Q/Q) acts geometrically.

Yah, have you looked much into dessin d'enfants?

yep :)

Is finding the center and commutator subgroup of $S_n$ trivial?

Well not trivial

But it is easy?

center of S_n is trivial.

commutator subgroup of S_n is A_n.

and its not that difficult to show that

(the only normal subgroup of S_n is A_n)

I don't know A_n

One sec looking up

look up alternating groups

I was reading about these things called "constellations", which have relations to drawing maps on Reimann surfaces, which of course has relations to dessin d'enfants, about a year ago..., forgot most of it I think

Is it easier or harder than finding the commutator of GL_n(R)