point-set-topology

[quaint agate]

I understand, well could I ask a question then?

Just a general question

[sleek thicket]

sure

[quaint agate]

Ok, so would an easy to compute knot invariant be useful?

in any way

[sleek thicket]

I mean, I guess? There are lots of knot invariants and some are easier to compute than others

[quaint agate]

how about a complete knot invariant?

[sleek thicket]

that would be useful

[quaint agate]

Ooh

I have that

I finished it earlier today

[sleek thicket]

neat! I'm not that interested in knot theory though, sorry

[quaint agate]

Oh yeah np

just sharing

[sleek thicket]

Okay so

for some reason we have $\langle [Y, [X, Y]], X\rangle = \langle [X, Y], [X, Y] \rangle$

[gentle osprey]

shamrock:

[sleek thicket]

I should be able to turn this into a purely algebraic problem but I'm not sure how helpful it is

i don't know for sure that this holds for any lie algebra with an inner product (there's this extra condition about the adjoint representation of G being bounded that tells you the metric is bi invariant)

actually this makes me wonder

say you have a lie group map $G \to H$ which is a local diffeo (equivalently, the differential gives an iso between lie algebras$

I'm thinking of the universal cover really

let $V$ be their common lie algebra

[gentle osprey]

shamrock:

[sleek thicket]

identified via the differential of this map

How can we relate the adjoint representations of $G$ and $H$? Can I say something like if $\mathrm{Ad}(H)$ is precompact then $\mathrm{Ad}(G)$ is? Or even better, $\mathrm{Ad}(G) \subseteq \mathrm{Ad}(H)$

[gentle osprey]

shamrock:

[sleek thicket]

oh you can probably maybe say that like, $g$ acts the same way as $\varphi(g)$

Yeah by definition of the identification (I think)

[gentle osprey]

shamrock:

[sleek thicket]

oh yeah uhh this makes sense

this is naturality of the adjoint representation

in the special case where the map is a local diffeo

[fading vale]

hello shamrock

ok since it looks like u figured it out

what exactly are the maps phi and psi here?

I think I get that psi sends a + b -> (a, b)

[sleek thicket]

I did not lol

[fading vale]

oh

[sleek thicket]

I got distracted by another shiny problem

[fading vale]

sorry haha

[sleek thicket]

Np you're good

okay lemma think about it

so like

[fading vale]

but anyway yea im not sure what phi is supposed to be here

[sleek thicket]

Let's look on the level of cochains

Not cohomology

[fading vale]

yeah

[sleek thicket]

I don't understand your definition of psi

I think it's a restriction

[fading vale]

am i misunderstanding something about C^n(A + B)?

"the map psi has coordinates the two restrictions to A and B"

[sleek thicket]

right

So like

first off C^n(A+B;G) is a free module on a certain set of functions

No

Sorry

It's a module of functions from a free module

Is what I wanted to say

[fading vale]

hurb

yea

[sleek thicket]

So the elements aren't like a + b

[fading vale]

ohhh

its

[sleek thicket]

They're functions on things like a + b

But also

[fading vale]

the restriction o the function to chains in A

and to chains in B

[sleek thicket]

a+b is a nonunique representation in C_n(A+B)

Yes exactly

okay so ψ is a restriction

[fading vale]

so what would the difference of those restrictions be then?

[sleek thicket]

Well you just further restrict and subtract

Right?

[fading vale]

ohhhhh

[sleek thicket]

That's gonna be zero iff they come from some overall map

[fading vale]

okay yeah

that makes sense thanks : )

cuz applying phi psi to a map f basically just restricts f to A cap B twice and then takes the difference of those two restrictions which r the same

[sleek thicket]

Right

And if the difference of two restrictions is zero you can glue the original maps

yeah?

If you have cochains α, β then define γ(a+b) = α(a) + β(b), γ is well defined since α, β agree on chains with image contained in A cap B

This isn't quite right but it's close enough

[fading vale]

mhm

[sleek thicket]

okay so back to my thing that I got distracted from

[fading vale]

lol sorry

[sleek thicket]

No!! I got distracted on my own

you're totally fine

[fading vale]

_>

ok go do ur hw!!

[sleek thicket]

So I want to show $\langle [Y, [X, Y]], X\rangle = \langle [X, Y], [X, Y] \rangle$

Chmonkey suggested I embed into a matrix algebra where the lie bracket is the commutator

[gentle osprey]

shamrock:

[fading vale]

and gl : )

[sleek thicket]

right exactly moth

I'm embedding in $\mathfrak{gl}(n)$

:^)

[gentle osprey]

shamrock:

[gritty widget]

secXY?

[sleek thicket]

lol

Yes I am computing secxy

so then $\langle [Y, [X, Y]], X\rangle = \langle Y(XY - YX) - (XY - YX)Y, X \rangle = \langle YXY - Y^2X - XY^2 + YXY, X \rangle = \langle 2YXY - Y^2X - XY^2, X \rangle$ versus $|[X, Y]|^2 = |XY|^2 + |YX|^2 - 2 \langle XY, YX\rangle$

[gritty widget]

how can you compute yourself

[sleek thicket]

hmm I do not see it

[gentle osprey]

shamrock:

[sleek thicket]

I heard that if you compute yourself too much you go blind

Maybe I can rewrite $\langle [X, Y], [X, Y] \rangle = \langle \mathrm{ad}_X(Y), \mathrm{ad}_X(Y) \rangle$ using the fact that $\mathrm{Ad} $ lands in the isometries of $Lie(G)$

[gentle osprey]

shamrock:

[sleek thicket]

AHHH

Secret magic exercise

From forever ago

Exactly the adjoint representation thingy

[fading vale]

shamrock moment

what book is this from?

[sleek thicket]

IRM

[fading vale]

neat

[sleek thicket]

Introduction to riemannian manifolds

It's a good book

[gritty widget]

nice copy

give it to me

[fading vale]

wtf

tterra stealing my rightful inheritance

[gritty widget]

moth moment

[fading vale]

hurb

ok i finished hatcher 3.1 time to go do all my english hw

[sleek thicket]

lol

I'm not giving either of you this book

I need to keep my trilogy together

[exotic root]

Bumpedibump, does anyone see what triviality I am missing?

[sleek thicket]

Ah sorry lejoon I didn't see that

In your question, are A, B, Z topological spaces or schemes?

[nimble flower]

are a path and its inverse not homotopic to one another?

ah no they arent relative to end points

what if it's a loop?

[night parrot]

is the second fundamental form always positive definite? or are there no guarantees about that

[gritty widget]

it doesn't have to be

iirc

1st fundamental form is positive definite yeah but not necessarily the 2nd

[night parrot]

ok yeah

im just trying to understand why they say this in this paper

if this doesnt make sense out of context, no worries, just wanted to quickly share on the off-chance anyone knows what they are talking about

[nimble flower]

ah - ofc it isnt. otherwise pi_1(S^1, 1) would not be isomorphic to Z

[night parrot]

am i like mis-interpretting what they are saying above then? btw, the whole point of that paper that they reference, Rusinkiewicz [2004], is to estimate the second fundamental form across all vertices of a discrete mesh, which ive already implemented

[nimble flower]

oh sorry - not responding to you mike

just to what i said earlier regarding loops

[night parrot]

oh oops, no worries

[gritty widget]

ignore that

i can't take any more differential geometry today LOL

my riemann geo class has me bent over

[night parrot]

this stuff is driving me insane. i am also super new to diff geo like i said the other day, but it is really challenging so far

[gritty widget]

yeah for "second fundamental form" i was thinking of something that takes in two vectors and spits out a normal vector, in which case "positive definite" doesn't make a lot of sense. however it seems some people define it as a legitimate form (taking real values) in which case it makes sense to say whether it's positive definite or not

but in that case it does not have to be positive definite it seems

[night parrot]

yeah thats what i thought

[gritty widget]

differential geometry hurts my soul

[elder yew]

differential geometry gives my life meaning

[sleek thicket]

This is relevant to the next/final problem on my homework!

I'm supposed to show that if the scalar second fundamental form is negative semidefinite and the sectional curvatures of the big manifold are bounded below by c then the sectional curvatures of the submanifold are bounded below by c

[meager python]

Can be seen as top spaces @sleek thicket

[sleek thicket]

Kk I don't want to embarrass myself lol

So let g be the map Z ×_B A -> Z

If I'm reading the orange correctly, it's saying that the closed subsets of Z ×_B A are exactly those of the form g^-1(C) for C closed in C

so it suffices to show g(g^-1(C)) is closed

oh wait g is injective yeah? Because if z = g(z, a) = g(w, a') = w then z = w and i(a) = f(z) = f(w) = i(a'), so a = a'

Then the claim is that g is a topological embedding

and the image of g is exactly f^-1(A)

So g looks like the inclusion f^-1(A) -> Z, and f^-1(A) is closed by the property they mention

@exotic root I didn't understand the orange bit but I think i see why the claim holds

basically like, the orange is saying g is a topological embedding, and im g = f^-1(A) is closed since A is closed in B

[sleek thicket]

I am confused.........

the formulas for the Kulkarni nomizu product on Wikipedia and IRM are flipped by a sign

But

They have the same formula $Rm = \frac{S}{4} g~\wedge! !! ! ! ! ! ! ; \bigcirc ~g$

[gentle osprey]

shamrock:

[sleek thicket]

In dim 2

so one of those two is stated wrong

right!?

literally like, kulkarni and nomizu can go fuck themselves

so basically let $h$ be a negative semi-definite symmetric bilinear form. I want i to be true that $(h~\wedge! !! ! ! ! ! ! ; \bigcirc ~h)(v, w, w, v) \geq 0$

[gentle osprey]

shamrock:

[gritty widget]

sorry shamrock

[elder yew]

we're taking over.

[gritty widget]

i am taking over

we

[elder yew]

3fast6u

[fading vale]

hello moonbears and tterra and shamrock

[elder yew]

alright, so you got this point and wanna show you can't conjugate

[gritty widget]

this was the main one iirc

[elder yew]

this is baby do carmo, right?

[gritty widget]

do carmo's riemann geo

[elder yew]

oh ok

[gritty widget]

baby do carmo basically has all the same stuff as do carmo's RG

[sleek thicket]

lol

[gritty widget]

but in the surface in R^3 case

[elder yew]

lemme find a PDF

which chapter is this?

I think I got a pdf now

[gritty widget]

the one i just posted is in chapter 5

this problem is also given in chapter 7

what i tried: if i take $f(t,s) = \exp_p(\frac{t}{a}v(s))$ as in the proposition, then i get geodesics $t \mapsto f(t, s)$ which start at the vertex $(0,0,0)$, and then a previous homework problem (long ass computation iirc) tells me these are all meridians

[gentle osprey]

TTerra:

[gritty widget]

where a meridian is...

time to flip my book back and forth

[sleek thicket]

lol

[elder yew]

Ok conjugate definition is on page

116

Let's get our definitions in line

[gritty widget]

let me grab some water first

[elder yew]

Can you post the question again?

[gritty widget]

sure

[elder yew]

Sorry flipping through a PDF is horrible

[gritty widget]

the definitions in RG and in curves and surfaces are the exact same

and here are conjugate points

[elder yew]

So what are the geodesics on the paraboloid?

parabolas?

[tepid depot]

at least the ones coming out of p should be

i think

[gritty widget]

if they start at the origin, then they're either constant or parts of parabolas yeah

[tepid depot]

so i believe there shouldn’t be conjugates

[gritty widget]

showing that was on the previous homework

[elder yew]

There might be an easy contradiction

[gritty widget]

[elder yew]

sorry wife is asking questions. Distracting me

[gritty widget]

i don't want to just say the geodesics from the vertex "don't get close again"

what i tried attempted to formalize that

[tepid depot]

i think that’s the idea though

oh ok

[gritty widget]

i can repost it

[elder yew]

Lemme chase down the definition of jacobi field

[tepid depot]

this is potentially a stupid question/line of thinking

my intuition is that you have some family of geodesics

they all start at p

and q is conjugate if they all come back to q

so like, there would need to be at least two geodesics which start at p and eventually end up at q

but there aren’t?

[elder yew]

Is it just one that satisfies that differential equation

[gritty widget]

re moonbears: yeah lol

[elder yew]

Sorry this is how I do math, just write everything relevant down on one piece of paper

and stare at it

[gritty widget]

re doubledual: i'm not exactly sure they must meet up again but that sounds very plausible

[tepid depot]

lol i do that plenty too

[elder yew]

Jacobi field just satisfies the jacobi differential equation

Is that a defining differential equation?

[tepid depot]

ok looking at pg. 369 of baby do carmo, my intuition is not quite right

[elder yew]

Sorry this book is fucking confusing

[gritty widget]

here's what i attempted earlier; one problem with this is that r(t) may actually depend on s, i think (gimme a moment to post the relevant prop 2.4)

[tepid depot]

but the “geodesics getting infinitely close” thing should be good enough here

[elder yew]

Ok I gots it right

[tepid depot]

i just have to make sense of the intuition he’s describing...

[gritty widget]

the curves and surfaces book basically defines jacobi fields as the things in proposition 2.4 (variational fields of variations all of whose curves are geodesics)

ree too many definitions

[elder yew]

Wife is really taking up some time here

Ok

Have you computed the jacobi field for the paraboloid explicitly?

[gritty widget]

there's an attempt in the picture

i could write down the jacobi equation

[elder yew]

What do jacobi fields tell us?

[tepid depot]

oh wait im a moron my intuition is totally right

[gritty widget]

re moonbears: they tell us how geodesics spread

[tepid depot]

im using baby do carmo to do this, but here’s my thinking

start with a variation of the geodesic you’re interested in

take the derivative wrt that variation

that derivative J is the jacobi field

the J(0) term corresponds to everything starting at p

and at the end, J(s) = 0 means that the curves are ending at the same point q

or something that is almost like that really since you’ve only got the derivative at 1 point

[elder yew]

Can you use rolle's theorem?

something something both = 0

[tepid depot]

that’s what this “infinitely close curves” idea is about

[elder yew]

you get a max-min type thing

But if you compute out the jacobi field for this thing

that can't happen?

Like you can immediately throw out the constant case

For your geodesic, right?

[gritty widget]

i like the rolle's theorem idea, i'm playing around with it

yes

[elder yew]

So you have the form of a parabola that lies on the surface

What's the general form of that curve?

That's just my first idea after looking at the definitions

dunno if it will work

if you switch to polar coordinates, it's easier to write down a parabola, no?

Isn't a curve on the surface like

[gritty widget]

something of the form $$ \gamma(t) = (t \cos u_0, t \sin u_0, t^2) $$ up to parametrization

[gentle osprey]

TTerra:

[elder yew]

$$ x^2 + y_0 ^2 $$

[gentle osprey]

MoonBears-C-:

[elder yew]

For some fixed y_0

since the graph is just

$$ z = x^2 + y^2 $$

[gentle osprey]

MoonBears-C-:

[gritty widget]

re doubledual: that's kind of what i was trying to do in the solution i posted (started with a variation, use it to compute J, show J won't vanish at more than one point and be nontrivial)

one maybe useful fact is that if a jacobi field along a geodesic vanishes at two points then it's orthogonal to the geodesic's tangent vector along the curve

[elder yew]

What is the jacobi field in this case?

[tepid depot]

ok here’s a thought

arclength parameterized geodesics coming out of p are 1-1 with unit vectors in the tangent space at p (i would think intuitively)

[gritty widget]

sounds right

since they parallel transport their tangent vectors

[tepid depot]

like i have an intuitive picture here where it’s like

the curves in the variation at the end converge to q

but also they’re slowing down to speed 0 if it’s a conjugate point

but to slow down as you’re hitting q, you’re just slowing to the speed you shift your angle

[gritty widget]

right, that's the more difficult part i feel

i see what you mean

[tepid depot]

oh i think i can make this argument more formal in a good way

all the curves in the variation are arclength parameterized

they travel the same amount of time

so they’re landing in a circle

[gritty widget]

can you always make sure the curves in the variation are arc length parametrized? i guess with some magic reparametrization

[tepid depot]

i think so?

oh not by definition 😦

sad

[gritty widget]

😔

let me compute something and see

[tepid depot]

ah but!

jacobi fields are always orthogonal to the tangent vector of the geodesic you’re considering

for the parabola, they’re pointing up always

[gritty widget]

right

[tepid depot]

so the variation is approximated by a variation where you just change the angle

but also at the end of the curves angle isn’t changing

[gritty widget]

the variation is approximated by a variation where you just change the angle
if i can show this it's basically done, yeah

well it'd be nice if i could start off with one like that

but ill let you continue

[tepid depot]

so you know near you’re geodesic gamma the derivative is horizontal

everywhere

and it’s also zero at the end

[elder yew]

therefore constant?

Is that what you're getting at?

[tepid depot]

well more like impossible

[elder yew]

Oh it won't be on the surface if the derivative is constant

cuz then it's linear

??

[gritty widget]

[elder yew]

My brain hurt

[gritty widget]

same

i would like to avoid extreme computation

[tepid depot]

oh wait i think i got a really easy solution

for real

[elder yew]

(That's the only idea I had with rolle's lol, that would suck)

[gritty widget]

i very much like what doubledual is putting forward

[tepid depot]

take your variation h(s,t)

[gritty widget]

truly the adv helper role

[tepid depot]

s is moving along the curve, t is changing curves

and s_0 is the end of the curve

h(s_0,t) traces out some points in the surface

now there’s one geodesic from p that hits each h(s_0,t)

[gritty widget]

so by uniqueness

[tepid depot]

you use this to define a function from t to the angle of that geodesic!!!

[gritty widget]

👁️

[tepid depot]

now if there’s a conjugate point

the derivative at the variation is 0 at the end

which should implies the derivative of this angle function is 0

but if the angle function stops, the variation stops everywhere

so it’s not a conjugate point

[elder yew]

hrmm

[tepid depot]

so to recap

when you vary the curves in the variable t, consider the function that gives you the angle you go out from at t

if you have a conjugate point, the derivative of the jacobi field is 0 at the end of the curve

at t = 0

but that should mean the angle function has vanishing derivative at t=0

but if that’s so, then the jacobi field is trivial

[gritty widget]

give me a moment to take this in

[tepid depot]

the existence of this angle function is predicated on the fact that’s there’s one geodesic between p and some other q

[elder yew]

How're you defining this angle function?

[tepid depot]

that’s why this argument fails on the circle e.g.

h(s,t) is the variation

look at the geodesic s |-> h(s,t)

that geodesic corresponds to a unique angle

map t to that angle

[elder yew]

OH

I SEE

I SEE

AHH

You can sweep along the geodesic to the point

Huh that's interesting

[gritty widget]

sorry i got up to get a snack

let me read over what happened

[tepid depot]

yeah that’s basically what conjugate points are always about as far as i can tell

[elder yew]

Would never have thought of that

[tepid depot]

you need to sweep over a family of geodesics towards the same point, in a non-trivial way

[elder yew]

My brain just bash computation to get contradiction

[tepid depot]

i don’t think i understood this the first time i read the book lol

[gritty widget]

this stuff is kinda hard

ngl

[tepid depot]

look at this tonight i’m thinking to myself “wow i really just did all the computations and proofs without trying to figure out what was going on geometrically”

this shit is hard because the computations are hard and understanding the geometric meaning is hard

[gritty widget]

ah i see it

well

i think i see it

it's easy to doubt myself at 4 am

[tepid depot]

there’s details in here that gotta be worked out

and i’m tired and wouldn’t immediately be able to tell you how to do it

but i think this idea is solid

[gritty widget]

i think this is a great idea

nothing i'd come up with lmao

[tepid depot]

nah don’t say that

idk how i came up with this

i mean i just banged my head against the geometric intuition i had until i could make it formal enough to actually make sense

[gritty widget]

i took moonbears' earlier advice and looked at what other people in the class were saying about the problem and it seemed like they were just as confused as i was

[tepid depot]

it’s a hard problem imo

[gritty widget]

probably the most difficult one we've had thus far

[tepid depot]

also it’s a hard topic, don’t feel bad about struggling with it at all

[gritty widget]

all the other ones on this pset are kinda easy

[tepid depot]

understandable

ok i definitely gotta sleep now, so goodnight y’all!

[gritty widget]

yeah goodnight

thank you and @elder yew so much for helping

tomorrow i'll try to write this out in some more detail

[elder yew]

I feel I didn't do anything

but glad I was here for the ride

lo

lol

[gritty widget]

this course got me feeling like a first year seeing epsilon-delta proofs again

it seems there's a solution that uses some facts about killing fields

i would like to avoid this

[elder yew]

My RG class was much more abstract, just kinda generalized calculus on manifolds AT

[sleek thicket]

god how I wish that was me

[gritty widget]

ah that reminds me moonbears

i should see if petersen's book has some good stuff on this

[fading vale]

does shamrock hate RG again

[sleek thicket]

yes

[gritty widget]

i have like a million rg textbooks downloaded and it never occured to me to check all of them

[fading vale]

my predictive powers are unmatched

[sleek thicket]

Inequality is going the wrong way >:(

[elder yew]

@sleek thicket everyone in my class for RG got an A for doing nothing cuz he didn't want to grade

[sleek thicket]

lol

[elder yew]

I missed 2 weeks of class to deal with bureaucracy

[sleek thicket]

Cutoff for a 4.0 for my class is 80%

There are 5 homework sets and no tests

[elder yew]

That's not too bad

[sleek thicket]

the homework is actually meaty but like, graded generously

[gritty widget]

what the fuck thats basically my rg class

[sleek thicket]

lol

[gritty widget]

4.0 cutoff is 85%, there are 4 homework sets, no tests

[elder yew]

I thought the UW grading system would be much more rigid based on what I saw online

[sleek thicket]

i mean like

this is post-quals

quals classes can still be kind of ass

[fading vale]

such is the way of sham

[elder yew]

(I'm skipping pre-qual classes)

[sleek thicket]

(aahh gotcha)

[gritty widget]

quals

[elder yew]

First year exams to prove you're not an idiot

[gritty widget]

lol

[elder yew]

And you actually paid attention

[gritty widget]

feels good to be an undergrad

not

[sleek thicket]

feels good to be an undergrad

yes

[elder yew]

I had two Masters quals. My topology prof didn't write my topology qual, got a B

[sleek thicket]

🐱

[fading vale]

feels good to be in

[elder yew]

Real qual got an A

so hell ya

[fading vale]

high school

[sleek thicket]

srsly tho

[gritty widget]

yeah but then i get slow ass algebra classes that take 3 months to cover 4 chapters of d&f

[sleek thicket]

not having quals

oh yeah lol

undergrad taking undergrad classes is painful

[elder yew]

I actually prefer quals over class exams

[sleek thicket]

undergrad taking grad classes but with no quals is nice

i like my week long take home tests

thats my comfort level

[fading vale]

shamrock did i tell you about the pace of my AT class

[sleek thicket]

no

[elder yew]

The quals are great cuz you can just study on your own over the summer and nail down every problem

[fading vale]

last week we spent the entire two hour lecture discussing the basics of group presentations

[elder yew]

Look things up

[fading vale]

theres a month left in the semester and we havent even finished proving van kampen nor have we touched covering theory

[sleek thicket]

huh

[fading vale]

second sem doesnt even try to touch pi_n generally

[sleek thicket]

neat

[gritty widget]

prove that if the sectional curvature is identically zero then the manifold is locally isometric to R^n

[sleek thicket]

that is uh

[fading vale]

we spent more time on point set than we did AT

[sleek thicket]

less than we did in my manifolds course

[elder yew]

Proof: By picture

[fading vale]

[sleek thicket]

first quarter

lol

like

that sounds like a normal topology class

with no focus on AT

[fading vale]

this is the GRAD CLASS!!!!

[elder yew]

Different profs do different paces

[sleek thicket]

yup, undergrad course doesn't cover pi_1

[elder yew]

For knots we crash coursed AT in a week

[fading vale]

it would be fine if he didnt spend the first like 5 weeks on point set

[sleek thicket]

undergrad top at uw does pi_1 as an extra thing at the end, not covered in lecture

[fading vale]

i wish he had just told ppl to review at home

we spent so much time on it : (

[sleek thicket]

there is 1 optional homework on it

[fading vale]

wait what

[sleek thicket]

yes

[fading vale]

how do u do AT without

oh ok i misread

[sleek thicket]

then yo uget 2 quarter of diff geo

yeah no at

undergrad intro top

[fading vale]

i was rly confused

lol

[sleek thicket]

anyways, someone fix this inequality pls 😦

make it work

why is it

= sec(v, w)

and not

<= sec(v,w)

[fading vale]

cuz ur dumb

obviously

[sleek thicket]

convex means the scalar second fundamental form $h$ is negative semi-definite

[gentle osprey]

shamrock:

[sleek thicket]

$h$ is defined by $II(X, Y) = h(X,Y) N$

[gentle osprey]

shamrock:

[gritty widget]

[sleek thicket]

pls help

inequality

should go

[gritty widget]

kulkarni nomizu

[sleek thicket]

other way

bro i hate it

so much

it is

the worst

fucking

thing

https://en.wikipedia.org/wiki/Kulkarni–Nomizu_product

oh except

[gritty widget]

gross

[sleek thicket]

this is the negative of the one in IRM

[gritty widget]

why should i care about the kn product

[sleek thicket]

and they have the same formula in one plsae

so

wtf

they cannot both be right

[fading vale]

sadrock

[alpine rain]

Sorry to peek in here but I wonder, why is so advanced field like geometry coupled with topology which is introduced in undergrad

[sleek thicket]

can you stop trolling pls

[fading vale]

hurb

[alpine rain]

Eh this is not trolling

[sleek thicket]

oh TTerra we have $Rm = \frac{S}{2} g \text{ kn-prod } g$

for surfaces

this is one reason to care

[alpine rain]

I was genuinly wondering, but sorry for interrupting when you were solving problems

[sleek thicket]

lol

[gritty widget]

Rm is the <R(x,y)z,w> right

[elder yew]

how is

[gentle osprey]

shamrock:

[sleek thicket]

eah

*yeah

[elder yew]

sec(v,w) - something bigger than sec(v,w)

[sleek thicket]

it is

and also

[gritty widget]

y'all just shut that guy down like that

[sleek thicket]

there's a formula for hypersurfaces in a riemannian manifold

@gritty widget they were trolling last night about algebra in the same way

posting galois theory in #prealg-and-algebra

[gritty widget]

[fading vale]

annoying

[gritty widget]

polynomial 2.0

[fading vale]

and unfunny

[alpine rain]

That was genuine misunderstanding tbh

[sleek thicket]

lol

@elder yew so wait uh

i may be confused

please tell me you spotted a mistake

[elder yew]

you have sec(v,w) - something \geq sec(v,w)

[gritty widget]

[alpine rain]

I'm too bored that I'm making problems, sorry

[sleek thicket]

yes

[elder yew]

Is that thing negative?

[sleek thicket]

because that thing is proven to be nonpositive in the previous line

[elder yew]

shucky darns

[sleek thicket]

lol

[gritty widget]

why is geometry like this

[sleek thicket]

anyways i really want that inequality to go the other way lol

and am confused

mood

oh yeah ttera just do AG

then it's mindfuck comm alg

and awful category theory

[gritty widget]

like holy fuck geometry hurts me sometimes but i love it at the same time

[sleek thicket]

instead of this bullshit

different flavor of awful

[elder yew]

where did you use the bound?

[sleek thicket]

[gritty widget]

bro i dont even know group theory don't ask me to do comm alg

[elder yew]

on sectional curvature?

[sleek thicket]

@elder yew this solution isn't complete

if the last inequality is flipped

[elder yew]

I see

[sleek thicket]

then it gives the result

becaues you just c <= ...

[elder yew]

Hrmm

Hrmmm

[sleek thicket]

but like

if M-tilde has a minimal section curvature

this just feels wrong lol

i guess the kn product could vanish

but it seems like something is backwards??

[gritty widget]

[sleek thicket]

Oh I think I figured it out

here's what I was missing

oh wait wrong image

[gritty widget]

shitposting in #point-set-topology

[elder yew]

I got nothin'

[sleek thicket]

i ma just

[elder yew]

On this problem

[sleek thicket]

so frustrated with this class

like

[alpine rain]

I started it sorry

[sleek thicket]

blaaagh

[elder yew]

Have you looked in Peterson?

[sleek thicket]

@alpine rain no i shitpost in these channels constantly

[elder yew]

That thing is a mammoth

[gritty widget]

dont worry shamrock i am also frustrated with my class

[sleek thicket]

@elder yew nope

[alpine rain]

O

[sleek thicket]

the issue is

like

IRM has a different convention for that product

[alpine rain]

High IQ guy

[gritty widget]

@sleek thicket read #❓how-to-get-help

[sleek thicket]

lmaooooooo

[alpine rain]

Shitposting in geometry

[gritty widget]

geometry is high taste shitposting

[sleek thicket]

well geometry is an advanced graduate topic

only taught to 10th graders and up

jk I took it in 9th grade

[alpine rain]

Tru

[gritty widget]

"yeah i do geometry"
"ok, draw two columns..."

[sleek thicket]

lol

[elder yew]

what subject is this under in jack's book?

[alpine rain]

Euclidean geometry is a hoax

It does not exist

[gritty widget]

jack

[elder yew]

like topic, is it in sectional curvature?

[gritty widget]

moon on a first name basis with his profs

[elder yew]

(I was told by a prof to use his first name at a conference and I couldn't do it)

[sleek thicket]

this is problem 8.18, in the Riemannian Submanifolds chapter

I use jack when talking with others

but "Professor Lee" when talking to him

because I am a coward

he's old!!

[elder yew]

My mentor has told me multiple times to stop calling him professor

[sleek thicket]

lollllll

[elder yew]

so I call him by his first name now

That took months

[sleek thicket]

I called my algebra prof Sándor Kovács Sándor

and my AG prof Max Lieblich Max

[gritty widget]

imagine talking to your profs ||||

[elder yew]

I've called Tao terry

[sleek thicket]

like every prof other than jack lee

i call by first name

but

he is OLD

@gritty widget simple remove your shame module

[elder yew]

Riemannian Submanifolds eh

[gritty widget]

its harder with online

[elder yew]

let's see what peter peter has

[alpine rain]

Module? Is this CS

[sleek thicket]

definitely ttera

@alpine rain it's algebra

sorry, you've probably forgotten by now

been so long

[alpine rain]

But how do you remove module in algebra

[sleek thicket]

modules are gay vector spaces

[gritty widget]

i had a 2-1 chat with my riemannian geometry prof and another student, that's basically the most i've gotten with online

[sleek thicket]

you quotient? duhhh

oof

[gritty widget]

gay vector spaces

[sleek thicket]

that is a bummer

I ask questions in class a lot

[gritty widget]

i dont attend class

recordings 😌

[sleek thicket]

and like asked him for a lie theory recc

[gritty widget]

"my book"

[sleek thicket]

also I was in class with him irl for 6 years lol

[gritty widget]

6

[sleek thicket]

6 months**

lol

[elder yew]

I'm the same way shammy, I'm super obnoxious

[sleek thicket]

whoops

[elder yew]

I once asked my Random Matrix Theory prof. if his winter break went well

[sleek thicket]

@gritty widget he doesn't have a lie theory book

aww

that's nie

[elder yew]

He was known for being cold

[sleek thicket]

*nice

[elder yew]

He stared at me said "No" giggled to himself

and then started lecture

[sleek thicket]

omg

adorable

[gritty widget]

[elder yew]

the giggle was probably a scoff

[sleek thicket]

i prefer to imagine

giigle

[elder yew]

but I have a way of warming up professors

even the cold ones

[gritty widget]

snuggle up to em

literal warming

[sleek thicket]

yess

glomp them

[elder yew]

@gritty widget manifolds without conjugate points

[sleek thicket]

uwu

[elder yew]

is in Peterson

[gritty widget]

i see

i know one fact about those

manifolds with nonpositive sectional curvature do not have conjugate points

[elder yew]

Page 162

[gritty widget]

thank you

i will look now

[sleek thicket]

YO

it has the berger sphere shit

i was doing

last night

ugh

[gritty widget]

burger sphere

[sleek thicket]

marcel 🍔

[gritty widget]

is such a good emote

[sleek thicket]

it is

oh woah

that's pog

[gritty widget]

wait i didnt see this generalization

what the fuck

that's cool

[sleek thicket]

that is

very cool, ttera

[gritty widget]

in do carmo it's stated for simply connected manifolds

[sleek thicket]

this book has a whle chapter on metrics on lie groups

[gritty widget]

so like this is obv stronger

[sleek thicket]

pog

wait um hm

say you have a universal cover M -> N

and N is complete

[gritty widget]

yes M is complete

with pull back metric

[sleek thicket]

nice

cool

good

[gritty widget]

you just project a geodesic to N and then lift it

and it exists for all time

so ur good

[sleek thicket]

right, that was what I was thinking

neat

[gritty widget]

this is actually on my RG homework lol

[sleek thicket]

oh lmao

[gritty widget]

already did it though

[sleek thicket]

i was wondering if yo ucan reduce to the simply connected case

like uhh

[gritty widget]

in fact M is complete iff N is

[elder yew]

I'm sure this is somewhere in peter peter shammy

It might take some digging

[gritty widget]

this needs it to be a covering map btw

[elder yew]

Does this result have a name?

[sleek thicket]

if you have a universal cover M -> N with the pullback metric

[elder yew]

Your exercise sham

[sleek thicket]

oh uhh

not that I know of

i stopped looking lol

[elder yew]

minimal sectional curvature lower bound remains a lower bound for other sectional curvatures

(By definition of minimum QED)

[sleek thicket]

lol

[elder yew]

God I wish it were that easy

[gritty widget]

i skimmed a little bit after the page you said moonbears

didnt find much

😔

gonna look a little more though

[sleek thicket]

peterson doesn't have the term convex in the appendix

tfw

[gritty widget]

honestly

one absolutely terrible thing i could do for my problem

is compute the jacobi equation for the paraboloid

it would suck so much ass

[elder yew]

And use rolle's theorem

how bad would that be?

[gritty widget]

yes but do i really want to compute curvature

[elder yew]

not gonna get into McGill with that attitude

[sleek thicket]

wait uh

of the paraboloid?

i did that recently

[gritty widget]

i have the christoffel symbols written somewhere on my previoushomework

[elder yew]

LOL

[sleek thicket]

i can just post lol

[gritty widget]

please do

[elder yew]

doit

you won't

[sleek thicket]

this is z = x^2 + y^2 yeah?

[gritty widget]

yes

[elder yew]

yA

[gritty widget]

i will put on my homework "credit to discord user @sleek thicket for these"

[sleek thicket]

yesss

heya

[elder yew]

lol

[sleek thicket]

basically like

its a surface of revolution

away from the origin

finding a unit speed param is ass

but

[gritty widget]

our prof unironically said "you will probably have to look things up for the problem sets" and said "as long as the final thing's in your own words i don't care"

👀

[sleek thicket]

you don't need to know it explicitly

you can just differentiate

oh wait

when you said curvature

what kind of curvature did you mean lol

oh it doesnt mmatter

gaussian determines the rest

for surfaces

[elder yew]

He's looking for the jacobi field

[gritty widget]

$$ R(X,Y)Z = \nabla_Y\nabla_XZ-\nabla_x\nabla_YZ+\nabla_{[X,Y]}Z $$

[sleek thicket]

so like

[gentle osprey]

TTerra:

[elder yew]

lol

[sleek thicket]

this is where the KN product comes in

[gritty widget]

[sleek thicket]

from wikipedia

and irm

Scalar curvature is twice gaussian

So this lets you compute Rm in terms of K

[gritty widget]

and once i know Rm

[sleek thicket]

And R is just Rm but you raise/lower/idk an index

[gritty widget]

ya

whats it called

musical iso

[sleek thicket]

Yeah

You can google how to get the curvature endomorphism of a surface in terms of gaussian curvature

That's standard

Like not paraboloid dependent

[gritty widget]

so like Rm(X,Y,Z,W) = <R(X,Y)Z, W> and that's.... dropping the index?

[sleek thicket]

bro I have no idea

[gritty widget]

[sleek thicket]

It's one of the two

I have algebra brain worms

You just say

The isomorphism induced by

And that's enough

[elder yew]

Only one of them should work

[gritty widget]

god i remember when i was in highschool and everything was easy

now this is not the case

bro let me go back to factoring polynomials

[sleek thicket]

Just do AG

Anyways ttera

My pdf computes gaussian curvature

By looking at it as a surface of revolution

Idk if you know the gaussian curvature of a surface of revolution but it ends up not being so bad

[elder yew]

I found a MSE post where someone tried to find a counter example to jack lee on conjugate points

Jack lee responded

[gritty widget]

post

[sleek thicket]

ope

[elder yew]

https://math.stackexchange.com/questions/1966424/counter-example-of-conjugate-points-of-geodesics

[gritty widget]

nice

[sleek thicket]

If (a(t), b(t)) is a unit speed parameterization of the curve then the curvature at (a(t) cos(θ), a(t) sin(θ), b(t)) is -a''(t)/a(t)

[gritty widget]

fun thing, all of the related posts i've clicked on trying to figure out things for my previous/current problem

all clicked on

[sleek thicket]

lmao "first of all..."

That's an mse mood

[gritty widget]

what if i just posted my q on mse

ive only ever posted one question

[sleek thicket]

[sweet wing]

if it isnt algebra you wont get a answer unless you're lucky

[gritty widget]

https://math.stackexchange.com/questions/3889622/existence-of-fredholm-operators-of-a-given-index-in-a-non-separable-hilbert-spac

instant answer

[sleek thicket]

I wonder if Jack would care

Like

He did last year

Feels like he'd get pissy about it

man I want to learn functional analysis

It seems good

[elder yew]

Nobody answers my posts on MSE

I have to answer my own

[sleek thicket]

I say this like I'm not taking a class on it next quarter lol

[gritty widget]

i feel like im learning absolutely nothing in my functional analysis course

[sleek thicket]

Awww

That's a bummer

[gritty widget]

this is because of the prof's... unusual style

[sleek thicket]

oh?

[gritty widget]

let me get some water

this might take a while

[elder yew]

story time at 5 am

with tterra

[gritty widget]

i dont mind

im not very tired

[sleek thicket]

oof

[gritty widget]

doubledual's huge brain idea woke me up

[sleek thicket]

wait

For some reason I thought you were in pacific time lol

[elder yew]

No I am

[sleek thicket]

Your tiredness makes more sense now

No I know you are

[elder yew]

No

I am

[sleek thicket]

Wait

[gritty widget]

,ti

[gentle osprey]

The current time for TTerra is 04:41 AM (EST) on Sun, 22/11/2020.

[sleek thicket]

Brain exploded

anyways

Point is

Shamrock knows nothing

[gritty widget]

ok im gonna post something and i want you to tell me if this is a red flag

or not

[sleek thicket]

uhoh

I'm getting worried

[elder yew]

Red flags just look like regular flags with rose tinted glasses

[gritty widget]

ill upload a copy of the book in a moment

"prereq: calculus, linear algebra, mathematical maturity"

instant shitshow guaranteed

[sleek thicket]

Yeah 100%

[sweet wing]

ikr

[elder yew]

I dunno. this seems solid

[sleek thicket]

Did u see the course description for the Kahler manifolds course at uw in spring

[sleek thicket]

@elder yew no knowledge of measure theory? Or like, any prior exposure to analysis?

[gritty widget]

tell me if this takes just

calc and la

[elder yew]

I'm not one to ask about skipping pre-reqs

since I did it every term in college

[sleek thicket]

imo skipping prereqs is good on the students part

But listing incorrect prereqs is bad as a prof

like

If a student knows what they're getting into and believes they're prepared

great!

[elder yew]

where did you get to Tterra?

[sleek thicket]

If a prof lies about the necessary background and ends up with students who are unprepared

Not great

[elder yew]

I think calc + linear would be fine

[gritty widget]

re moonbears: i skipped chapter 1, currently finished 4.1

[sweet wing]

should one list like more prereq than prob required or less 🤔

[gritty widget]

but the thing is

i lack the requisite integration theory

so a lot of stuff just goes over my head and is there as "there are examples of this abstract thing you just read about"

[elder yew]

yA you pick them up

[gritty widget]

so the prof doesn't have a lecture plan at all, there are no assignments except for vague "essays." the work you submit is entirely up to you; you just email the prof some problem solutions at the end of every week

lectures are just

[sleek thicket]

oh yeah anyways

[gritty widget]

streams of conciousness

[sleek thicket]

Related

Prereq list is sus

[gritty widget]

hmm

[elder yew]

are you taking it sham?

[sleek thicket]

No, because of that and because I heard bad things about the prof

[elder yew]

ah

[gritty widget]

mat247 is the linear algebra course (in terms of coverage basically all of axler) and mat257 is spivak's calculus on manifolds

[elder yew]

If I wasn't poor I'd take geometric measure theory next term

[sleek thicket]

Not like bad person bad, bad lecturer bad

GMT is in spring! They changed it

[elder yew]

Oh is it?

[sleek thicket]

I'm not sure whether I'm taking it tho

[elder yew]

What're they offering in the winter?

[sleek thicket]

https://math.washington.edu/annual-course-overview

[gritty widget]

doxxed

[sleek thicket]

@gritty widget oh calculus on manifolds feels like a more reasonable prereq

[elder yew]

hyper bolic geometry

[sleek thicket]

Then calc 3

Still a little sus imo

[elder yew]

that looks interesting. I just saw it

[gritty widget]

these courses look so nice

wtf

[sleek thicket]

Lol

[elder yew]

is that the same one

[sleek thicket]

The fancy fancy ones?

[gritty widget]

why cant uoft offer stuff like this

[elder yew]

you're talking about

[sleek thicket]

Like special topics?

Because I'm not taking them

@elder yew which one?

[elder yew]

hyperbolic geometry and teichmuller

Rhode/Artheya

[sleek thicket]

I'm not taking that if that's what you're asking

Oh you meant the syllabus I posted?

[elder yew]

Is that the one

You're talking

yeah

[sleek thicket]

That's for 549

Spring

Geometric structures

It's dumb that they don't have course descriptions here

https://sites.math.washington.edu/~novik/proposed-grad-courses-2020-21/

Here is where I know to find course descriptions

And ofc irl