Mathematics - 2021-02-20 (page 2 of 3)

Wolgwang

18:01

@anakhro What you think? :-)

anakhro

18:14

@Wolgwang We don't? If A is a set, A is in the powerset of A, always.

Ted Shifrin

Watch out. $\{\emptyset\}$ is different from $\emptyset$.

anakhro

@MikeMiller I see that Thorgott gives a counterexample to just substituting in x-a.

Ted Shifrin

No, @Thor is doing a different function.

DETAILS matter!

Wolgwang

@anakhro O_o

So Power set of $\{\phi\}$ is $\{\phi,\{\phi\}\}$?

Ted Shifrin

Is this any different from considering $\{a\}$?

Wolgwang

18:19

Yes...

Ted Shifrin

What is the power set of $\emptyset$?

anakhro

@TedShifrin But it's still a counterexample to just substituting in x-a into a power series about zero to get one around a.

Ted Shifrin

But that is not what you're talking about.

Wolgwang

@TedShifrin $\{\phi\}$

Ted Shifrin

Of course. $z= a+(z-a)$, so if $f(z)=z$, then $f(z)=a+(z-a)$.

OK, agreed, @Wolgwang.

anakhro

18:23

Maybe I am not sure what I am talking about at this point, Ted. I am trying to understand what is written in a book and I am grasping at straws at where they are getting this Laurent expansion around $a$ from (when at this point, the only thing we are given is the expansion for around $z=0$).

Thorgott

ahh, stop using phi for the empty set

use \emptyset

anakhro

Thorgott is first to break.

Thorgott

lol

Ted Shifrin

No, I just used the correct symbol :P

They're not looking at the Laurent expansion of $\wp(z)$, @anakhro. They're looking at the Laurent expansion of $\wp(z-a)$. This is why I said you have to think about the composition of functions.

IMHO, you're making a mountain out of a speck of dust.

Thorgott

What's the name for an element in an abelian group that can not be written as a proper multiple of another element in the group?

anakhro

18:26

Yes, I probably am. I just feel so out of my element with this elliptic function stuff.

It's been a frustrating week trying to understand some of this stuff.

Ted Shifrin

Just think about $f(x)=x^2$. Then of course $f(x-a) = (x-a)^2$. This is different from writing $f(x)=(a+(x-a))^2 = a^2 + 2a(x-a) + (x-a)^2$.

We're talking definition of a function. NOTHING to do with elliptic functions.

I have no idea of the context. I'm just paying attention to the literal text you quoted.

anakhro

I think I see what you mean. Because an expansion around z near a is f(z) = ... rather than expansion of f(z-a) (near zero) which is f(z-a) = ...

Roughly the right idea?

Ted Shifrin

Right.

Different functions.

The issue is at the level of precalculus, not at the level of graduate complex analysis.

anakhro

So that's what Mike meant about right answer to the wrong question. The substitution thing is the right answer to the question of finding the expansion of f(z-a), but not f(z) near a.

Ted Shifrin

Well, if you do the latter, you have to do the algebra. I just did two examples for you.

anakhro

18:32

Okay, now let me see if I can understand what the book says. Thanks for being patient and helping me, all.

Thorgott

@Thorgott I think the term is primitive

Wolgwang

@Wolgwang I still don't know which one is right :-/

anakhro

@Wolgwang what does it mean for B to be a subset of A?

Wolgwang

@anakhro It means that all the elements of B belong to A.

anakhro

Great, so let's test to see if your teacher is wrong. You claim that $\{\emptyset\}$ is not a subset of $\{\emptyset,1\}$. What are the elements of the former?

Wolgwang

18:39

Pink elephant

anakhro

Well when you want to try, feel free to ping me.

Wolgwang

@anakhro I had read it somewhere sorry :-(

Hint: Every element of the empty set is a pink elephant. Or an element of $X.$ (No joke) — Dan Christensen Jan 30 '14 at 4:59

BigSocks

so let's say I got this limit and it doesn't commute with a colimit I like. Let's say the category is nice enough for us to do interesting things (probably we'll need it to be Abelian for reasons I can't remember). How do I figure out "how bad" they don't commute and is there a way to do it in general?

maybe I haven't given enough details

Ted Shifrin

locks @BigSocks in the category room with @Thor

BigSocks

I am finally safe

Mike Miller

18:46

Somehow I think your textbook is a clearer source than a 0-upvote comment on a 7 year old thread.

Ted Shifrin

Was that supposed to make sense to the casual observer?

Mike Miller

He's making a joke which you will understand when you understand the empty set ... the point is that there is no such thing as a pink elephant ...

Ted Shifrin

Ohhhh ... that.

My elephants usually arrive purple.

Wolgwang

@anakhro No element is present in it...

anakhro

@Wolgwang are you sure? What are the elements of {x}?

robjohn

18:49

@TedShifrin with yellow flowers

Wolgwang

@anakhro x only

anakhro

Great. Now let $x=\emptyset$ and what do you notice?

Ted Shifrin

@robjohn Soon we should be at e e cummings.

Wolgwang

Element of {x} is $\emptyset$ ..

Ted Shifrin

This is correct. So $\emptyset$ has no elements, but $\{\emptyset\}$ has one element.

Astyx

18:55

A box containing an empty box is not an empty box

robjohn

And so does $\{\emptyset,\emptyset\}$

Wolgwang

:-o

Thanks :-)

anakhro

@TedShifrin do you like ee?

Emolga

Why if $\lambda$ is in the spectrum of some continuous linear transformation $T$ then $\lambda ^n$ is in the spectrum of $T^n$?

Mike Miller

Start by spelling out what the words mean...

Emolga

19:00

I know that $T-\lambda^n I$ is of the form $T-\lambda I$ times something

Mike, I don't assume finite dimensional

So it's not clear to me why (non-invertible) times (something) cannot be invertible

Mike Miller

Yes, this was clear from the word continuous. It still fails to be surjective or injective, you just now have to do a case analysis instead of focusing on kernel as in the f.d.case. Maybe what you're missing is that $B = T^{n-1} + \lambda T^{n-2} + \cdots + \lambda^{n-1} I$ has $B(T - \lambda I) = (T - \lambda I)B = T^n - \lambda^n I$.

Emolga

I am not happy with this because in reality I need it in a general Banach algebra

The book claims that if $\lambda \in \sigma (x)$ then $\lambda ^n \in \sigma (x^n)$

I agree that $B(x-\lambda I)=x^n-\lambda ^n I$

Balarka Sen

Suppose you randomly pick a collection of balls on the Poincare disk $\Bbb H^2$.

Choose a point not in this collection. Would you be able to see the ideal boundary from that point?

Mike Miller

If you don't ask the question you really mean then obviously you aren't going to get the answer you want. I don't immediately know the answer to the Banach algebra question, but I suspect you just want to argue that $(ab)^{-1}$ commutes with both $a$ and $b$ separately when $ab$ is invertible and $ab = ba$.

From which it will follow that both $a$ and $b$ are invertible with inverses $b(ab)^{-1}$ and $(ab)^{-1} a$ respectively.

Could be the wrong approach though, I don't know yet that this is true.

Balarka Sen

i.e., can random obstacles in your view clog the sky?

Ted Shifrin

19:16

I presume the collection is infinite?

Balarka Sen

@TedShifrin Yeah

Astyx

You don't need that much, it's sufficient to argue a has a right and left inverse whenever ab has a right and left inverse and a and b commute

Ted Shifrin

I think I've run into something like this before, a @Balarka, many years ago. Cool question.

Mike Miller

Yeah but can you argue that

I'm stumbling

Balarka Sen

That's interesting! It occurs to me I need to say something like this if I want to do percolation in negative curvature spaces

Astyx

19:18

ab invertible means there is a c s.t. abc = 1 = cab = cba, so cb anc bc are respectively left and right inverses

Mike Miller

Oh of course, the point is that if an element in a ring has a left and right inverse those elements must coincide

Which is in fact how you'd prove my commutativity claim

Astyx

right

geocalc33

0

Q: looking for $3D$ vector field satisfying certain projection conditions

I'm searching for a $3D$ vector field $V$ in $(0,1)^3$ whose parallel projections onto the boundary of $[0,1]^3$ are the following $2D$ vector fields: The parallel projection of $V$ onto the $x-y$ plane will yield the vector field $\{x\log(x),-y \log(y)\}.$ The parallel projection of $V$ onto the...

I'm open to hints!

Ted Shifrin

@Balarka: So if the countable collection of balls is dense, real trouble.

Mike Miller

@Emolga See the above discussion. This is pure algebra, no analysis. If a,b are elements in a monoid so that ab is invertible and ab = ba --- meaning (ab)c = c(ab) = 1 for some element c --- then in fact a, b are invertible. To see this, observe that a has a right inverse and a left inverse: a(bc) = 1 and c(ab) = c(ba) = (cb)a = 1. Now observe that because cb = (cb)(abc) = (cba)(bc) = bc in fact the two inverses coincide and a is invertible. Similarly b.

Balarka Sen

19:22

That should be a 0 probability event

Mike Miller

Notice the essential use of commutativity, which is why this doesn't reflect examples like the shift operators where a,b can fail to be invertible even though ab is.

Astyx

You also need to argue that B is a bounded operator

Emolga

@Astyx @MikeMiller I understand now, thank you very much!!

Mike Miller

????? In our explicit case $b = a^{n-1} + \lambda a^{n-2} + \cdots + \lambda^{n-1}$

no argument needed...

@Emolga For sure. Notice that the POV one uses when thinking about B(H) is quite different than the POV one uses when thinking about an abstract Banach algebra :)

if you're working with C* algebras by Gelfand-Naimark the two POV's coincide but that's using heavy machinery

Balarka Sen

What's an analogue of Stone-Weierstrass theorem for topological Banach algebras?

Emolga

19:26

Yes, I now realize that

Balarka Sen

There must be one, right

Mike Miller

TIL Naimark's name is "Mark Naimark"

Troll parents

Emolga

lol didn't know that either

Astyx

@MikeMiller Yeah sure, but you still need to keep that in mind

If ab is invertible but b is not bounded, then you can't say anything about a

Mike Miller

Sure but nobody ever said any sentence in a level of generality which suggested b was an unbounded operator :) That would be unnatural

Balarka Sen

19:29

Astyx is actually German, not French

Mike Miller

I get your point but I think it's pedantry

@BalarkaSen They're the same, mathematically

Balarka Sen

Thurston school bro

Pierre Pansu, Francois Laudenbach

et al

Astyx

Perhaps it is :)

Mike Miller

That's uhhhh uhhhhhh the exception that proves the rule

Balarka Sen

apparently Thurston was an inspiration for Grothendieck. What exactly he inspired Grothendieck with I do not know

I just know it is true

Ted Shifrin

19:32

Wow. I thought Grothendieck was mostly done by the time Thurston came along.

Balarka Sen

Yeah he was thinking about Galois-Teichmuller Yoga lmao

that must be the inspitation then

Mike Miller

Maybe he was inspired to stop writing down his results

Balarka Sen

loool

good one

ok yeah it is GT

archives-ouvertes.fr/hal-01286494/document

(one of the authors, Athanese Papadopoulos, is from the Thurston school - i forgot to mention him above)

Mike Miller

Most greek name I have ever heard

Balarka Sen

yeah but i think he was born and brought up in the french school mathematically

i could be wrong

geocalc33

19:38

he has huge 3 volume books on Teichmuller theory

I saw them in the library

Balarka Sen

yeah student of Laudenbach

Mike Miller

Just amused

Balarka Sen

i always think of the tintin character when i hear papadopoulos

oh lol rastapopoulos is also greek

Ted Shifrin

Rastafarian !

geocalc33

are there any good math podcasts out there that everyone can recommend?

Mike Miller

19:41

I think Papakyriakopoulos whenever I hear papa

Balarka Sen

LOL

impossible name

Mike Miller

had to edit twice

Ted Shifrin

I remember that!

Balarka Sen

greatest proof of all time in basic topology

Mike Miller

"Papa's theorems"

Balarka Sen

19:42

@geocalc33 honestly there should be

Mike Miller

I struggled for a long time to understand that for my reading course before getting started on research

Balarka Sen

i straight up did not understand it for ~3 years

i think i understood it very recently.

for a long time i wanted to modify the self-intersecting disk to make it an embedded disk

Mike Miller

It's very clever! It's one of these things that makes no sense until you understand it when it is very clear.

Balarka Sen

papa avoids it completely

he just takes covers and just gets a new guy in the cover

philosophically it is like unwinding but thats not how you say the proof at all

Mike Miller

I distinctly remember being asked to explain this and giving some handwaving thing because I didn't really understand it. I think my explanation only makes sense if the tendril goes around once.

Balarka Sen

19:45

hahah

Ted Shifrin

<--- definitely NO topologist

Mike Miller

Very trial by fire summer

Balarka Sen

speaking of summer, I hear things happened in Texas

Futurism finally?

Ted Shifrin

Opposite of summer.

Balarka Sen

lol

Mike Miller

19:46

Learnt 3-manifolds, kirby calculus for 4-manifolds, symplectic topology, knot theory, a little contact topology

Balarka Sen

good, wish i could read that many things at the same time

i read so little

Ted Shifrin

I know none of that.

Balarka Sen

there should be undergraduate courses on 3manifold topology because otherwise no one will learn it

thats the point of undergraduate courses, teach dead subjects

Ted Shifrin

Hard to do that without algebraic topology first.

Balarka Sen

yeah, that's annoying

Ted Shifrin

19:49

I aim to please (being annoying, that is).

Mike Miller

It was fun but those were 80 hour weeks. It's not something I could ever do again.

Balarka Sen

this was your 1st year grad school i assume?

geocalc33

found a pretty good episode of a podcast

It's called "Oxford Mathematics Public Lecture: Henry Segerman"

Mike Miller

Summer after first year

Balarka Sen

Cool

All I have been reading is probability. Feels bad man

Mike Miller

19:58

One day I'll take a year off and just read combinatorics for a year with no human contact

Balarka Sen

why "no human contact"?

Thorgott

I think that comes automatically when you do combinatorics

Balarka Sen

What nonsense

Mike Miller

@BalarkaSen I'll catch up on my human contact the next year after I truly understand combinatorics

Thorgott

is understanding combinatorics even a thing

Mike Miller

20:03

Yes

Though I do think we should rename all intro courses to "Counting 101"

Balarka Sen

Count the number of self-avoiding walks of length $n$ in $\Bbb Z^2$

Or $\Bbb Z^d$ in general

If you can do it you get Fields medal

Thorgott

should we also rename intro topology courses to "Wobble 101"

Balarka Sen

unige.ch/~smirnov/slides/slides-saw.pdf

This is where they do it for the honeycomb lattice

Mike Miller

@Thorgott No, call them Introduction to Rubber

Thorgott

also good

the continuation will be An Introduction to Glue

Mike Miller

20:08

[0,1] is a quotient of the Cantor set? more like you can glue a cloud of particles together into a line

Thorgott

obvious

Alessandro Codenotti

Just glue together the endpoints of every removed interval

Mike Miller

@BalarkaSen Get me a teaching free job at ISI and we will do it

geocalc33

Why do they say "don't work with the square lattice self avoiding walk?"

Balarka Sen

@MikeMiller Too hard, let's rob a bank instead

We split the money. You don't have to teach and I don't have to do grad school

Mike Miller

20:16

Well now that you've posted about it the jig is up

Balarka Sen

They don't know that we know that they know

Ted Shifrin

takes notes furiously

geocalc33

Yeah now I am racing to solve this problem

If I were a random walk how would I move?

Thorgott

randomly walk towards a bank and rob it

Balarka Sen

Gambler's ruin

but its the bank

Mike Miller

20:21

Solid bit Thor

geocalc33

hahaha

Balarka Sen

geocalc33

1

Q: Parametrization of curve in $\Bbb R^3$

I’m trying to find the parametrization of a curve in $\Bbb R^3$ satisfying $p_1=(0,1,1)$ and $p_2=(1,0,0)$ that parallel projects to the x-y plane onto $\log(x)\log(y)=1.$ I think that maybe I can do a rotation on the x-y plane curve and then scale its length to satisfy the endpoint conditions? A...

algebra-precalculus projection

can someone upvote so they can talk in chat?

Hi @user889830

Thorgott

am I understanding correctly that the total spaces of S^3-bundles over S^4 still haven't been fully classified up to difeomorphism?

Mike Miller

Doubt it

arxiv.org/abs/1406.2226

robjohn

20:35

I try not to get the phrases "Rob the bank I'm at" vs "I'm Rob at the bank" mixed up.

@Thorgott Not a bed time story for horses...

Mike Miller

people.oregonstate.edu/~escherc/nov06.pdf @Thorgott

Every classification you could possibly want

The diffeomorphism classification is in Corollary 1.6 and has you reduce mod 56n. :)

Thorgott

the Crowley-Escher paper doesn't have a complete diffeomorphism classification, but it seems the Crowley-Nordström paper may have it

Mike Miller

Oh, you aren't satisfied with oriented diffeomorphism?

Thorgott

1.6 only tells when $M_{m,n}$ and $M_{m^{\prime},n}$ are diffeomorphic, but it doesn't seem to say anything about $M_{m,n}$ versus $M_{m^{\prime},n^{\prime}}$

Mike Miller

The n can be read off from H_4 of the total space

Or at least |n|

H_4(M_{m,n}) = Z/n

Thorgott

20:45

duh, I forgot about that

thanks

Mike Miller

And then for orientation-reversal, the orientation-reversal of M_{m,n} is M_{-m-n,n} from Rmk 1.1

It's not so obvious to me how the cases of n and -n relate admittedly

Here are some combinatorics questions to chew on if you want

p is prime.

(1) Give a combinatorial proof that C(pa, pb) = C(a,b) mod p (This is easy, I had a proof in a few minutes)

(2) Give a combinatorial proof that C(pa,pb) = C(a,b) mod p^2. (Still working on this)

(3) p >= 5. Give a combinatorial proof that C(pa, pb) = C(a,b) mod p^3. (I think this is open)

Thorgott

but you just said how they relate?

Mike Miller

Oh I see I should have read their remark more carefully. Notice that my n is unchanged there. But their remark implies that $M_{m,n} \cong M_{m+n,-n} \cong M_{-m, -n} \cong M_{-m-n, n}$ unorientedly

Whaat I wrote above is an oriented diffeomorphism I think

Oh I've been foolish it literally says the orientation reversing case right there, and then these formulas tell you how to see if M_{m,n} and M_{k, -n} are oriented diffeo or not --- check whether M_{m,n} and M_{-k, n} are or-reversing diffeo using Cor 1.6

Ted Shifrin

@BalarkaSen He turned the bank inside-out?

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$Mathematics$ Mathematics