Mathematics - 2016-01-06 (page 1 of 2)

Mike Miller

00:00

instead of orbifolds better to think of it as them studying instantons which are "singular along a graph with specified singularities". it's a very mild generalization of instanton knot homology, where you study things with specified singularities along the knot

They specify precisely what's left to prove in section 1. I don't have a good idea how hard that should be or what it should entail. Should hav's a better idea at the end of the quarter

Semiclassical

hmm

Balarka Sen

I don't understand what an instanton is, so probably I won't open the links. :P

Mike Miller

don't need to to read section 1

Semiclassical

i know what it is in physics!

Mike Miller

"instanton homology" are two words in the name of my project, @SemiC, so I would say "extremely close"

Semiclassical

00:02

hmmmmm

Mike Miller

a majority of gauge theorists nowadays work on monopoles instead of instantons. I'm the "instanton guy" in the dept

the last instanton guy graduated

Balarka Sen

... i thought monopoles do not exist.

Mike Miller

they're just words.

Semiclassical

monopoles = certain kinds of solutions

Balarka Sen

ah, ok. i thought it was something to do with magnetic monopoles.

Mike Miller

00:05

I think it is relayed to that. But they mathematically make perfect sense

the question is: d they exist IRL?

Semiclassical

well, it does. but whether or not magnetic monopoles exist in the real world, monopoles as field configurations are certain valid in various models

Mike Miller

if you prefer I could have said "most gauge theorists do stuff related to seiberg Witten Floer homology"

or maybe other sets of equations. instanton is less common

Semiclassical

that actually sounds kinda-sorta sensible, given what i remember about SW

namely that )physicists, anyways) talk about monopole regime, electric regime, etc.

Mike Miller

to me monopole and SW are synonymous

instanton and Yang-Mills are synonymous

in the sense that "instantons" and "Yang-Mills theory" mean the same thing to me

Semiclassical

hmm

out of curiousity, does the acronym BPS mean anything to you?

Balarka Sen

00:11

look up in urban dictionary maybe

Semiclassical

oh, i know what it means. i encountered it on the physics side in connection with seiberg-witten stuff

Balarka Sen

pretty sure you'll get lots of meanings there

Semiclassical

so i wondered if it's something Mike had run into

Balarka Sen

ah, ok.

Mike Miller

nope!

@Balarka: I looked again and section 1 is entirely readable.

I first read it before I knew what instanton homology was, if that helps.

Semiclassical

00:13

nLab's entry on BPS states has an interesting paragraph

"Several mathematical theories in geometry are interpreted as counting BPS-states in the sense of integration on appropriate compactification of the moduli space of BPS-states in a related physical model attached to the underlying geometry:

most notably the Gromov-Witten invariants, Donaldson-Thomas invariants and the Thomas-Pandharipande invariants; all the three seem to be deeply interrelated though they are defined in rather very different terms. The compactification of the moduli space involves various stability conditions."

Balarka Sen

@MikeMiller Ok, bookmarked it. Will read tomorrow.

Mike Miller

I once read their article on gauge theory, where it told me that it's a cute special case of $(\infty,2)$-differential topoi or something like this.

loled. closed tab.

Semiclassical

snerk

Balarka Sen

what isn't a special case of $(\infty, n)$-topoi?

Semiclassical

an $(\infty,\infty)$-topoi?

Mike Miller

00:17

I don't think those exist

Balarka Sen

^

Semiclassical

then it can't be special case of something that does exist :P

Balarka Sen

there is an $(\infty, n)$-category of $(\infty, n)$-categories, iirc. that's possibly something which is not an $(\infty, n)$-category.

Semiclassical

it's turtles, all the way down

Mike Miller

meh

Bye_World

01:32

I was writing up a nice comment on the question about gravity pointing up or down about how the question was entirely ill-posed. But it was migrated to phys.SE before I finished. *Sigh.

Michael

03:10

is it possible to have a vector function in 4 d?

Forever Mozart

well

Michael

well...

I guess the stupidity of my question just rendered you confused

uhhh

It should be possible, right?

Stan Shunpike

03:36

0

Q: Is the Monty Hall problem easier to understand in terms of complements?

I am very confused by the Monty Hall Problem. See Wikipedia for the set up. Suppose we have the three doors $A,B,C$. Suppose $t$ indicates rounds of choosing. Round 1 Prior to round one, no doors have been opened. Thus, $$P_{t=1} (A)= P_{t=1} (B)=P_{t=1} (C)=\frac{1}{3}$$ The contestant cho...

probability

@MikeMiller @BalarkaSen ^ any thoughts on this?

Mike Miller

03:50

@Stan: Donaldson-Kronheimer, geometry of 4-manifolds; Donaldson, Floer homology groups in Yang-Mills theory; Gompf-Stipsicz, Kirby calculus and 3-manifolds; Joyce, compact manifolds with special holonomy; Gaddis, JR

Balarka Sen

o_O

Forever Mozart

04:03

@BalarkaSen

Balarka Sen

Yep?

Forever Mozart

i got some good stuff

Balarka Sen

ok?

Forever Mozart

working on a strange space

Balarka Sen

what is it

Forever Mozart

04:06

every space generated this tree-like space

Balarka Sen

what do you mean?

I don't know what "generated" means in this context.

Forever Mozart

you know like the binary tree

but instead of $2$, you use a given space

Balarka Sen

yes, very cute idea, have used it in different contexts. can you elaborate on how you're constructing this space?

Forever Mozart

what did you use it for?

the topology would take a while to define

Balarka Sen

this. probably you're doing something more exotic than this though.

i'm interested in hearing about your space if you want to tell.

trees are dear to me.

woods are lovely, dark, deep, and p-adic.

oh wait.

Forever Mozart

04:10

haha

you have to realize I am more interested in set-theoretic topology and not algebraic

in fact I know very little algebraic topology

Balarka Sen

sure.

Forever Mozart

I know the algebraic stuff in Munkres, but that's about it

Semiclassical

many sets before i sleep

Balarka Sen

I like to think about un-nice spaces occasionally. Algebraic topology mainly works with nice things.

(although Cech, Bing, etc have applied alg top to study bad spaces)

@ForeverMozart So, gonna define this tree like space for me?

Forever Mozart

some topologists can do everything

Jan van Mill is one that constantly impresses me

Balarka Sen

04:14

haven't heard of him

Forever Mozart

lots of papers

and interesting ones

after 4 beers it would be hard for me to define the topology... maybe tomorrow

Balarka Sen

definitely.

Forever Mozart

it it still is a topology tomorrow ;)

I hate waking up and realizing my "theorems" are nonsense

Balarka Sen

@Semiclassical many *geodesics before I sleep.

Forever Mozart

I wish we had hats all year long

Balarka Sen

04:18

there won't be any point of them then though

Forever Mozart

yeah then I wouldn't look forward to christmas

Balarka Sen

:P

Forever Mozart

I had some good ones this year

my points always go up during December cause I come here so often

almost to 3000 now!

Balarka Sen

great!

@ForeverMozart The real reason I am interested in tree-like structures is actually not just because of p-adics, but the geometry of them. You know, hyperbolic stuff. Dunno much.

Michael

Isn't hyperbolic geometry mandatory to learn before topology?

Balarka Sen

04:26

not really.

Michael

I'm pretty sure you can learn it really quickly

OH I have a question. Do you ever use numbers in topology?

Forever Mozart

I have come to view topology as a game

very formally

Balarka Sen

@ForeverMozart I'd think everything is.

Life itself is a game.

Michael

So what role does a mathematician play in this game?

Forever Mozart

game within the game

Balarka Sen

04:30

oh no an (\infty, 1)-game

anything but this

the horror

Forever Mozart

for some reason I think you are from india. is that correct?

Balarka Sen

i have that written on my profile

Forever Mozart

oh

Michael

Sphere eversion

In differential topology, sphere eversion is the process of turning a sphere inside out in a three-dimensional space. (The word eversion means "turning inside out".) Remarkably, it is possible to smoothly and continuously turn a sphere inside out in this way (with possible self-intersections) without cutting or tearing it or creating any crease. This is surprising, both to non-mathematicians and to those who understand regular homotopy, and can be regarded as a veridical paradox; that is something that, while being true, on first glance seems false. More precisely, let be the standard embedding...

Forever Mozart

I am on the other side of the planet

canada

Michael

04:34

Oh we are neighbours

Forever Mozart

candle wax is so wierd

Balarka Sen

why

Forever Mozart

i was playing with it

lol

sorry I'm on beer #5

Michael

can you change your beer cup into a donut?

Forever Mozart

no!

unless I smash it

Michael

04:40

but but but... topology

Balarka Sen

Forever Mozart

i would have to cut a hole in the bottom and glue its boundary to the hole in the top

haha, that's me today

Michael

but isn't that a tear?

Forever Mozart

but I haven't drank in a month, so my tolerance is down

Balarka Sen

can't turn cup into donut in real life. they are homeomorphic, but not isometric.

Forever Mozart

04:42

when I was an undergrad, 5 beers was nothing

Michael

with 5 beers, anything is isometric if you want it to be

Forever Mozart

why are they not isometric?

Semiclassical

oh hey tintin

i used to read that a lot as a kid at the local library

Balarka Sen

@Michael in fact, even the Banach Tarski holds in real life

too much axiom of choice bro

@Semiclassical tintin is great.

Michael

@BalarkaSen Haha I get that one!

Forever Mozart

04:44

i dont get it

Balarka Sen

@ForeverMozart dis.

@ForeverMozart only isometries of $\Bbb R^3$ are composition of translations and rotations.

Forever Mozart

oh I see

Balarka Sen

I'd give you 10 million dollars if you can just translate and rotate your cup to be a donut.

Michael

Any good books on set theory?

Really?

Ok I need you to close your eyes really quickly

Balarka Sen

but only when you're not drunk

The Substitute

04:47

Hello, off topic. In ODE, why can't we call $y(x)=1/x$ a solution of $xy'+y=0$ on $\mathbb{R}-0$ instead of on, say, $(0,\infty)$? I.e., what's the reason for restricting the domain to an interval?

Semiclassical

whenever i see one of those kind of double vision things i can't help but remember this simpson's bit

Balarka Sen

@TheSubstitute That's probably the most on topic thing you have said in the past 15 minutes

Forever Mozart

i have the hiccups

who wants to play a drinking game with me

?

The Substitute

@BalarkaSen it's the only thing I said ;)

Balarka Sen

@TheSubstitute lol no I mean that message is probably the most on topic thing in this chat in the past 15 minutes

Forever Mozart

04:50

its 11pm here but like 11am in india now... crazy

Balarka Sen

only 10:20

Michael

what's the weather there like?

it was -20 Today here in Toronto,Canada.

Balarka Sen

@Forever: You should stop drinking if this happens:

Forever Mozart

around 30 in thunder bay

The Substitute

@BalarkaSen that was my first interpretation lol

Forever Mozart

04:53

wow

are you allowed to drink in india?

Balarka Sen

i dunno lol

with probability 1, yes

L33ter

@BalarkaSen is it true that if $V \cap W = \emptyset$ then $Cl(V) \cap W = \emptyset$?

Balarka Sen

$V, W$ are?

Forever Mozart

ok I looked it up. it says beer at 21 but all other forms of alcohol at 25

thats insane

L33ter

are open sets of a space X

Balarka Sen

04:56

No, false.

Michael

I came here to ask a differential equations question, but then I got distracted by your topology

Forever Mozart

no its not trie

hell no

Balarka Sen

Look at $\Bbb R$. $V = (0, 1)$ and $W = [1, 2)$.

Forever Mozart

nice edit :)

Balarka Sen

Oops, $V, W$ open.

L33ter

04:57

V,W open

Forever Mozart

oh

then yes

Michael

What is a definition of a node in mathematics?

L33ter

why ?

Forever Mozart

because $X\setminus W$ is closed containing $V$, thus containing $\text{cl} V$

sorry

simple topology is hard to me after five beers

Michael

What are the pre-reqs for topology ?

L33ter

05:00

nothing

Balarka Sen

If $V$ overlaps $\text{cl} W$, it has to overlap $\partial W$. The rest is clear, I think.

Forever Mozart

common sense

Michael

How much set theory should I know

L33ter

what do you mean overlaps ?

Forever Mozart

its good if you know real analysis

Balarka Sen

05:00

intersects.

Forever Mozart

no set theory is needed

Michael

R.I.P

L33ter

well cl(W) = W since W is open

Michael

Real Analysis is...Interesting to say the least

Balarka Sen

@L33ter Huh?

Forever Mozart

05:01

i hated it

Michael

I hate it too

L33ter

I am sorry

Balarka Sen

@ForeverMozart You need to know what a set is.

L33ter

closure not

Balarka Sen

:P

Forever Mozart

05:01

or rather I hated functional analysis

L33ter

the interior

Michael

Why are there never numbers in topology, or atleaest the stuff I see in M.S.E

L33ter

$Cl(V) \cap W = (V \cup V`) \cap W$

Forever Mozart

I guess it would help to know a few things about ordinals and cardinals

Balarka Sen

@L33ter Right.

L33ter

05:03

which is $(V \cap W) \cup (V` \cap W)$

we don't know if $(V` \cap W) = \emptyset$ though

Balarka Sen

Well, $V \cap W$ is null.

L33ter

yes

Balarka Sen

And we have assumed $\text{cl}V \cap W$ is not null.

So $\partial V \cap W$ has to be not null too.

Michael

What do you do after you learn linear algebra @ForeverMozart . Is there any course that follows up on the stuff you learn?

L33ter

why did we assume that Cl(V) \cap W is null

how do we know that

Forever Mozart

05:04

@Michael Abstract Algebra

Michael

@ForeverMozart Is that a hard area?

Forever Mozart

you study vector spaces in linear algebra

Michael

yup

L33ter

I don't understand @BalarkaSen

Forever Mozart

in part of abstract algebra you study modules, which are generalizations

Balarka Sen

05:05

@L33ter Here's what I am trying to say, I am parsing it wrong. We have $V \cap W = 0$. We want to prove $\text{cl} V \cap W = 0$, right?

L33ter

yeah

Michael

Then what's advanced algebra?

Forever Mozart

that's abstract algebra

Michael

All I see them doing is limits

L33ter

advanced algebra is module theory,field theory, and galois theory

Balarka Sen

05:06

Ok, $\text{cl}V \cap W = (V \cap W) \cup (\partial V \cap W)$, as you have proved. But this union is just $\partial V \cap W$, as $V \cap W = 0$ as given.

Michael

hard,scary limits

L33ter

yes

Michael

oh wait

advanced calculus

L33ter

exactly @BalarkaSen

Forever Mozart

but most schools call it abstract algebra

Balarka Sen

05:06

Can $\partial V \cap W$ then be nontrivial?

L33ter

I don't see why not ?

I don't see any restriction of why it can't be trivial.

Forever Mozart

@L33ter when you are trying to determine if these simple statements are true or not, you should think about simple spaces

Brody

Hello!

Forever Mozart

like think about intervals in the reals to see if there is an obvious counterexample

L33ter

@ForeverMozart I m just trying to understand some proof in my book about locally compact spaces

Forever Mozart

05:08

what theorem>

?

Balarka Sen

@L33ter If it is nontrivial, there is an $x \in \partial V$ so that $x \in W$. As $W$ is open, there is an nbhd $U$ of $x$ in $W$. But $U$ intersects $V$ by definition of bd point.

But then take a point in $U \cap V$ to get a interior pt of $V$ contained in $W$.

There is no such pt, contradiction.

L33ter

Let X be a hausdroff space. Then X is locally compact iff given x in X, and given nbhd U of X, there is a nbhd V of X such that Cl(V) is compact and Cl(V) subset U.

I am proving this statement by myeslf without looking at book I needed that result

Michael

IS there a topological space that deals with complex numbers?

Balarka Sen

I mean, this is pictorially clear. Topology is all about thinking up the right picture. Then you can translate that back to math.

An open set can never intersect boundary of another open set without intersecting the interior. What's unclear about this?

L33ter

I see

I understand

Balarka Sen

05:13

Did you see the proof above?

L33ter

yes I understand @BalarkaSen I was reading it after I posted the statement above

Balarka Sen

ok, good.

L33ter

yes ok good I understand

oke good I get it

brb

Balarka Sen

ForeverMozart's argument is much more concise though. But definitely not something I'll think up at first look.

L33ter

gotta eat

Balarka Sen

05:15

@L33ter Watched Shutter Island.

@Michael That looks just as random as your other questions.

L33ter

how was it @BalarkaSen pretty good right?

Balarka Sen

@L33ter Very good.

I think it's one of the best I have seen.

L33ter

yeah I told you that you would like it

Balarka Sen

Yep. Thanks.

L33ter

this is the kinda of movies I like

Balarka Sen

05:16

@Michael Well, even before stupid, I don't know what "moebius triangle" means.

L33ter

my semester starts tomorrow

I can't wait :D

Balarka Sen

How can you ask if a mathematically undefined object exists or not?

Michael

I was thinking of something like this en.wikipedia.org/wiki/Penrose_triangle

L33ter

brb

Michael

I guess I used the wrong wording*

Brody

05:17

If I can add to the (previous) convo @Michael, several subjects follow up on linear algebra ... assuming "linear algebra" means an undergrad course in elementary linear algebra.

Forever Mozart

what

Balarka Sen

Still not sure what your question is.

Forever Mozart

what is ha

Balarka Sen

@ForeverMozart ha? ha ha. hahahaha!

Michael

You know what I am running on 2 hours of sleep

Forever Mozart

05:18

2 hours? JESUS

Michael

I'm going off before I drive you crazy with something even stupider

Forever Mozart

you cant drive me crazy

Balarka Sen

you're already crazy

Michael

You haven't seen my abilities @ForeverMozart

I drove ted crazy

Forever Mozart

exactly lol

aw dont be mean to ted

Michael

05:21

Poor guy was trying to explain something, then I asked about something that makes no sense

Balarka Sen

@ForeverMozart The torus, $S^1 \times S^1$, lives in $\Bbb R^4$, right? Because $S^1 \subset \Bbb R^2$. Standard fact: cylinder with ends glued is homeomorphic to this, and is embedded in $\Bbb R^3$. We all know this.

Forever Mozart

I havent seem him here in a while

Michael

he was like "what are you trying to say".

I said "I have no clue"

Balarka Sen

Question is, can $S^1 \times S^1$ be isometrically embedded in $\Bbb R^3$?

Michael

I love ted, he helps everyone with everything

Balarka Sen

05:22

Gluing ends of cylinder stretches the thing a lot, so is not an isometric embedding.

Brody

Depending on your college, as far as pure math you might find... advanced linear algebra, linear algebra II (trivially), abstract vector spaces, modern aka abstract algebra, so-called "advanced" algebra, etc...

Forever Mozart

@BalarkaSen whaaat

Balarka Sen

?

what's unclear

Michael

@Brody What about advanced calculus? What is that

Forever Mozart

oh ok your question

advanced calculus is undergraduate real analysis

Balarka Sen

05:24

What is unclear about my question?

Forever Mozart

I think the answer is no @BalarkaSen

Michael

then what do you do in grad school?

Balarka Sen

Me too! But the answer is yes!!!

Forever Mozart

@Michael you drink beers and prove theorems

its great

Michael

Do conjectures count?

Balarka Sen

05:25

@ForeverMozart You can in fact $C^1$ isometrically embed $S^1 \times S^1$ in $\Bbb R^3$.

Brody

@Michael, typically the undergrad course Advanced Calc is basically intro. real analysis, or often an intermediate between basic first-year calc stuff and (relatively) rigorous real analysis

Forever Mozart

yes, some people are better at coming up with new conjectures than answering old ones

Michael

@Brody so What do you do once you are done with advanced calc II?

Forever Mozart

what is $ C^1$

Balarka Sen

continuously differentiable

The embedding looks like this:

Forever Mozart

05:26

oh

Balarka Sen

$user image$

Forever Mozart

whaaat

Balarka Sen

[ref : math.univ-lyon1.fr/~borrelli/Hevea/Presse/index-en.html ]

Forever Mozart

my intuition said no

Balarka Sen

pretty pathological, isn't it?

@ForeverMozart it's super-surprising

or at least, it is to me

Michael

05:27

Do they sell those on E-bay?

Forever Mozart

i will save that link and read it when I'm not drunk :)

Balarka Sen

iirc Nash-Kuiper is a generalization of this of some sort. I think Mike told me about this, but I forgot.

@ForeverMozart Sure.

Brody

@Michael, I don't know tbh. depends what topics that course entailed + how rigorous and formal the math was. Also, basically what your college uniquely offers and what the math dept. (aka your instructors/advisors) recommends

Balarka Sen

thought you might like

Forever Mozart

third prize is you're fired

Brody

05:33

heyy ppl, "In an ideal phonemic orthography, there would be a complete one-to-one correspondence (bijection) between the graphemes (letters) and the phonemes of the language, and each phoneme would invariably be represented by its corresponding grapheme"

perhaps I'm being dull right now, but doesn't the first clause imply the latter?

Forever Mozart

are you reading godel escher bach?

Brody

me? nope. and if the question wasn't for me, I'm just interjecting. don't mind me!

L33ter

07:58

@BalarkaSen this looks super cool

Isaac Solomon

Hey, looking for a quick reality check here. Suppose $u$ is a function on a manifold $M$, equipped with two conformal metrics $g,h$. If I want to compare the norms $|\Delta_{g} u|$ and $|\Delta_{h}u|$ will this just be $|h/g|^2$?

Mary Star

11:42

Hello @robjohn !!

Could you take a look at my question http://math.stackexchange.com/questions/1593141/unit-normal-of-the-surface-s and tell me if it is meant to consider the notation from the example only for $K$ or for $K$ and $\sigma$ ?

iwriteonbananas

12:49

It's great to see you diversiyfing your portfolio by learning analysis, @BalarkaSen

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