#point-set-topology

1 messages · Page 203 of 1

gritty widget
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times tensor^sigma

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in the summand

sleek thicket
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(-1)^σ is the sign

gritty widget
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oh

sleek thicket
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it's common notation!!

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Oh no why is it a different shade?

gritty widget
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i did not even take a second to think about what that could have meant

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i have no alts

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im already alternating hmm

sleek thicket
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hmmreturning is a lighter shade

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That's awful

gritty widget
sleek thicket
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(!)

gritty widget
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:3

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it means my alt is myself

sleek thicket
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🐺 sully

gritty widget
cedar pebble
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everyone talks about alt accounts but nobody talks about sym accounts

gritty widget
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hmm

sleek thicket
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They do but the terminology is a little corrupted

gritty widget
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sym(p)

sleek thicket
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Due to a translation error they're called "simp" accounts

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GOD DAMMIT

gritty widget
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sniped

tight agate
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lol

cedar pebble
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damn killed it

gritty widget
cedar pebble
gritty widget
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(legosi) sym(p)

sleek thicket
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petthecat

gritty widget
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every fucking time i see this

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i think of that stupid fucking picture

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of legosi

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with the joint

sleek thicket
gritty widget
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this isn't even a topology channel anymore

cedar pebble
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tonight I am thinking about the following scary theorem:
let X be a smooth variety over C (or maybe some weaker assumptions). Let E be a complex vector bundle over X. then there exists a canonical isomorphism between any two fibers of E

sleek thicket
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Uhh fuck someone ask a topology question

gritty widget
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sniped!!!

cedar pebble
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the canonical part is uhhh

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actually there's a stronger version: E admits a canonical real-analytic trivialization

sleek thicket
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huh

gritty widget
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hmm

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neat

cedar pebble
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yea it seems kinda

gritty widget
cedar pebble
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illegal

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when you state it that way

gritty widget
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yeah

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you're going to jail

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🚔

tight agate
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doesnt the chern connection do this?

cedar pebble
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Daniel Litt set me up this was a sting

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no it doesn't

gritty widget
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well what does!!!!

cedar pebble
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well

gritty widget
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bundle-pill me

cedar pebble
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actually yea maybe it does but that requires E to have Hermitian structure

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this requires no such assumption on E

tight agate
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connection ----> parallel transport ----> identification of fibers no?

cedar pebble
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yes that's the idea

sleek thicket
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are you worried that if you choose a hermitian structure you might get a noncanonical iso?

cedar pebble
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I guess you can think of it as like a real-analytic version of the Chern connection that requires no additional structure whatsoever

sleek thicket
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it might just end up not depending on the chosen metric? That would be sus though

sleek thicket
cedar pebble
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idk these theorems I'm proving are fun

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need to finish this damn paper

sleek thicket
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I'm a bundle fan

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Started doing char class stuff today

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was neat

cedar pebble
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Daniel scolded me harder than he has before today because this was the N-th week in a row I got literally nothing done on the paper

gritty widget
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lee's bundles book...

sleek thicket
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Oof :/

tight agate
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I will be reading some fun papers this weekend petthecat

tight agate
cedar pebble
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it is but not during a pandemic

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I mean I've been procrastinating because I've been working on other papers too

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and like

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dying

sleek thicket
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ah that reminds me

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I need to finish this application

cedar pebble
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although I feel motivated again since I finished these two talks this week

tight agate
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chicago?

sleek thicket
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Nah, ucsb

cedar pebble
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and found a million typos while preparing the slides

sleek thicket
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I submitted Chicago like an hour ago

cedar pebble
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so at least I fixed the first half of the paper

sleek thicket
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And then lost of motivation

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Hoping may sees my app and just ignores everything but the word operad

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@maxj rejoin the server so I can bug you about this

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smh

cedar pebble
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MaxJ is gone?

tight agate
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does peter may read all the applications?

sleek thicket
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Probably temporarily, he comes and goes

cedar pebble
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ahh

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lmfao Sham did you see the massively gay twitter thread the other night

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wake up
Max has liked 47 of your tweets

sleek thicket
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oh I don't think so lmao

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There was one with a lot of gay people that I muted at one point so maybe it was that one

cedar pebble
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it was in one of Sarah's threads and Zack said like "this is now a skincare thread"

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and I posted like

sleek thicket
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Oh yes lol I did see that

cedar pebble
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I miss going to the Sauna

sleek thicket
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So long...

cedar pebble
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yeaaa lmfao that was a blast of a thread

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Alex's sauna recommendation

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(totally not as advertised on the site but flonshed )

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"come to Atlanta once we're all vaccinated I'll take you to all the fun parties ;)"

sleek thicket
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Okay yes this was the thread I was thinking of lmao

cedar pebble
sleek thicket
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I jumped in like 10 posts deep

cedar pebble
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yea that thread went on for too long lmfao

sleek thicket
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I was looking forward to spending last summer in Atlanta :/

cedar pebble
sleek thicket
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hopefully things are looser this summer

cedar pebble
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man I hope so :/

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who looking forward to March rolling around and celebrating 1 year of lockdown

sleek thicket
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I have the date on my calendar

cedar pebble
sleek thicket
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My start date was when I flew down to California

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so it's easy to keep track of

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:/

cedar pebble
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my start date was the flight home from Arizona Winter School

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we got on the plane and UGA announced that we would be staying open

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sat next to our postdocs on the plane and we ranted about it

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by the time we landed they rescinded the decision

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then we like

sleek thicket
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LMAO

cedar pebble
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drove back to Athens with the postdocs and we all went on a big grocery run together since the stores were starting to run out of things

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kinda started to feel real at that point

sleek thicket
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I was on a plane two days after uw announced they were going online I think. We were the first college in the nation to go online iirc?

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I thought I was starting my spring break early (was already planning to go down and stay with parents)

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Here i am 11 months later lol

cedar pebble
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man I haven't seen my parents in ages

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had to cancel so many flights we bought in advance

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fuck

sleek thicket
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It has not been a great year

cedar pebble
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it really has not WAAAAH

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literally has been one of the worst years of my life

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just was starting to recover from horrible family emergency back in November/December and then pandemic

sleek thicket
long hornet
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I meant the arguments should be simpler. We can't rule out the possibility right away, because it's easy to prove that every vector space has a basis..

sleek thicket
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wait what

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I guess I don't see what you mean by more direct if "every vector space has a basis" counts

trim cypress
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projectivities are fun to play with

long hornet
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I mean Tychonoff's theorem works no matter how many spaces I have, so it's natural to assume that in this particular case there should be a simpler proof.

shut moat
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if you consider the closed ball of radius 1 with the origin removed, this is what Hubbard calls a "piece-with-boundary", which is a subset of a manifold that is 1. compact, 2. the surface area of the smooth boundary is finite 3. the non-smooth boundary points have 1-dimensional volume zero. The nonsmooth boundary is the origin, here, and it certainly has 1-dimensional volume 0

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now consider the vector field $\vec{E} = \frac{(x\hat{e}_1 + y \hat{e}_2 + z\hat{e}_3)}{(x^2 + y^2 + z^2)^\frac{3}{2}}$. It is smooth on the above piece-with-boundary, and it is obviously compactly supported. the only "misbehavior" is that it isn't bounded, which isn't a condition for stoke's theorem to hold

gentle ospreyBOT
shut moat
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and if stoke's theorem holds, then $\int_{\overline{B_1(\mathbf{0})}-\qty{\mathbf{0}}} \div{\vec{E}}dx\wedge dy\wedge dz = \int_{\partial\overline{B_1(\mathbf{0})}-\qty{\mathbf{0}} } \Phi_{\vec{E}}$

gentle ospreyBOT
shut moat
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however, it's well known that for the above vector field, stoke's theorem breaks- the vector field is divergenceless on R^3\{0}, but the flux is 4pi (I've removed some constants but you probably recognize this as just an electric field)

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so why exactly does this fail? this seems to match all the conditions for stoke's theorem

sleek thicket
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What exactly is your statement of stokes?

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Could you post a pic?

shut moat
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(implicitly here ∂X is actually the smooth boundary of X)

sleek thicket
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hmm

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Okay, digging deeper: what's the definition of boundary, smooth boundary, and piece with boundary in Hubbard?

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Sorry for making you find this stuff I just want to be really certain

shut moat
sleek thicket
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A piece with boundary must be compact

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Right?

shut moat
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yeah

sleek thicket
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But the ball minus a point is not compact

shut moat
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oh fuck

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I'm dumb lmao

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ty 😂

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yeah that's probably it

sleek thicket
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This ensures the form will be bounded

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Because it's a continuous function on a compact set

shut moat
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right, that makes sense

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I was suspicious about it being unbounded

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ig it's time to learn currents realshit

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in E&M we resolve this by defining the divergence to be a dirac delta lmao

sleek thicket
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Yeah so the general statement I remember is that the manifold just has to be a manifold with boundary but the form is compactly supported

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Because you can't integrate forms with noncompact support in general

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Like it's just not defined

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For the statement I'm saying

shut moat
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couldn't you take the lebesgue integral of the pullback to R^n? (assuming the form decays fast enough)

sleek thicket
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I'm not sure, the issue is making sure everything patches together

cedar pebble
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assuming the form decays fast enough
this is essentially what the compact support is doing, in a rather crude way

sleek thicket
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Like you have a lot of charts

cedar pebble
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of course you can integrate things in general that don't have compact support but you need something that captures this rapid decay

sleek thicket
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Maybe on each chart its lebesgue integrable

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but you still need to sum over the charts, which might cause issues?

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I'm genuinely not sure, sorry

cedar pebble
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^

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yea that's right

shut moat
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i think it's sufficient to use a chart that's dense in the manifold

sleek thicket
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That might not exist though

shut moat
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every manifold admits a weak parametrization in this way, according to hubbard

sleek thicket
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oh

shut moat
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let me find the statement if you like

sleek thicket
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Maybe I am lying

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Yeah I'd like to see it

cedar pebble
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there's still issues though, as sham says

sleek thicket
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Also yeah take the identity function on R^n

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This is locally L^1

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So you can choose charts where it's L^1

shut moat
sleek thicket
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But the total integral is infinity

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And by changing signs you can screw it up really badly

shut moat
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so if you choose a single large "weak" parametrization, do you still run into the "local L^1 but not globally" issue?

cedar pebble
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yes

sleek thicket
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I'm not sure

shut moat
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how come? you're not breaking it up into small regions

sleek thicket
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I mean if by locally L^1 you mean L^1 on that chart you shouldn't have issues

shut moat
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it's effectively a global parametrization

sleek thicket
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Does such a parameterization preserve integration of forms under pullback?

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I think so

shut moat
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I'm not sure if this is generally how it's done but integration of forms is defined as the integral of the pullback by a weak param (in this book)

sleek thicket
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3+4 should tell you it's a diffeo on U \ X, right ?

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Oh no

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It might not be surjective

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But the point is you only miss out on a subset of measure zero

shut moat
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yeah

sleek thicket
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So it shouldn't affect the integral of the pullback to excise it

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Weird

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I'm really not sure

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I guess my thoughts are like

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Does being L^1 depend on the weak parameterization?

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Hmm actually I think I'm thinking about this all wrong

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You can take the absolute value of a form and get a thing

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I'm trying to remember the name

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It stars with a d lol

shut moat
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densities?

sleek thicket
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Yes!

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Right so

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Any nonnegative (measurable?) density should have a well defined integral

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Over the whole space

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I think you can define the sigma algebra of measurable sets on M

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You can detect measure zero because smooth maps are locally lipschitz, so transition maps preserve being measure zero

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and you can detect borel sets

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So take the union of all null sets and all borel sets

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This gives you a notion of measurable function

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You should be able to turn that into a notion of a measurable distribution

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And a nonnegative measurable distribution should have a well defined total integral

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And we can define L^1(M) to be all measurable forms ω such that |ω| has finite integral

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I'm not really sure, sorry

shut moat
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no ty this is interesting

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this makes sense, it's like an analogy with normal lebesgue integrals on R^n

sleek thicket
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I think it might be better to think about riemannian manifolds

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Where you have a canonical density

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And now you're talking about integrating functions

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and then the stuff I said should definitely work out

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oh dear

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This is relevant

shut moat
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lol yes

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this is where I first saw this

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was just about to bring it up

sleek thicket
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Comments are saying to look at any book on global analysis

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I happen to have a book like that at the top of my reading list

shut moat
#

:o

sleek thicket
shut moat
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presumably I'd need to know a bit more analysis than I do now to approach it lol

sleek thicket
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¯\_(ツ)_/¯

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What do you know now?

shut moat
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Rudin and Hubbard basically lol

sleek thicket
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Ah okay gotcha

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I am debating getting into this kind of thing next year

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Diff top/geo which is very analysisy

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Kind of want to write my senior thesis on the atiyah singer index theorem

shut moat
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oh i've heard a lot about that :o

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physics people seem to like it

sleek thicket
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It has a lot of really important theorems in geometry as consequences

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Riemann roch, gauss bonnet, something else which I forgot

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Mostly I just like things which mix algebra analysis and geometry/topology

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And this seems like a neat capstone on my degree

shut moat
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all the math realshit

sleek thicket
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Absolutely not, no number theory or combinatorics allowed

shut moat
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oh true, all the *interesting math kekw

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btw what's the title of that book you were considering? I'd like to libgen it and take a look anyway

sleek thicket
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I think global calculus

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This is it

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Also would be a nice way to teach myself sheaf cohomology

shut moat
sleek thicket
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Without having to get over my fear of schemes (yet)

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oh right I was going to schemepill you

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😈

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Do you know what a commutative ring is

shut moat
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I know very little algebra, but I think a commutative ring is like (field minus a property) that's commutative?

sleek thicket
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Nope, a commutative ring is still commutative

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Luckily

ivory dragon
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that would be a very confusing name

sleek thicket
#

It's a field but without multiplicative inverses

ivory dragon
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^

shut moat
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oh I worded that badly, I meant like a (field without a property) + commutative

ivory dragon
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well, possibly without multiplicative inverses

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fields already commute

sleek thicket
#

I'm just going to call it a ring

shut moat
#

gotcha

sleek thicket
#

so here are the top infinity rings you should know

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Z

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k[x] where k is a field

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k[x, y] where k is a field

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k[x, y, z] where k is a field

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k[x1, x2,...,xn] where k is a field

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okay so question

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What is this square bracket notation?

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Have you seen it before?

shut moat
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idt so

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well I've seen some people use it to refer to polynomials of some sort, I think

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so my guess is polynomials with coefficients in that field

sleek thicket
#

yeah!

shut moat
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pog

sleek thicket
#

With these prescribed variables

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so here is the moral of algebraic geometry, part 1

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Let k be an algebraically closed field, meaning any polynomial with k coefficients (in a single variable) is a product of linear factors

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question: what's an example of an algebraically closed field?

shut moat
#

C?

sleek thicket
#

Yes!

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So algebraic geometry classically mostly happened over C

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I'd draw you a parabola and write y = x^2 but secretly it's some kind of fucked up surface in R^4

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That's not super important but it's worth mentioning

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So here is the moral of algebraic geometry

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Let A^n be the set of n-tuples of elements of k (this is set theoretically k^n but we use a new notation for reasons)

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The ring k[x1,...,xn] of polynomial functions on A^n captures "all" the geometric information about A^n

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namely, this ring encodes all the information about "subvarieties" of A^n

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These are sets cut out by polynomial equations

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So { (x, y) : y - x^2 = 0 } is a subvariety of A^2

shut moat
#

makes sense

sleek thicket
#

Right, so what does this mean more specifically

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Actually I don't want to get into that lol

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but there's this correspondence between "irreducible" subvarieties and prime ideals

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An irreducible subvariety is something which can't be written as the union of two smaller varieties

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So y^2 = x^2 is not irreducible, because it's the union of y = x and y = -x

shut moat
#

interesting ok

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so if you have an algebraicly closed field the only irreducible subvarieties are the linear ones, I think

sleek thicket
#

No

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there's like y^2 - x^3

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You're only thinking in 1 dimension

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And you're thinking about only one equation

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We might have several

shut moat
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oh ic

sleek thicket
#

Right so like

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Anyways, point is that varieties and polynomial rings are closely connected

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And you can eg discover whether a variety is smooth by looking at the ring

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the singularity of y^2 = x^3 is reflected in the fact that k[x, y]/(x^3 - y^2) is not a unique factorization domain

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,w plot y^2 = x^3

gentle ospreyBOT
shut moat
#

what does k[x, y]/(x^3 - y^2) mean here?

sleek thicket
#

It's like the polynomial ring but we've collapsed x^3 - y^2 down to zero

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So the equality x^3 = y^2 + (x^2 - y^2) means we force x^3 and y^2 to be equal

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This is a little hard to get formal about, quotients are confusing

shut moat
#

ok cool

sleek thicket
#

right so like

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We have these spaces (varieties) and functions on them

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And classical ag is basically understanding the geometry in terms of the algebra

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Okay so modern AG is this but we no longer have an algebraically closed field or polynomial rings

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If A is any ring we construct a ""space"" where the ""functions"" on that space are elements of A

shut moat
#

I'm assuming the quotes are carrying a lot here KEK

sleek thicket
#

Yeah lol

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But that's almost what a scheme is

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That's what an affine scheme is

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The space dual to A

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The one with ring of functions A

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A scheme is something which is locally an affine scheme

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Like how a manifold is something which is locally an open subset of R^n

shut moat
#

oh that seems reasonable

sleek thicket
#

it gets very hairy

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It's sort of like if you took differential topology and then replaced the local theory of "calculus in R^n" with "commutative algebra"

shut moat
#

wait so is this "locally an affine scheme" a homeomorphism or is there additional structure that you're trying to preserve

sleek thicket
#

Lots of additional structure

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It all comes down to functions

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So here's a cool theorem

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let M, N be manifolds

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We have rings A = C^infty(M) and B = C^infty(N)

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Yeah?

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you can multiply pointwise

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and add pointwise

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And get a ring structure

shut moat
#

right

sleek thicket
#

Okay well also if you have a smooth map F : M -> N you get a pullback map F^* : B -> A

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F^*(f) = f ° F

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Yeah?

shut moat
#

yeah

sleek thicket
#

Since everything happens pointwise, this is a ring homomorphism

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Ring homomorphism means it preserves addition and multiplication

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And 1

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So eg F^*(fg)(x) = (fg)(F(x)) = f(F(x)) g(F(x)) = F^*(f)(x) F^*(g)(x) = (F^*f F^* g)(x), so F^*(fg) = F^*f F^* g

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So we have a function from {smooth maps M -> N} to {ring homomorphisms B -> A} taking a smooth map to its pullback

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Theorem: this map is a bijection

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Categorically this says that the "ring of smooth functions" functor is fully faithful

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anyways point being like, even in manifold land the functions capture all the information about your space

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so the extra structure we have is a "sheaf", which can be thought of as assigning a set of "functions" to each open subset of your space

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And so we want both the space to be locally homeomorphic to an affine scheme but also the "functions" to be the same under this identification

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You can actually define smooth manifolds like this, instead of thinking about transition functions on overlaps being smooth

shut moat
#

oh wow pandaWow

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ty Shamrock! I should def read more about this once I learn more math

sleek thicket
#

It's very cool! Definitely wouldn't recommend jumping in unless you have sufficient algebra background though

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Geometry is cool in lots of differential forms

shut moat
#

Definitely wouldn't recommend jumping in unless you have sufficient algebra background though
yeah it seems to have a giant pile of prerequisites lol

sleek thicket
#

Learn from my mistakes lol

shut moat
#

do people still learn about "classical" alg geo?

cedar pebble
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yes

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especially at the beginning of learning about algebraic geometry

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it's hard to learn about schemes without learning about the classical motivations first

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people still study classical AG in the sense that they use modern tools but are still interested in problems which would be interesting in the classical context

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I suppose people no longer work on classical problems in AG using classical techniques

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but lots of people work in classical problems in AG using modern techniques

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and of course lots of modern problems in AG that don't even make sense in the classical setting

shut moat
#

that makes sense

tough imp
astral cedar
#

do people still learn about "classical" alg geo?
Yeah, most first courses don’t even name schemes

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Mine used Shafarevich, which is very nice but does everything in the context of quasi-projective varieties

trim cypress
#

whats the best way to get the algebra background for algebraic/diff geo books?

gritty widget
#

diffgeo
linear algebra will most likely suffice

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and judging by recent personal experience, familiarity with group actions might help opencry

trim cypress
#

yeah thats kinda more what i was targetting

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i think my linalgs fine for diff geo

gritty widget
#

we'll lock you in a room with a copy of atiyah-macdonald and you can only come out when you've solved all the exercises

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then you're ready for AG

trim cypress
#

oh no

#

hartshorne

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in all seriousness, probably vikals notes

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*vakil

river granite
tight agate
#

it might not hurt to start reading hartshorne

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and learn whatever comm alg you need on the go

trim cypress
#

havent taken a dedicated course, just know vector spaces (from linalg), groups, basic category theory

tight agate
#

no rings and modules?

trim cypress
#

heh, pretty informally just for cryptography

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so ill say no

tight agate
#

ok yeah it probably would make more sense to learn about rings and modules first

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it might also help to know stuff about manifolds

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and some diff geo

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not essential, but a lot of the motivation comes from those areas

trim cypress
#

makes sense

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ive heard some things about dummit/foote being boring before, is it really that bad and should i look elsewhere

tight agate
#

aluffi might also be a good option on the side

trim cypress
#

brb libgen

tight agate
#

aluffi + dummit/foote is a good combo

trim cypress
#

946 pages... ive got time i guess

tight agate
#

you don't have to read all of it ig

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but it is stuff worth knowing

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if youre interested in the algebraic side of things

trim cypress
#

okay cool, thanks

gritty widget
#

make sure to learn the sylow theorems in incredible depth

tight agate
gritty widget
#

prove that a group of order 3495739857239573958673249563249862347632587436589 is simple

trim cypress
#

lmao

tight agate
#

is simple monkagigagun

trim cypress
#

maybe ill finish coxeter first

gritty widget
#

i'll start

#

,w prime factorization 3495739857239573958673249563249862347632587436589

gentle ospreyBOT
gritty widget
#

trivial

sleek thicket
#

is this true? feels like you still need a good deal of comm alg

sleek thicket
#

I mean like classical alg geo works due to the nullstellensatz

#

Step #1 already involves a pretty big comm alg investment

trim cypress
#

is there a way of saying this out loud besides just "X and Y are perspective with center P"

📎 unknown.png
gritty widget
#

X wedge equals P Y

obtuse meteor
#

professor assigns midterm and homework in same week
professor: "Don't worry, the homework will be short this week"
homework: 1 question shorter than regular homework, includes proof that the fundamental group of CW boi depends only on 2-skeleton

#

why even

sleek thicket
#

I would simply take classes without midterms

#

:smug:

#

actually I have a French midterm tonight but at least no math

obtuse meteor
#

that would be big brain Shamrock

obtuse meteor
#

@sleek thicket you just made me realize we only defined the fundamental group 10 days ago

sleek thicket
#

lol

obtuse meteor
#

scary shit

#

defines fundamental group 15 days before midterm

#

study problems for the midterm from previous Qualifying Exams

sleek thicket
#

So many words

obtuse meteor
#

it's basically like

#

take the side of the torus you can see

#

that's like a disky boi

sleek thicket
#

right

#

a little square

#

Curved square

obtuse meteor
#

identify boundary of disky boi with circle

sleek thicket
#

Okay, this makes sense

obtuse meteor
#

quotient out disky boi and union that with a new disky boi

#

identify boundary new disky boi with boundary of old disky boi

#

by twisting new disky boi twice

#

with z^2

#

now calculate the fundamental group of this garbage

sleek thicket
#

That seems annoying

obtuse meteor
#

the solution probably starts something like

#

"We clearly have a CW complex structure of this form"

sleek thicket
#

right

obtuse meteor
#

but the amount of meme in that

sleek thicket
#

I was thinking of just doing seifert van kampen but you've probably built up more theory for cw complexes

obtuse meteor
#

doing van kampen considering we're quotienting stuff

#

seems bad

sleek thicket
#

(more theory in that I don't remember the way to compute π1 of a cw complex)

obtuse meteor
#

and I would rather cw complex

sleek thicket
#

well in my head it's like, you have a torus minus a point and a torus minus a point and they intersect in a circle

#

but the inclusion of one circle is doubled

#

And the other isn't

obtuse meteor
#

hmm idk

sleek thicket
#

like you have this nice closed cover where the intersection is S^1 and you just thicken the sets a little so they're open

obtuse meteor
#

I can't see that picture but it's probably right

#

it's really funny like how different the difficulty of these problems can be

#

imo

#

like

#

This is just really simple

sleek thicket
#

CW complexes are probably nicer but I don't remember the nice way to compute the thing for them the nice way

#

Yeah

obtuse meteor
#

because pi1 of a graph is trivial

sleek thicket
#

Right

obtuse meteor
#

and pi1 of Sigma_2 is just polygon and van kampen

#

then you say "abelianizations disagree"

sleek thicket
#

What's Σ_2?

obtuse meteor
#

surface of genus 2

sleek thicket
#

Ah okay

#

Then yes that makes sense

#

And seems easy

obtuse meteor
#

I have learned the way to pass my qualifying exams in a few years

#

take them during the right year

sleek thicket
#

Hmm so the abelianization of π1 of Σ_2 is like Z^4 right?

#

and then the fundamental group of the graph is like... blech I forget how it works exactly all I remember is you take a minimum spanning tree

#

And then it's like...free on all the edges not in that tree?

obtuse meteor
#

the easiest way to think of it for me is just to contract the subcomplexes

#

so you contract the left edge

sleek thicket
#

Right okay that is easier

obtuse meteor
#

and you end up with like wedge sum of circles or something

sleek thicket
#

Yup

#

So it's free on 6 generators, right?

#

π1(K5) I mean

obtuse meteor
#

it's K4

#

so you're free on 3 generators

sleek thicket
#

Oh misread lol

obtuse meteor
#

I got this as well by contracting

#

but I'm not sure I did it correctly

sleek thicket
#

Okay yes I agree you're free on 3

#

so this rules our any retraction at all

#

neat!

obtuse meteor
#

yea

sleek thicket
#

chm had a cool problem on his topology final last quarter

#

Of the "compute π1" variety

#

it was to determine the fundamental group of $(S^2 \times S^2) \setminus \Delta$

gentle ospreyBOT
sleek thicket
#

Where Δ is the diagonal

obtuse meteor
#

S^2 x S^2 is scary

#

that's in 4D

sleek thicket
#

yes

obtuse meteor
#

too big for my small brain

sleek thicket
#

And then you cut out a weird bit

#

👻

obtuse meteor
#

but yeah

sleek thicket
#

it turns out that $(\mathbb{S}^n \times \mathbb{S}^n) \setminus \Delta$ is secretly a well known space

gentle ospreyBOT
obtuse meteor
#

good news at least: of the students who come to office hours I seem to be like pretty decent

sleek thicket
#

(but this doesn't actually make the problem much easier)

#

Nice!

obtuse meteor
#

but there are a lot of students who don't

#

and IDK how they're solving some of the homework without Jenny helping them with the solutions a bit lol :P

#

Ex. proving CW complexes are Hausdorff is a fucked up mess

sleek thicket
#

Oh lol I thought you meant for the course you were TAing (I think you said you were doing that?)

#

One of my classes has that vibe

obtuse meteor
#

Ah yeah I am TAing a course

#

that's really fun

#

reminds me

tough imp
obtuse meteor
#

I need to grade

tough imp
#

6D?

sleek thicket
#

I seem to be the only person asking questions in my bundles lecture

#

It's a 4d space chm

obtuse meteor
#

^

tough imp
#

I mean okay yes

#

But idk I can’t think about S^2 except as being in 3-space

sleek thicket
#

bro

#

you literally

#

Spend all day thinking about P^n

#

Or some shit

tough imp
#

Yeah so

sleek thicket
#

aren't you an algebraic geometer???

tough imp
#

Yeah so

sleek thicket
#

tfw

obtuse meteor
#

I just think about S^2 as fancy D^2

tough imp
#

S^2 isn’t AG

obtuse meteor
#

where you just glue

tough imp
#

You see heres the difference okay

sleek thicket
#

I think about S^2 as RP^2 but you untwist it

#

:smug:

tough imp
#

To think about S^2 I attempt to visualize

#

I do not do that when I do AG

sleek thicket
#

1 was me being a dumbass

#

and 1 was just me being curious about something

tough imp
#

I am Chmonkey

sleek thicket
#

but I watched the recording of lectures I missed and nobody said anything

#

It was just the prof talking

tough imp
#

Maybe ur the chosen one

#

O_O

obtuse meteor
#

yeah I ask more questions in lecture than most of the others

sleek thicket
#

you taught me not to have fear

tough imp
#

You should let it get to your head

sleek thicket
#

And to ask stupid questions

tough imp
#

Lol

#

With 334?

sleek thicket
#

it's true!

#

Yeah

tough imp
#

🥴

#

Apparently ppl read this as being horny

#

:(

sleek thicket
#

yes

tough imp
#

This is not the first time I have interpreted horny emotes as not being horny

obtuse meteor
sleek thicket
#

I try to mix it up

#

ask a balance

#

Keep them vigilant by sometimes pointing out subtle mistakes and sometimes forgetting extremely basic things

#

nobody will ever know which it is until we resolve it

#

ugh I do not want to study for this french class. I better read EGA after all this to make it worth it

tough imp
#

Lmao

#

I’ll have you read the parts of SGA ppl keep telling me answer my questions

obtuse meteor
#

I have neglected both grading / my manifolds class homework because of REU apps

#

and I got in but am now suffering the consequences

sleek thicket
#

it is really really tough

#

It was the thing that finally made me give up on my AG class last year

obtuse meteor
#

it might end up being that drops are made for when I have midterms in alg top

#

🧠

sleek thicket
#

:(((((

#

@tough imp you know what rocks

tough imp
#

Hmm?

obtuse meteor
sleek thicket
#

we're going to have class with Sándor next quarter

tough imp
#

Deciding you don’t want to do REUs

#

True!

sleek thicket
#

while trying to like

#

Read a chapter of Hartshorne a week and do 10 problems

#

Or whatever nonsense standard I had set

tough imp
#

Indeed

obtuse meteor
#

ah

tough imp
#

I have accepted that stacks and moduli class is goinf to be stacks class for me

#

And I am O K with this

obtuse meteor
#

at least like I'm done managing REU apps

#

which is a big advantage

tough imp
#

?

sleek thicket
tough imp
#

I mean that

obtuse meteor
#

bc now I can just alg top my way to victory

tough imp
#

I am not going to able to follow the moduli of curves shit

sleek thicket
#

tfw I'm halfway done

#

ah gotcha

tough imp
#

So just focus on

obtuse meteor
#

you can do this sham <3

tough imp
#

“Okay try to at least understand one thing”

obtuse meteor
#

actually yall will both get into fields tomorrow

sleek thicket
#

Hopefully someone will admit me and I don't have to do the late ones

tough imp
#

I have ascended above REUs

sleek thicket
#

Oh word is it coming out tomorrow? Like have they announced it?

obtuse meteor
#

I have no idea lol

cedar pebble
#

Faye good luck in the AT exam I believe in you WAAAAH

sleek thicket
#

Oh lmao

obtuse meteor
#

I know it should be around this time

obtuse meteor
cedar pebble
#

Also always happy to chat if you want to review

obtuse meteor
#

it's big meme

cedar pebble
#

No I didn’t

obtuse meteor
#

one sec

sleek thicket
#

a boy (me) can dream (be admitted to the fusrp tomorrow)

obtuse meteor
#

let me show the two wildly different difficulties

tough imp
#

Compute pi1 S^1

sleek thicket
#

Holy fuck s

obtuse meteor
cedar pebble
#

estrogen.fun

obtuse meteor
#

based

cedar pebble
#

Based domain

obtuse meteor
#

keep in mind

sleek thicket
# tough imp Compute pi1 S^1

I would simply use Whitney approximation to reduce to smooth paths and develop the theory of winding numbers from complex analysis

obtuse meteor
#

You see that date at the top of the page

cedar pebble
#

the homeomorph

tough imp
#

Of course

cedar pebble
tough imp
#

Lol

obtuse meteor
sleek thicket
#

Just apply seifert van kampen?

tough imp
#

I would simply quote ITM

obtuse meteor
#

that we will prove later

tough imp
sleek thicket
#

(for groupoids)

obtuse meteor
#

topology and groupoids monster

tough imp
#

Sham

#

Shoutout manifolds TA

obtuse meteor
#

also sham I ended up doing the cone thing with pushouts

sleek thicket
#

bro I haven't checked the reply to my regrade request yet

#

nice!!!!

obtuse meteor
#

and proving that pasting pushouts works

tough imp
#

He broke me to the point I have managed to let go of the crushing feeling of always worryjnf what grad schools think of me

#

And now I do what I want

sleek thicket
#

That's extremely based faye

tough imp
#

😎

#

Ayyy I also used pushouts and my TA asked why

obtuse meteor
#

I also just like said "these spaces are the pushouts of these diagrams. I talked with Jenny in office hours and she said I could just say this"

tough imp
#

Because I can

sleek thicket
#

lmaooo

#

That's so good

#

It's frustrating how like, one of my classes is super strict about what you can use and another I just cited things from Hatcher without blinking (for a course with no AT prereq)

tough imp
#

Chod

cedar pebble
#

proof by appeal to authority chad

obtuse meteor
#

@cedar pebble so like one problem is "apply the fact that you can contract cw complexes and then it's obvious"

#

and the other problem is

sleek thicket
#

I got points off for not proving the dual map is linear

obtuse meteor
cedar pebble
#

BIG NOZOMI

obtuse meteor
tough imp
sleek thicket
#

I hate this fucking class so much Faye

#

like

tough imp
#

Bruh...

obtuse meteor
sleek thicket
#

ugh

tough imp
#

I remember someone asked Max

obtuse meteor
#

yeah that sucks

sleek thicket
#

I'm not gonna rant about 525 again

cedar pebble
#

I need to stop doing my grad topics homework like right before it's due

tough imp
#

“What can we use” on the hw

sleek thicket
#

I've done it a lot of times on this server

#

Lmao

cedar pebble
#

I have this amazing skill where I have perfect knowledge of how long an assignment will take me

sleek thicket
#

I can't imagine lol

tough imp
#

And he was just like “fucking whatever lol”

sleek thicket
#

He didn't read those ever

cedar pebble
#

so I will leave it until exactly before the deadline and I get it in a minute before the deadline

tough imp
#

He read them once

cedar pebble
#

looks at quivers homework

sleek thicket
#

Oh right

cedar pebble
#

this will take me exactly 35 minutes

#

starts 40 minutes before deadline, stressed the whole time

#

😢

obtuse meteor
#

bad nG

sleek thicket
#

I genuinely wonder if I could convince this prof to just let me take the final tomorrow

#

like

#

hnnng

cedar pebble
#

like I'm amazed by how good my estimates are every time

#

it's still like

#

aaaAAAAA

sleek thicket
#

It's important to know your skills

tough imp
#

How many weeks are left in the quarter?

cedar pebble
#

too many

sleek thicket
#

4? Or 3? Idk

cedar pebble
#

I'm like 4 weeks into the semester and I already want to curl up in bed and die in my sleep

obtuse meteor
#

lol

sleek thicket
#

Finals starts March 15th

#

I'm not sure if I just finished or started week 6

obtuse meteor
#

I'm still continuing to be astounded that we've gone from def of pi1 to fundamental group of all cw complexes so fast

#

how could someone who has never been exposed to alg top keep up?

cedar pebble
#

yea that's kinda wild

tough imp
#

Ah you forget

cedar pebble
#

these problems are decently tricky too!

#

but very good practice

tough imp
#

The intro classes are not designed as intros

sleek thicket
#

lol

obtuse meteor
sleek thicket
#

lots of very discussion about uw classes

cedar pebble
#

you got this though seriously smol_nozoomi

tough imp
#

I wonder who’s teaching AG next year

sleek thicket
#

if it's sandor I might take it

tough imp
obtuse meteor
tough imp
#

I’ll do the first quarter

sleek thicket
#

I think I'm planning to ask for a letter

obtuse meteor
#

I just have to hope that there's not a wild jump in difficulty

tough imp
#

And maybe the rest if I wanna hang out

cedar pebble
#

is this a grad course?

sleek thicket
#

My main interests are like, AT, AG, diff top

obtuse meteor
#

but these like study problems she posted are scary

#

yeah it is

#

it's prep for a qual exam

cedar pebble
#

I suspect the exam will be curved pretty heavily, as they are in basically every grad course

tough imp
#

I thought u were AT god

#

Or something idk

sleek thicket
#

And I have letters in the other subjects

#

but need one for AG

tough imp
#

🧠

cedar pebble
#

when Polishchuk taught Hartshorne AG at my undergrad he made the mistake of giving exams and they got curved so heavily that like 30% was an A

obtuse meteor
#

lol

tough imp
#

Wait sham you aren’t gonna ask for 3 AG letters?

sleek thicket
tough imp
#

Cringe

sleek thicket
#

There were several 100s

cedar pebble
#

oh god

sleek thicket
#

on every exam

obtuse meteor
#

Also she said that the homework this week was gonna be short bc of the midterm

#

right

cedar pebble
#

what the fuck

obtuse meteor
#

and um

tough imp
#

Never me tfw

sleek thicket
#

They were take home

tough imp
#

195/200

sleek thicket
#

always me tfw

cedar pebble
#

oh takehome is valid

sleek thicket
#

Yeah

cedar pebble
#

okay that's different

tough imp
#

I’m mad cuz one of them I wasn’t actually wrong

cedar pebble
#

this was a Hartshorne exam with 1 hour in class

obtuse meteor
# obtuse meteor and um

let's just say we're doing the entire thing about computing fundamental group of cw complexes on homework

tough imp
#

But my proof was written confusingly

#

A Hartshorne exam?

#

They’re like here’s one problem

cedar pebble
#

yea the average was like 15% kek

tough imp
#

Solve in an hour

obtuse meteor
#

and then also computing pi1 of like RP^n and CP^n (this is the gimme problem which is good)

cedar pebble
#

oh yea that's a good one

obtuse meteor
#

and then there's also computing fundie group of genus g surface. Thankfully we can use van kampen and assume fundamental polygon so not that bad

#

but it's still like

#

the whole cw complexes thing

#

just

#

vibing

#

while we have a midterm this week

#

is very good

cedar pebble
#

yea that one isn't so bad with Van Kampen

tough imp
#

I feel like profs all somehow lose touch on how hard the problems are

sleek thicket
#

Holy shit chm I just remembered my grades in 504/5/6

cedar pebble
#

you'll do fine I'm sure

obtuse meteor
#

the exam is only an hour

tough imp
#

Wdym sham?

obtuse meteor
#

so I am suspecting heavy curve

tough imp
#

4.0/4.0/4.0

cedar pebble
#

yea sounds like it

sleek thicket
#

my lowest grade across all of the classes was 99.5%

obtuse meteor
#

and only like 2 problems

tough imp
#

Lmfao

sleek thicket
#

no I mean the total score

obtuse meteor
#

maybe a good few problems to select from

sleek thicket
#

and that's unweighted

cedar pebble
#

2 problems can be either woj5 or monkaGIGA

obtuse meteor
#

but only have to do like 2

sleek thicket
#

for exams vs homework

tough imp
#

Meme

obtuse meteor
#

yeah

sleek thicket
#

I probably shouldn't have taken thst class

#

And done analysis instead

cedar pebble
#

I've always joked about a grad AG exam that's one questions, true or false

#

so you either get 100% or 0%

obtuse meteor
#

🧠

tough imp
#

Yeah but last year was also trash

#

For analysis

#

Also you wouldn’t have met Sándor

sleek thicket
#

True

obtuse meteor
#

I am like between on whether I should skip the undergrad algebra course

#

and just take the grad one

#

my algebra is really weak

tough imp
#

Tfw

cedar pebble
#

take the undergrad one

sleek thicket
#

Also I do not think my analysis experience would've been worse

cedar pebble
#

if it's too easy, just do more work in the class

obtuse meteor
#

fair

cedar pebble
#

even undergrad algebra has like

#

a really really high ceiling

sleek thicket
#

I don't agree with the philosophy

obtuse meteor
#

I could also switch too tho

sleek thicket
#

can't really speak to your particular case

little hemlock
#

I took undergrad and audited grad

cedar pebble
#

true

#

I guess it depends on how the grad algebra course is taught

sleek thicket
#

maybe I'm biased by ug algebra here being really bad

tough imp
sleek thicket
#

and grad algebra starting from ground 0

tough imp
cedar pebble
#

like when I took grad qual algebra here it was taught in a self-contained way just went insanely fast

#

like we got through the Sylow theorems within the first 3 classes

little hemlock
#

That was basically my conclusion chm opencry

tough imp
#

...

obtuse meteor
#

my algebra is weak but I can also properly just bash and exploit universal properties to get the answer

obtuse meteor
tough imp
#

Thomas’s Monty class

sleek thicket
#

I think it's worth asking other people who've taken the course faye

obtuse meteor
#

yeah

sleek thicket
#

(not the prof profs lie and don't understand difficulty)

obtuse meteor
#

being not in person sucks tho for all this

tough imp
#

Indeed

cedar pebble
#

yea honestly I think the better option without knowledge of this would be to take undergrad algebra but challenge yourself with harder problems as they come up

obtuse meteor
#

I would just like walk up to noah and be like "hey so what's up with this"

little hemlock
#

Wait I took undergrad and audited grad at the same. That’s what I meant

cedar pebble
#

like I'm sure the class will likely be too easy but e.g. you will come to Galois theory and if you're bored you can punish yourself with harder and harder computations

#

sky is the limit

tough imp
#

I mean nG at least at UW

sleek thicket
#

Faye if you want study materials to diagnose/prepare for grad algebra

tough imp
#

The topics you cover are so limited

sleek thicket
#

I have problem sets written up

tough imp
#

Indeed

sleek thicket
tough imp
#

This is field-tested

sleek thicket
#

lol

tough imp
#

Chmonkey approved

sleek thicket
#

uhoh dox

#

Real names on there

#

me and who?

#

You'll never know

cedar pebble
#

wait sham and chomkey were both TAs for that course hyper

#

dream team

sleek thicket
#

Well there was no prof

#

It was just us

cedar pebble
#

yOOOOOOOO

sleek thicket
#

and a bunch of students

cedar pebble
#

king shit

sleek thicket
#

yeah so like

#

Story is

obtuse meteor
#

very much chad energy

tough imp
#

Even freshman
“Why is |sum a_nb_n| <= sum |a_nb_n| not cauchy’s inequality”

sleek thicket
#

In high school I did a reading group on group theory

#

At my community College

tough imp
#

Became Chad algebra man from this

sleek thicket
#

and I was super lost

#

Like

cedar pebble
#

these high schoolers are getting too good what the fuck

sleek thicket
#

Very much did not get it

#

but then uw didn't let me take ug algebra

#

because I didn't have intro to proofs

tough imp
#

dab

#

Sham consider this avtually

sleek thicket
#

Despite the honors analysis course not teaching you profs

tough imp
#

They did u a favor

sleek thicket
#

so I was like

obtuse meteor
#

I started reading Hatcher in high school and didn't understand it at all

sleek thicket
#

"hey does anyone want to do a reading group"

tough imp
#

Because then we got to experience Thomas Algebra

sleek thicket
#

and then the analysis ta offered to help us

cedar pebble
#

lmfao funny story similar to this, I went to take intro to proofs first quarter I transferred to my undergrad, thinking I needed to take the class because it was a prereq

sleek thicket
#

l o l

cedar pebble
#

went to office hours since it was taught by a well known homotopy theorist and asked about homotopy theory nonsense

#

and he basically said

sleek thicket
#

l o l

cedar pebble
#

please drop the class and register for UG algebra you don't need my class

obtuse meteor
#

fine meme you got there

sleek thicket
#

I did that in my calc 3 class because I was an fp nerd and people were talking about HoTT

#

It only gets funnier the more math I learn

obtuse meteor
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why is every person in math now originally an fp nerd

sleek thicket
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the real kicker is the prof did his PhD in homotopy theory before heading to community College

obtuse meteor
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fucking dum

cedar pebble
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god that year of undergrad was my real nlab brain blunder year

tough imp
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fp?

sleek thicket
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Functional programming

tough imp
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Lol

sleek thicket
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it's like functional analysis but cringe

tough imp
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Don’t group me with you ppl

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I was a pure boy who just went “hahaha integral go brrr I like math”

sleek thicket
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Yeah chm has a background in computational geometry

tough imp
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Oof

sleek thicket
tough imp
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Sadge

little hemlock
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I started as a physics nub

sleek thicket
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lmao fucking

tough imp
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Not even a lie sadly

sleek thicket
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"honors English"

tough imp
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Hahahaha

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Honors 9th grade English

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Tyvm

obtuse meteor
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me: spent 30 minutes talking on discord instead of grading
the poor 297ers:

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"I am sorry children, for I have failed you"

sleek thicket
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I have a midterm due noon tomorrow

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It's been 3 hours since I said I would start it

cedar pebble
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go to office hours regurgitating nlab shit thinking I was smart and getting the "you stupid shithead please stop" look from professors

sleek thicket
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lmao

obtuse meteor
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ah relatable

tough imp
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Tfw
All of you took evil paths

obtuse meteor
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I had at least gotten past the worst of that phase

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by the time I was in college