Mathematics - 2014-06-20 (page 1 of 5)

Pedro Tamaroff

00:00

@AlexanderGruber Maybe Alex and Gruber and Peter should hang out.

Com Truise

Team Gruber

Karl Kronenfeld

@AlexanderGruber You'll find the deep connection between finite superdupersolvable group theory and infinite cardinals, leading to the important Gruber's Theorem of set theory.

Com Truise

what does that theorem say?

Karl Kronenfeld

It would be Kronenfeld's Theorem if I knew

Alexander Gruber

@KarlKronenfeld and spend the rest of my life trying to disprove it, like Planck did with Quantum Mechanics.

Karl Kronenfeld

00:01

:D

skullpatrol

Planck was too stubborn, like Einstein

Alexander Gruber

I have a problem @PedroTamaroff.

robjohn

@BalarkaSen it has.

Kaj Hansen

@Comtruise, Ramsey theory is concerned with finding how large/complex a given structure must be in order to guarantee a certain degree of low-level order. In that sense, Ramsey's theorem is definitely part of Ramsey theory.

skullpatrol

@robjohn Are you considering posting something?

robjohn

00:11

@skullpatrol at some point probably

skullpatrol

cool

Kaj Hansen

In it's least general form, Ramsey's theorem takes complete, 2-colored graphs (the given structure) and asks how many vertices they must have such that you are guaranteed a monochromatic clique of a certain size (which is a certain degree of low-level order).

Mike Miller

@BalarkaSen Looks like the proof is very similar in flavor to the standard proof. Looks interesting nonetheless, though, I'll check it out sometime.

Com Truise

Theorem. Let X be some countably infinite set and colour the elements of X(n) (the subsets of X of size n) in c different colours. Then there exists some infinite subset M of X such that the size n subsets of M all have the same colour.

Mike Miller

@coffeemath how do you take your coffee?

Kaj Hansen

00:16

Plus, Ramsey's theorem is used to prove a large number of theorems within Ramsey theory. Of the top of my head, the Erdos-Szekeres conjecture and Schur's theorem can be proven with its help.

skullpatrol

@KajHansen How many courses have you taken taught by Dr. Shifrin?

Com Truise

are you talking about the same theorem I just gave?

coffeemath

@MikeMiller Usually I take coffee with math...

Com Truise

lol

Mike Miller

ah, you're preventing me from starting a riot

effectively too

Alexander Gruber

00:18

@MikeMiller

How do YOU like YOUR coffee

Kaj Hansen

@skullpatrol, I've taken two courses that are a combination of linear algebra, vector calculus, and manifolds from Ted (the second of which has been posted to YouTube). I also took differential geometry from him. So, all told, 3 courses.

skullpatrol

nice

Alexander Gruber

@KajHansen anybody ever told you you look like Jason Bourne?

Mike Miller

@AlexanderGruber i like it black, and generally the darker the roast the better.

also i think those who have diferent tastes are cowards

Com Truise

or good will hunting?

Karl Kronenfeld

00:21

@MikeMiller You puritan....

Welcome to the club!

Com Truise

I have been told I look like Liam Neeson

Mike Miller

@KajHansen has anyone told you you look nothing like matt damon?

Kaj Hansen

@AlexanderGruber, not until just now. I'll take that as a compliment though. I get Dicaprio a ton though.

Com Truise

I have a very particular set of skills...

Alexander Gruber

@KajHansen I think it's the hoodie. You look low profile.

@KajHansen Dicaprio, I see it. I hope you're using that.

skullpatrol

00:24

I have been told I look like Rubén Cortada.

Kaj Hansen

@AlexanderGruber, supposedly it's quite uncanny when I have my hair combed back. Think Inception, lol.

Alexander Gruber

@ComTruise I will find you. And I will teach you Sylow theory.

Com Truise

:D

Alexander Gruber

@KajHansen You should do it for Halloween.

Kaj Hansen

Sounds like a plan @AlexanderGruber.

So where are you guys with your math education? I feel like I'm an undergrad surrounded by grad students/professors in here, haha

AWertheim

00:29

@KajHansen: two months ago I was an undergrad. Does that count?

Com Truise

grad student trying to find something new to research

Kaj Hansen

@AWertheim, more or less. Are you headed to grad school, or trying to find work elsewhere?

Pedro Tamaroff

@AlexanderGruber !!!!!!!!!!!!

I am green with envy.

AWertheim

A bit complicated, @KajHansen. But the short is that I'll be a software dev for a year before applying to grad school.

skullpatrol

@PedroTamaroff that would match the color of your avatar :D

Alexander Gruber

00:32

@KajHansen I'm a grad student. I came into the game late- did a bachelor's and went back to school for another one in math, then went to grad school.

Kaj Hansen

Sounds like a plan. Personally, I'm a bit worried about getting into grad school. I'm dead set on it, but GPA is on the low side.

Alexander Gruber

@KajHansen My GPA could've been better. You'll probably get in somewhere. Apply to a variety of backup schools just in case.

Mike Miller

@KajHansen what year are you?

AWertheim

@KajHansen: I wouldn't worry too much. Not only do you seem very competent, but solid graduate coursework and excellent recommendations carry far more weight than GPA (though see if you can't do well in your graduate courses). For what my advice is worth though (not much, lol): prepare well for the subject test.

Pedro Tamaroff

Is GPA really that important?

Alexander Gruber

00:34

@PedroTamaroff i found a place to get all these little ink samples too, these things are like heroin.

Mike Miller

@PedroTamaroff it's a cutoff. schools like to toss out apps with gpas belwo their 'cutoff line' and otherwise it's nigh completely unimportant.

Kaj Hansen

@Mike, I'm starting my 3rd year this fall

Alexander Gruber

i think rec letters are most important, then stuff like papers, then math GPA, then other stuff.

Kaj Hansen

@AWertheim, that's what I'm hoping. I'm trying to get into graduate complex in the spring, and hopefully some more after that.

Com Truise

yes rec letters are very important

Mike Miller

00:35

@KajHansen I'm starting at grad school in the fall, so I've done plenty of paranoid thinking about this topic. Feel free to email me, email's in my profile

Kaj Hansen

@MikeMiller, I'll keep that in mind for when I start applying. Thanks!

Alexander Gruber

My GPA was not stellar because I doubled in math and physics and that means 6 courses per semester of not easy things. i'm pretty sure i only got in based on the paper I wrote, and the rec letter associated with it.

AWertheim

That'll be good. Complex analysis is beautiful too, a good place to start. :)

Com Truise

My GPA was sort of wrecked by my first year as an undergrad... I was a little out of control then ;)

Mike Miller

@KajHansen I can't imagine I'd say much there I wouldn't here, so: you probably shouldn't worry about your GPA. mine was decent but not stellar and I think I did fairly well for myself, and unlike UGA, my school isn't even in the AMS's top 50 thing. Take grad classes, try to do research (maybe a senior thesis? or if it's not too late try to work with someone this summer)

Kaj Hansen

00:38

Yeah, mine is hovering around a 3.3 because I dove right into math and I've been taking 3-4 math courses each semester. It's slowly rising now that I'm getting the hang of things.

Alexander Gruber

@KajHansen one thing I wish I'd have done is gone to a conference sometime between my junior/senior year. those things are great for networking.

Mike Miller

get to know a good number of professors and work with them (and take their classes)

@AlexanderGruber omg i'm still mad I didn't go to the joint meetings in january

after I got back to my school they told me they would have funded me to go if I asked, even though I had nothing to present

Alexander Gruber

@MikeMiller were they near you?

Mike Miller

and it was like 'why didn't you tell me"

Alexander Gruber

@MikeMiller Oh yeah, they love giving travel stipends. Tax break.

Mike Miller

00:39

@KajHansen Yeah you're fine. Wait, you're starting your 3rd year? You're in way better shape than I thought

Kaj Hansen

Thanks for the advice Alex. I wish I could go to MAA this summer, but it's all the way in Portland.

Mike Miller

I definitely had less than a 3.3 when I started my third year

Kaj Hansen

That's very encouraging @Mike.

Mike Miller

You have a lot of time to learn a lot of math and more importantly do a lot of math

The fact that you're interested now and already learning a lot says good things, and the amount you learn every year (from experience) grows essentially exponentially

AWertheim

I'd second that. Not to give stupid advice, but actually doing a lot of math is probably the best thing you can do.

Pedro Tamaroff

00:41

►

skullpatrol

I third that.

Kaj Hansen

Great song @PedroTamaroff

Com Truise

lol

Mike Miller

The absolute best advice I can give is to do research with some faculty member (and apply for an REU next year!) and then put your whole heart into it. You'll get great letters, you'll get research experience (both of which are good to admissions committees)

but far more important is that you'll really be doing math, and that's an experience a class can never give you

ok I'm done

Kaj Hansen

Sounds good @Mike. I applied for some REUs this summer without much luck. I'm absolutely dead set on getting into one next summer, even if that means filling out dozens of apps.

Com Truise

00:43

Steve Miller cheesiest band?

Kaj Hansen

Anyways, we don't have to clog up the chat with my anxiety any longer :P

Mike Miller

The point is not to have anxiety :)

It's too early for that! And you sound like you shouldn't be anxious anyway.

Alexander Gruber

ehh

One should expect to be anxious.

skullpatrol

@KajHansen Feynman wrote on his last blackboard "Solve every problem that has been solved."

Mike Miller

@AlexanderGruber as a human he'll probably be anxious

Alexander Gruber

00:45

I think applying to grad school (and REUs and all that) and waiting for the responses were some of the most stressful times in my life, I wasn't quite mentally prepared. One should be try to be ready for some stress.

Mike Miller

as a human he should also try not to be

imo

Com Truise

solving every problem that has been solved would leave no time for original research

AWertheim

@AlexanderGruber: for sure. Stress from submission onward lol.

Alexander Gruber

@MikeMiller I guess you're right.

@AWertheim Yeah exactly. I didn't realize how stressful it was going to be after I finished the apps.

AWertheim

@AlexanderGruber: to my shame, I think I checked those results (e.g. gradcafe) sites several times every day. I would NOT recommend that, LOL

Kaj Hansen

00:48

gradcafe is a main source for my anxiety re: GPA! I think that's some sound advice.

Com Truise

I already had a job when I applied, so it wasn't that stressful for me

Mike Miller

oh

get off gradcafe

those people are insane

Alexander Gruber

@AWertheim my approach was to smoke 2 packs of cigarettes a day, then get belligerantly drunk and yell at people

Mike Miller

@AlexanderGruber hey, mine too, roughly

Com Truise

smoking is bad for you

skullpatrol

00:49

2 packs of cigarettes a day is not good

Alexander Gruber

hahaha

AWertheim

@AlexanderGruber: haha, I'm dying laughing. Definitely some rough moments. Especially when everyone keeps asking =P

Alexander Gruber

I like how nobody says anything about the belligerent drunk yelling

Kaj Hansen

LOL

Alexander Gruber

but the cigs, gotta stay away from those.

skullpatrol

00:50

true^

alcohol is different

AWertheim

Btw, I'm late, but Mike is right @KajHansen. Stay away from gradcafe, you'll only make yourself needlessly anxious =P

Alexander Gruber

Whoever made that site must be such a sadist. Who thought that was a good idea?

skullpatrol

grads

Com Truise

what is gradcafe

Mike Miller

I'm 99.99% sure most of the stats and stuff on gradcafe are straight up lies

Its main use is checking whether school x has sent out results during admissions season

the rest of it is worthless garbage

Pedro Tamaroff

00:52

LOL "Take the Money and Run" is a rip off of "Sweet Home Alabama" or viceverse.

skullpatrol

99.99...%

Pedro Tamaroff

►

skullpatrol

STAAAAP^

anon

Chat guidelines | $\LaTeX$ in chat

14

Mike Miller

needs an H @skullpatrol

@anon no MSE map?

skullpatrol

00:53

once was enough

anon

@MikeMiller map? what?

Pedro Tamaroff

@anon Holies.

AWertheim

@MikeMiller: even then, it's still cause for nerves. I spent way too many hours worrying about when I would or would not hear from places based on whether people had posted results.

Mike Miller

@anon

anon

haha, jeez

Mike Miller

00:54

@AWertheim yup, it messed me up too

but I maintain it's not nearly as terrible as the "advice" they give you on the forum, or the stats people claim when they post their rejections or acceptances

Pedro Tamaroff

This is so sad.

AWertheim

For real. Trolls (or what I hope at least are trolls) abound.

Pedro Tamaroff

I have a headache, prolly because I need some coffee. I have to work tomorrow morning, so I had to settle for a "double tea".

Com Truise

yes, and then a chamomile to put you to sleep

Mike Miller

@AWertheim I think it's just psychology. You get rejected, you think you should have gotten in, maybe you're mad at the school, you modify your stats a little bit (lot) to make it look like they don't accept ANYONE (and that's why they didn't accept you).

If their stats were correct, I shouldn't have gotten in anywhere ;)

Pedro Tamaroff

00:57

Is it racist if I'm hearing "Fly like a Negro"?

►

skullpatrol

yes

you are a racist

if you hear that

AWertheim

@MikeMiller: that makes a lot of sense, lol. Clearly, you are very talented! UCLA is very exciting. :)

Pedro Tamaroff

It might also be delusional.

Everyone knows negroes don't fly.

Com Truise

I like some circa 1980 Steely Dan myself. Certain level of sleaze appeals to me.

Mike Miller

@AWertheim oh, I appreciate that :) I hope I'm not inadvertently #humblebragging here.

You're a graduating senior? Or graduated now, rather?

Alexander Gruber

01:00

@skullpatrol oh that reminds me of a joke

AWertheim

@MikeMiller: not at all. I'm listening keenly to your advice! UCLA is my dream graduate school. :)

Alexander Gruber

what do you call a black man who flies a commercial jet?

AWertheim

I just recently graduated. I would be your cycle, but I am in an odd situation (not to be overly cryptic, lol).

Mike Miller

@AWertheim I dig cryptic.

Pedro Tamaroff

@AlexanderGruber Hm...

blue

01:01

everyone in the us has heard that one

AWertheim

Without sounding needlessly cloak and dagger, I applied this past year with some success. But for (what I believe to be) good reasons, it is better for me to wait a year and do things a bit differently.

Alexander Gruber

somebody's gotta ask me what.

Pedro Tamaroff

what

AWertheim

@AlexanderGruber: a pilot, silly! ;)

Alexander Gruber

@PedroTamaroff a pilot. You racist.

Pedro Tamaroff

01:03

@AlexanderGruber LEL

You know what's not fair?

Com Truise

take your time @AWertheim. Grad school isn't going anywhere..

blue

@PedroTamaroff a biased coin?

Alexander Gruber

@PedroTamaroff what's not fair?

Pedro Tamaroff

A black man's arse.

uproxx.files.wordpress.com/2014/03/…

AWertheim

Thanks @ComTruise. It's good to hear that side of things lol :)

Alexander Gruber

01:05

haha

Com Truise

@AlexanderGruber what's the answer?

Mike Miller

@AWertheim you've now got a year in which you can do math, learn math, and also enjoy yourself, without any (much) external pressure

enjoy!

AWertheim

@MikeMiller: I will certainly be trying to, thanks! With any (read: a great deal of) luck, I will be seeing you next year :D

Mike Miller

i wish you the best, amigo!

Com Truise

i feel like I am stuck in a rut with my research... ugh

Mike Miller

01:10

@AWertheim oh, forgot to say: my email's in my profile for a reason, if y'ever wanna email me, feel free!

Pedro Tamaroff

@MikeMiller He cannot see it.

Mike Miller

@PedroTamaroff yes he can

Com Truise

any strong opinions on CH here?

Mike Miller

it's true

blue

no, it's false

AWertheim

01:12

@MikeMiller: thanks a ton! I def will. Hopefully with good news at some point!

Pedro Tamaroff

@ComTruise I like the model of the commercial a lot.

But I think HUGO has better fragances.

AWertheim

I'm off to work out friends. Bye!

Mike Miller

@blue do you want to take this outside?

Com Truise

@blue so you have a strong opinion

skullpatrol

later @AWertheim

Mike Miller

01:13

in fact the generalized continuum hypothesis is true

Com Truise

noooo!!!

skullpatrol

define: true

Mike Miller

the cardinals are the exact same thing as the ordinals

and that's the truth

skullpatrol

define: truth

Com Truise

it is true in V=L :)

Alexander Gruber

01:16

@ComTruise to my joke?

"A pilot, you racist."

skullpatrol

@AlexanderGruber not my answer?

Com Truise

do you believe V=L @MikeMiller

Alexander Gruber

@skullpatrol that's also pretty good. :P

Ha ha. You guys and your infinite sets.

Mike Miller

dunno what that means

skullpatrol

it means V equals L

Mike Miller

01:19

staaahp

skullpatrol

sorry

Pedro Tamaroff

@MikeMiller Define what it means for a B space to be reflexive.

skullpatrol

nice "h" placement btw

Com Truise

V=L is an axiom you can add which says sets must be constructable according to certain formulas... it is a restrictive view of sets, but you get well-ordering principle (AC), GCH

Mike Miller

@PedroTamaroff double dual of $X$ is iso to $X$

blue

01:21

a space with above-average but below-top grades, which has quick reaction time

Mike Miller

@ComTruise i tend to disbelieve that sets should be constructible, at least in my non-set-theorists understanding of the word constructible.

Pedro Tamaroff

@MikeMiller I figured. This book defines that $\widehat X=X^{\ast\ast}$.

Here $\widehat X$ is the isometric copy of $X$ inside $X^{\ast\ast}$ given by the natural evaluation map.

The book says there is a space $J$ such that $J\simeq J^{\ast\ast}$ but $\widehat J$ is a proper subspace of $X^{\ast\ast}$.

Book by Carothers.

Mike Miller

that's my definition, it's possible others disagree, I am not a banach spaceist

what's the space tough

Pedro Tamaroff

Well, note that what we're asking is that they are "naturally" isomorphic here.

Says the example is due to R.C. James.

Lemme see references.

Mike Miller

I don't care who it's due to :P

yeah I realize

the word you want is probably "canonically"

Pedro Tamaroff

01:25

No, natural as in the sense of categories.

James' example is in "Annals of Mathematics, 52 (1950), ps. 518-527"

Mike Miller

sure

Alexander Gruber

Oh. Human League.

Pedro Tamaroff

Oh, @AlexanderGruber?

skullpatrol

what about them?

Alexander Gruber

Just came on youtube

I still have Pedro's link open from earlier

Pedro Tamaroff

01:27

Oh.

skullpatrol

which song?

Alexander Gruber

@skullpatrol Don't you want me

skullpatrol

over played

Alexander Gruber

This song is... what is the right word?... this song is rad.

Pedro Tamaroff

Rad cool.

skullpatrol

01:29

27 secs ago, by skullpatrol

over played

Com Truise

anyone into synthwave / retro electro?

Alexander Gruber

@ComTruise yea man

Com Truise

pretty good stuff

skullpatrol

retro is my middle name

Com Truise

I listened to it over several months while writing my last paper

skullpatrol

01:30

skull retro patrol

Com Truise

made me feel like the Terminator

Alexander Gruber

@ComTruise My last publication was composed almost entirely to wu-tang clan

I don't think it shows though.

Com Truise

hear of them, but never listened

►

Mike Miller

@PedroTamaroff do you want the paper?

Alexander Gruber

@ComTruise this should get you started.

Mike Miller

01:34

I looked at it and it turns out I'm not interested at all

Pedro Tamaroff

@MikeMiller ERMAGHERD YES PLS.

Mike Miller

email = ?

Com Truise

I think I am incapable of listening to hip-hop

Alexander Gruber

@ComTruise most people have that response at first

skullpatrol

it's an acquired taste for most

Alexander Gruber

01:39

It's one of those things you have to get into the mindset for at first. if you listen to one song, you'll always hate it. If you sit it all the way through a wu-tang album, you will like hip-hop afterwards.

Mike Miller

@AlexanderGruber that album is better for the (mostly) lack of RZA's production

Alexander Gruber

You shut your dirty mouth @Mike. The RZA is a god. :P

skullpatrol

Go wash out your mouth :P

Hi Professor @TedShifrin

Pedro Tamaroff

@TedShifrin

Herro!

Do you know what is the font that is used in the Annals of Mathematics?

I love that font.

Ted Shifrin

Um, no, I don't remember what it is. Howdy @skull again.

@Pedro, it may have changed, but google Annals of Math fonts

Mike Miller

01:46

@AlexanderGruber i agree. but too much of a good thing...

Pedro Tamaroff

@MikeMiller Thanks!

Alexander Gruber

@MikeMiller ahhh, yeah. that is true.

I mean, the reason wu-tang has so many solo/duo/whatever albums is the same reason symphonies don't just have every instrument play at the same time.

Mike Miller

i like his production gimmicks a lot but sometimes i don't want to hear a track with bits interspersed about the shaolin style, dig?

Alexander Gruber

@MikeMiller Idk why you like this album then ;)

do you like meth and red's albums?

Mike Miller

@AlexanderGruber that was a separate comment than about this album

though it is one of my general complaints about the wu

Ted Shifrin

01:49

@Pedro: Did you ask your prof why he/she chose such a sloppy text?

Mike Miller

y'know I never listened to it, @Alex

Alexander Gruber

oh gosh, listen to the first one ASAP.

the second one is terrible

Mike Miller

imo ghostface was the best thing to come out of wu

Alexander Gruber

@MikeMiller it is impossible to not love this song

Mike Miller

@TedShifrin cute problem I'm working on (no hints, I'm just telling you the problem): show that if $n>m$, a map $f: \mathbb{RP}^n \rightarrow \mathbb{RP}^m$ induces the trivial map on their fundamental groups

I'm aware of a method of doing this with the cup product structure, but Bredon puts this after his section on borsuk-ulam, so I'm trying to show that such a map necessarily extends to a map $S^n \rightarrow S^m$ that's equivarient w/r/t the antipodal maps

skullpatrol

01:53

@AlexanderGruber What do ya think of this?

Mike Miller

(and thus, by the theorem he uses to prove BU, that necessarily $n \leq m$)

Ted Shifrin

That can't be right.

Pedro Tamaroff

@TedShifrin That's just a text I picked up from the internet. The argument is cool and generalizes to Kaj's remark that $R(s,t)\leqslant R(s,t-1)+R(s-1,t)-1$ when the first two summands are even.

Ted Shifrin

Ohhhh @Pedro ....

Mike Miller

@TedShifrin the thing I'm trying to show? or the original problem? original problem is totally right, I did it as an exercise in Hatcher back in the day

Alexander Gruber

01:54

i been listening to misdemeanor since I was a young man, @Skull.

Ted Shifrin

The argument is standard. Just not the lack of hypothesis. :)

skullpatrol

@AlexanderGruber I thought so :-)

Ted Shifrin

@Alex: You are a young man!

Alexander Gruber

I will have you know that I try to insert "put that thing down flip it and reverse it" into mathematics lectures as frequently as i can.

2

skullpatrol

:D

Ted Shifrin

01:56

@Mike: Your equivariant extension claim.

Alexander Gruber

@TedShifrin i should have probably said a little kid. :p

Mike Miller

@TedShifrin I haven't shown that such an extension exists yet, or convinced myself to give up on that approach, so shh

Ted Shifrin

Ok, @Alex :D

Alexander Gruber

i waited in line for Doggystyle with my cousin. this goes back a ways.

Mike Miller

or rather "equivariant lift"

Ted Shifrin

01:57

Right, my mistyping.

Mike Miller

oh, when I said "such a map lifts to an equivariant one", I meant that a map that induces a nontrivial map on fundamental groups

Ted Shifrin

Oh ...

Mike Miller

I don't think I'd ever read the proof of borsuk-ulam before... I think I prefer Bredon's approach to Hatcher's (nonexistence of equivariant maps vs odd maps have odd degree), even though they're essentially the same

Okie, got it

cute problem

Pedro Tamaroff

@MikeMiller Dude. This book by Carothers is fucking amazing.

Mike Miller

not as amazing as that problem

itw as a really neat problem

Pedro Tamaroff

02:13

Well, that was in the book, so...? =P

Mike Miller

@PedroTamaroff looks like a book I would have liked back when I was learning some functional analysis

from a glance it looks very well written

Alexander Gruber

I've been reading about combinatorial species lately.

First math I've done willingly in a while.

Pedro Tamaroff

@AlexanderGruber YAY

Alexander Gruber

@PedroTamaroff ever checked them out?

Pedro Tamaroff

@AlexanderGruber Not really. I do want to read Stanley's two volumes.

Alexander Gruber

02:18

all they are is endofunctors of finite sets & bijections

Like for example the species $\mathcal{G}$ maps finite sets to set of (simple) graphs on that set.

Pedro Tamaroff

endofunctors?

Alexander Gruber

functors from the category of finite sets and bijections to itself

the context i saw them in looks at species because they're a way of rigorously defining what "unlabeled" structures are, like unlabeled graphs, unlabeled linear codes.

Pedro Tamaroff

oh ok

Alexander Gruber

You can see how it might relate to functor-ness in that category - if you've got two sets of the same size $S$ and $T$, the set of graphs with vertex set $S$ and the set of graphs with vertex set $T$ are pretty much the same, right? it's just relabeling.

Pedro Tamaroff

OK

Alexander Gruber

02:28

in other words you can make a commutative diagram $$\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex}
\newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex}
\newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}}
\begin{array}{c}
S & \ra{\beta} & T \\
\da{\mathcal{G}} & & \da{\mathcal{G}} \\
\mathcal{G}[S] & \ras{\mathcal{G}[\beta]} & \mathcal{G}[T] \\
\end{array}$$ (where $\beta$ is the bijection)

Pedro Tamaroff

OK

Karl Kronenfeld

endofunctors are always cool anyway

is there anything interesting going on between a species and the identity functor?

Alexander Gruber

@KarlKronenfeld hehehe yeah. i'll get to it.

so, since finite sets of the same size are pretty much the same to species, we just call all finite sets of cardinality $n$ by the name "$n$"... and if $\mathcal{S}$ is a species, we write the size of $\mathcal{S}[n]$ as $\mathcal{S}_n$. So then we can define the cardinality of a species by way of a power series: $$\mathcal{S}(x)=\sum_{k\geq 0} \frac{\mathcal{S}_k}{k!}x^k$$

that's a little crazy, right? a functor with a cardinality.

But the thing that really blows my mind about these is this: there is a calculus of species. You can add them together, and multiply them, and differentiate them.

The identity functor ends up being an identity for one of those, I think.

Karl Kronenfeld

Yeah that's awesome

I suppose addition and multiplication are pointwise, right?

blue

...

Alexander Gruber

02:37

This is basically everything I wanted out of category theory... i remember when i first started reading about it, this is what I kept wanting to happen, but then everything went off into outer space with commutative algebra. it's making me really happy to finally see this realized.

Mike Miller

@KarlKronenfeld do you know the hopf index theorem?

Karl Kronenfeld

@MikeMiller no

Mike Miller

do you know that, if $M$ is a compact smooth manifold and $\chi(M) \neq 0$, then $M$ does not carry a nonvanishing (smooth) tangent field?

Karl Kronenfeld

yeah

Alexander Gruber

@KarlKronenfeld Haha, sort of. Addition is not too hard to define, $\mathcal{S}+\mathcal{T}$ is defined by $$(\mathcal{S}+\mathcal{T})[z]=\mathcal{S}[z]\sqcup \mathcal{T}[z]$$

Mike Miller

02:40

i knew that, but only as a corollary of the hopf index theorem (which says that in a certain sense $\chi$ counts the singularities of a vector field)

Karl Kronenfeld

@AlexanderGruber I looked up multiplication

it makes more sense that way

Alexander Gruber

with $\sqcup$ being the "forced" disjoint union, i.e. $\left(\mathcal{S}[z]\times \{1\}\right)\cup \left(\mathcal{T}[z]\times \{2\}\right)$

Mike Miller

@KarlKronenfeld but hopf index is complicated. the proof of the theorem i stated above is easy.

i never knew that theorem until now.

Karl Kronenfeld

@MikeMiller ah

Mike Miller

well, i knew the theorem. not the simple proof.

Alexander Gruber

02:43

multiplication ends up being $$\left(\mathcal{S}\mathcal{T}\right)[z]=\sum_{z=a\cup b}\left(\mathcal{S}[a]\times \mathcal{T}[b]\right)$$ where $\times$ is the regular old set Cartesian product.

but yeah - all that stuff works out. Cardinalities of added species are added, cardinalities of multiplied species are multiplied.

Mike Miller

(if it carries a nonvanishing vector field $\xi_x$, embed $M$ in $\mathbb R^n$ and give it a tubular neighborhood $\Theta$. recall there's a retract $r: \Theta \rightarrow M$. consider the map $f(x)=r(x+c\xi_x)$, where $c$ is a sufficiently small constant that $x+c\xi_x$ is always in the tubular neighborhood (possible by compactness)

Alexander Gruber

The point ends up being that you can decompose all kinds of really complicated combinatorial structures into smaller species, and use the calculus to find cardinalities and whatnot.

Mike Miller

this has no fixed points, because the tangent space at $x$ has trivial intersection with the normal space (is that a thing? normal space? i think you get my drift)

Alexander Gruber

Soon I'm going to make finite groups act on species, once I'm ready. and then we're gonna see some shit go down.

Mike Miller

but if $\chi(M) \neq 0$ then by Lefschetz FPT any continuous map has a fixed point. therefore, $\chi(M) = 0$.

Alexander Gruber

02:46

anyhow, sorry if I lost you @Pedro. That's what I've been looking at lately.

Karl Kronenfeld

@MikeMiller Ah, nice.

Mike Miller

it's super quick and elegant. I'm sad I never knew this before.

here's the hopf index theorem. i only know a proof for surfaces.

Karl Kronenfeld

@AlexanderGruber I know you implied you're not ready yet, but I'll ask anyway. What's an example of a group acting on a species?

@MikeMiller I must say, that's pretty remarkable.

Mike Miller

the index theorem? yeah, seriously.

Karl Kronenfeld

yeah

Mike Miller

02:59

if you asked me why the hairy ball theorem is true, before today i woulda said the index theorem

of course now I know it's true because of the simple proof above

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