in terms of physical activity, not really. I guess I like hiking/long walks but I don't really get the chance more than once or twice a month

Try to do more active stuff. Go for a walk, lift some weights, run around, play a sport. You don't have to be sweating and panting at the end. It is healthy for your brain and mood (particularly with motivation) to stay active.

@SamuelYusim I sympathize with your psychology somewhat. Feeling inferior goes down to a frustrating level sometimes: I stop doing math for days, maybe weeks. Completely understand having more productivity while in home with my family - I was away for like a month or so, about a thousand km away from where my home is, in a relatively well-known university to study math. I didn't feel more productive, although I probably did learn more. I didn't enjoy my time there either.

@BalarkaSen Hello.

Though I have realized it's more about being around my home than being together with my family or family doing stuff I feel lazy for. The familiar environment's the issue.

Hi @0celo7.

Currently pondering why a vector bundle deformation retracts onto its zero section

@SamuelYusim On the math side of the things, note that in Novikov's theorem you should add $n \geq 5$.

oh, did I forget that?

And Whitney only tells you manifolds embed in R^{2n+1}, not n-complexes in general.

@0celo7 Contract along fibers.

Whitney tells you it happens in $2n$, if you're clever.

Only orientable ones though.

@SamuelYusim Apparently so.

I thought all manifolds are orientable and compact ;)

@BalarkaSen I can say that too, but the details are eluding me.

Only compact.

there should be a way to extend whitney to complexes though, shouldn't there?

@0celo7 Oh, google says orientability is not needed.

@BalarkaSen I didn't think it was, but I wasn't going to say anything.

The proof is ugly enough for Hirsch to skip it.

@SamuelYusim I don't know an easy way to prove this. There is a "theorem" that any space of topological dimension $n$ embeds in $\Bbb{R}^{2n+1}$, but I don't know how to prove this.

I heard it from Mike. Maybe you'd find references if you google.

Simplicial complexes are nice enough to have top. dim. n.

oh well, I mean it's not the most important part of this, especially considering the citation I gave covers it

Fair enough. Yes, it's a point-set theorem, but I imagine the proof would not be exciting.

@0celo7 What's there to say? The fibers are contractible, just contract stuff along the fibers smoothly.

If you want to do this abstractly, use fiber bundle exact sequence + Whitehead theorem.

"The fiber bundles of the fibers are contractible"?

@BalarkaSen I don't know what that means.

I just woke up, writing nonsense.

Google is your friend.

What exact sequence

if $F \to E \to B$ is a fiber bundle, there is a long exact sequence in homotopy groups.

$\cdots \to \pi_{n-1}(B) \to \pi_n(F) \to \pi_n(E) \to \pi_n(B) \to \cdots$

This shouldn't require algebraic topology

For vector bundles, no, not really.

How do you write that map down

how do you write down a retraction of each fiber

You can't write down how to retract $\Bbb R^n$ to the origin?

I could like a week ago, dunno how to now

But I'm not asking that

Assuming I can do that, how do I write the map $E\to E$ that does that on each fiber?

The retraction is literally the projection map... the deformation retraction $E \times I \to E$ is $((x, v), t) \mapsto (x, v \cdot t)$, isn't it?

Starts at identity, ends at projection to zero section.

@BalarkaSen I think you mean $(1-t)$.

Whatever.

Well...I guess you define it in each trivialization?

But then you maybe have to show it does not depend on how you do that...

Or that you can stitch it together and get a globally defined retraction

But I wrote it down globally, on each point, didn't I?

Each point is not of the form $(x,v)$

To do that you're implicitly picking a trivialization

OK, fair, you need a local trivialization.

But scalar multiplication is independent of whatever you choose, so that's not a problem.

I'm not sure what you mean

Ah

Derivative makes the $t$ pop out.

The scalar multiplication commutes with transition functions

Da. I'm heading back to bed.

Night.

@BalarkaSen I meant linear transformation here, not derivative :P

@0celo7 Did you do any more of the Hirsch problems?

@BalarkaSen I solved 6.1.2, 6.1.4, 6.2.5 (after an epic battle), 6.2.9, and I will attempt 6.3.6 some time in the future.

Any interesting ones?

Those are the easy ones :P

6.1.4 is good.

It + Whitney's theorem is what allows Morse theory to actually be useful.

6.3.6 can be used to compute the homology of $\Bbb R P^n$.

6.2.9 is trivial if you know Riemannian geometry, I did it to feel good about myself :P

I should sit down and learn homology, but now I'm neck deep in vector bundle cohomology.

Also, I just realized Hirsch has a proof that the intersection theoretic Euler char equals the homological one.

@0celo7 Well, if you have a Riemannian metric, it boils down to linear algebra on a tangent space, doesn't it?

@BalarkaSen You have to be careful about how you define the gradient of the restriction.

With Hirsch, I do not feel bad using "outside material."

Oh, I just realized it says $C^2$. My proof probably fails...screw $C^2$.

lol

I used a "geometric" definition of the tangent space that probably only works in $C^\infty$.

Hmm?

Oh, collection of tangent vectors to smooth paths at $p$?

Yeah.

Because you can construct $T_xV$ by using smooth paths in $V$.

Does it only work for C^\infty?

So the restriction is immediate.

@BalarkaSen I'm not sure. There are certain constructions of the tangent space that fail for $C^r, r<\infty$.

This might be one of them.

@0celo7 I see.

...it seems you only need one derivative?

Maybe it works on $C^1$, who knows...

I'd think it works for C^1.

@BalarkaSen Have I told you that the algebraic definition fails for $C^r$?

I probably did.

Yeah, not finitely generated, I think.

Yeah, and you can't identify the tangent vectors with derivatives, period.

So there's no interpretation of the monstrosity you end up constructing.

It's not finitely generated because it's not Noetherian, right?

Hmm, no, I don't have an increasing ideal of things.

No, it's because you can find an uncountable set of "tangent vectors" which are linearly independent.

@0celo7 I have to figure out one day why the homological orientation agrees with smooth orientation.

@0celo7 Ah, ok.

Hi

Could someone provide intuition/motivation on the subject of Symbollyc Dynamics

@BalarkaSen The proof is in AMS Lee, btw.

welcome to my sexy party

ummm

Yay :)

oh HAHA

I am just very excited because I'm about to submit my first paper to a journal

excited / a nervous wreck

Sounds like a good time to send Spivak Vol2 to me

Can't think of a better time, honestly.

you have a PDF, right?

Piracy is a mortal sin, @ForeverMozart

I do not

I'm shocked that the 2nd book is out of print

[email protected] that is Spivak's email address

I should give a sob story

I am living in a box

literally

I would send you the book, but I was very attached to it at one time

I spent an entire summer using it to understand relativity

lots of writing on the pages

I highly doubt you need Spivak vol 2 to understand relativity

what relativity book?

but I do have a PDF if you want

I'm on the hive of scum and villainy looking for one

lol

dammit chrome

you can pretty much find any PDF if you look hard enough

why don't you show me downloads any more

there we go

its OOP, so it's not like you're taking revenue away from anyone

doesn't matter, it's intellectual property

@ForeverMozart so why did you need this for relativity

doesn't vol 1 cover Riem geo

I think vol 2 covers the fundamental forms

curvature tensors

I have too many PDFs open

computer lagging

and through reading vol. 2, I believe Riemann actually had relativity theory before Einstein

hmm, the first volume covers Riemannian geometry without connections??

general relativity, that is

@ForeverMozart Yeah, he had some ideas

but special relativity was completely missing

yeah, special is more physics, general is more mathematics

Without special there's no good reason to take the metric to be Lorentzian

how do we know that the Godel model is impossible?

I forget

Impossible?

It's a solution to the Einstein equations, so it's certainly possible

so everything can reoccur?

There are closed timelike curves though each point

So it's really screwed up causal-structure wise

so eventually we will have this discussion again

Depends how you define "again"?

It's not clear to me (or anyone?) how evolution works when there are such curves

oh

well we would have to devolve

and then evolve again

evolve in the PDE sense.

or be wiped out, and life start over

stop taking drugs, they're bad for you.

Godel is like a PCP hallucination

were you reading Hawking & Ellis?

no, I have never read Hawking. I'm skeptical for some reason

then what GR book were you reading??

Godel isn't forbidden on grounds of math, as I understand it, but rather on "the universe can't be that bizarre"

Hawking & Ellis is a landmark in mathematical physics, why be skeptical?

@Semiclassical Exactly.

you know how some great scientists promote crazy theories later in life, and people buy it because they were once great?

What's why I wanted clarification on "impossible"

@ForeverMozart Hawking & Ellis was written in his prime.

I learned relativity through a combination of Spivak and a tiny book by Faber

I know special relativity well enough by education, but general? Nope.

but it's been about 5 years since I thought seriously about it

I'm about to finish a PhD in topology

very abstract topology

can you tell me what a functor is

@Semiclassical like I said, Riemann pretty much had all the mathematics, and the Einstein just adapted it to re-do the Newton equation

functor is a mapping between objects in category theory

it is a map between sets that commutes with the category operation, or something

that's more abstract algebra

...or something?

I only spent like 2 weeks on category theory

I'm an engineer and I can do better :P

can you prove that a metric space is paracompact

you are a professional engineer or student?

are you that kind of topologist?

yes

@ForeverMozart student

@ForeverMozart I'd like to know the proof

that is a difficult one

I've got time

what does point set beyond Munkres even do

there are still open problems

like the D-space problem

that's the biggest

what's that

The D-space problem is, is every Lindelof space D?

See, not so technical.

You might think about why every compact space is D, as an exercise

which separation axiom is Lindelof?

Lindelof is almost compact

is it even a separation axiom

Lindelof means every open cover has a countable subcover

no, its not a separation axiom

so lots of people think about that

I think more about continuum theory

which is the study of compact connected spaces

why is it important

who knows

I admit I've never seen one of those spaces pop up

well, $[0,1]$ is a continuum

no

Lindelof

oh

clearly compact connected spaces are important.

the ordinal $\omega_1$ is Lindelof but not compact (in the order topology)

oh god

it gets very set-theoretic

@Semiclassical do you do any work with symplectic topology/dynamical systems stuff?

not symplectic topology

browses Amazon

wait that is wrong

Ok, look, every subset of the plane is Lindelof

some of the stuff i do presumably has a symplectic context, at least to the extent that symplectic manifolds = classical phase space as compared with quantum mechanics

but clearly need not be compact

(if I knew Hamilton-Jacobi theory I'd probably be able to say something brilliant here, but alas)

I should find a light intro to symplectic stuff

I think separable metric is Lindelof

Arnold was too informal

dynamical systems...well, I do a lot of PDE stuff lately

And I don't want a 400 page monster

including some stuff on integrability

dynamical systems is topology

like dendroids and fractals

I think most things are topology

@ForeverMozart that's the uninteresting side

noooo

I construct crazy pathological spaces

that is not uninteresting

you're one of those people

yes

it is, it must be said, a matter of taste

you write textbook exercises :P

the starred problems

possibly some day they will appear in textbooks

Mind if I throw a poll in here: strawpoll.me/10843492

i think equating dynamical systems to topology is rather reductive, frankly

it doesn't touch on the huge depth of stuff in the realm of integrability

I think everything can be reduced to topology

everything interesting

@HelkaHomba 5, easily

sorry set theory

clearly 5

2 would make me buy more things

i could see 1, but 5 is pretty straightforwardly great.

@ForeverMozart You'd get rid of your fridge for such a machine? What if you want to cook for yourself for a change?

i think i'll go with 1

I don't really cook, although maybe some day

I actually enjoy driving

@ForeverMozart I would own many books

many more

yeah I just got two today

which ones?

How to Solve It, and Being and Time

Heidegger

some math, some philosophy

I'm considering buying a short ~200 page book from AMS.

Some light reading

Currently battling through Bott & Tu, it's pretty terrifying.

differential forms - too much notation

what

let's browse the AMS book store

"Quiver Representations and Quiver Varieties"

Never heard of that

"Colored Operads"

@ForeverMozart learning a little math for cooking is one reason you can give those "what will I ever use this for" students ;)

Pretty sure that's a troll

"Introduction to Tropical Geometry"

sounds...warm?

many business students do not see why they need calculus though

> Undergraduate and graduate students and research mathematicians interested in algebraic geometry and combinatorics.

That's pretty broad.

I can't make them love it, as a teacher

is that a course description?

@ForeverMozart book description

i think the trouble i'd have is that i genuinely don't know what a calculus course would do for a business student

browsing AMS for a symplectic topology book

I give some business applications. Like how many units should you sell for maximum profit?

but mostly it is integrate this, differentiate that, find the area, ...

eh. is that really a business application, or just a certain kind of word problem?

I simply don't know enough about business to say.

well, it assumes that they have a simple function to model cost and demand vs price per unit

sort of superficial, but not completely useless

"Differential Algebraic Topology: From Stratifolds to Exotic Spheres"

oooooo

@BalarkaSen

exotic spheres?

i guess my point is that it's hard to create a good context-rich problem without knowing the context in some depth

Spheres not diffeomorphic to the usual ones

@ForeverMozart Would you agree with ted.com/talks/… ? (not that calc and stats are completely separate)

220 pages

I'll have to listen. Stat and calculus are fairly independent in my mind.

probably too advanced for now

eh, they're definitely linked when it comes to continuous random variables

Maybe IDK, I never took a lot of stats :I

@Semiclassical yes but for undergraduate?

Does the stats course talk about the central limit theorem? Then it probably needs some calculus.

Never heard of this dude

...apparently he's a prof here

huh

Applications of stats, on the other hand...maybe

Calculus is more abstract. Tangent lines, areas under curves, etc. Statistics and probability is more useful for most people, so I agree @HelkaHomba

calculus is very hands-on

I struggle to have enthusiasm and energy when I'm teaching

like the TED guy has

"An Introduction to Symplectic Geometry"

ah

you have too many books

@ForeverMozart you don't know how many I have

amazon.com/…

I like buying things

lol

at a certain point I stopped buying books and started reading research papers

unless they are books like I mentioned

:)

fun reading

what point

this book might be too short.

I'm disinterested in quantization

so there's only 100 pages there

amazon.com/How-Solve-Mathematical-Princeton-Science/dp/…

that is a famous little book that I'm reading

maybe I will get some inspiration

Polya was hungarian I think

@ForeverMozart He is pretty passionate - youtube.com/watch?v=e4PTvXtz4GM

@HelkaHomba yes lots of gestures

I just look at the board and write

like a schizophrenic

jeez

he describes me

can a professor drink beer with his students in his office?

@Semiclassical @0celo7

@arctictern

@CRAZYGAYSHERIFF this short but precise summary was actually very helpful. thanks.

@Huy hi

hi Mozart

i just submit my first paper

congrats !

thank you, it is emotional for me

you got a summary on what?

@ForeverMozart: click the arrow

i always forget what kind of math you do

mostly Mapping Class Group stuff atm

oh I never knew about the arrow

I really enjoyed abstract algebra

I did too, though I've never studied more than the very basics

you are taking a course now?

no

I was attempting to understand an answer given on MO to my question

to me it was interesting how complexity was generated from such simple definitions and results

you start with baby stuff, and then suddenly it's amazing

oh wow you ask on Overflow

that is for serious questions :)

Qiaochu Yuan is very brilliant

yes

he's answered many of my questions on MSE back when I was a freshman

you want to do research in algebra?

or something else?

no, I don't know anything about algebra

mapping class groups is actually a topological object

hmm

it's the group of isotopy classes of orientation-preserving diffeomorphisms on a manifold

I'm mostly reading things in Farb & Margalit's book on Mapping Class Groups that I find interesting, no research intended

in my world of topology, we only deal with homeo, not diffeo :)

in dimension 2, it's not much of a difference :)

but you could bend an object to have corners?

and then not be diffeomorphic

anyway

you are a PhD student so what do you think you will research?