#serious-discussion | Mathematics | Page 426

neat lintel Jul 2, 2021, 6:40 AM

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it's not necessary for spivak but it's something you'll deal with if you take an analysis course

cold needle Jul 2, 2021, 6:41 AM

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okey

neat lintel Jul 2, 2021, 6:41 AM

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what other exercises were there catThink

cold needle Jul 2, 2021, 6:41 AM

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idk i just did the first three

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i guess ill work on the next starred one for tonight

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well ok this one isn't starred but it caught my attention

neat lintel Jul 2, 2021, 6:42 AM

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stare

cold needle Jul 2, 2021, 6:42 AM

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its about norm preserving transformation

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and inner product preserving

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transformation

neat lintel Jul 2, 2021, 6:42 AM

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ah yes

cold needle Jul 2, 2021, 6:42 AM

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and i have to show that norm preserving is equivalent

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to inner product preserving

neat lintel Jul 2, 2021, 6:42 AM

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interesting

cold needle Jul 2, 2021, 6:42 AM

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i will do that now

neat lintel Jul 2, 2021, 6:42 AM

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hmmCat

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the proof for this is important

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well it's an important trick/technique

cold needle Jul 2, 2021, 6:44 AM

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hmm

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okey

neat lintel Jul 2, 2021, 6:46 AM

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stare

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oh btw

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problem 1-8 might have a typo

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i remember thisbeing one of those ones

cold needle Jul 2, 2021, 6:48 AM

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What

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oh

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what is the typo

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should i look up an errata pdf

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or sth

neat lintel Jul 2, 2021, 6:48 AM

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let me think about it

cold needle Jul 2, 2021, 6:48 AM

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okay

neat lintel Jul 2, 2021, 6:51 AM

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https://math.stackexchange.com/questions/361740/correcting-a-mistake-in-spivak

Mathematics Stack Exchange

correcting a mistake in Spivak

Spivak's Calculus on Manifolds asks the reader to prove this (problem 1-8, pp.4-5):
If there is a basis $x_1, x_2, ..., x_n$ of $\mathbb{R}^n$ and numbers $\lambda_1, \lambda_2, ..., \lambda_n$ ...

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i chose not to think

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and to simply google

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i am a Genius

cold needle Jul 2, 2021, 6:52 AM

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see one way is really easy

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going from inner product preserving

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just do y=x

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or

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okay i guess i should look at this

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is this the typo

neat lintel Jul 2, 2021, 6:53 AM

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are you thinking of problem 1-7 or 1-8

cold needle Jul 2, 2021, 6:53 AM

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oh this is 8

neat lintel Jul 2, 2021, 6:53 AM

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the typo is 1-8, the one after the inner product stuff

cold needle Jul 2, 2021, 6:53 AM

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i am working on 7

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okay

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good to know tho

neat lintel Jul 2, 2021, 6:53 AM

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yeah inner product preserving -> norm preserving is basically trivial

cold needle Jul 2, 2021, 6:53 AM

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i need to do some kind of trick

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with the other way around

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catThink

neat lintel Jul 2, 2021, 6:54 AM

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check out the results in the book catThink

cold needle Jul 2, 2021, 6:54 AM

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🤔 wdym results in book

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like the chapter theorms?

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okay

neat lintel Jul 2, 2021, 6:54 AM

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ya

cold needle Jul 2, 2021, 6:57 AM

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T is linear catThink

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smugsmug

neat lintel Jul 2, 2021, 6:57 AM

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hmmCat

cold needle Jul 2, 2021, 6:58 AM

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compute <T(x), T(x+y)>

neat lintel Jul 2, 2021, 6:58 AM

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catThink

cold needle Jul 2, 2021, 6:58 AM

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the result should follow if i canceled things out right

neat lintel Jul 2, 2021, 6:58 AM

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elaborate

cold needle Jul 2, 2021, 6:59 AM

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we are working in R^n so inner product is bilinear

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T is linear

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so then T(x+y) = T(x)+T(y)

neat lintel Jul 2, 2021, 6:59 AM

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you get <x, x> + <Tx, Ty> but what happens to this second guy?

cold needle Jul 2, 2021, 6:59 AM

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well originally you can write the equality as <T(x), T(x+y)> = <x,x+y>

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apply bilinearity

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cancel <x,x>

neat lintel Jul 2, 2021, 6:59 AM

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wait which direction are u doing

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norm -> inner right

cold needle Jul 2, 2021, 7:00 AM

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norm preserving -> inner

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yes

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wait

neat lintel Jul 2, 2021, 7:00 AM

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cold needle well originally you can write the equality as <T(x), T(x+y)> = <x,x+y>

this isn't a norm

cold needle Jul 2, 2021, 7:00 AM

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what am i doing

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hold on

neat lintel Jul 2, 2021, 7:00 AM

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stare

cold needle Jul 2, 2021, 7:00 AM

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okay this is fine

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i can get to this position easily

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square both sides of the norm

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er

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pain

neat lintel Jul 2, 2021, 7:00 AM

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maybe you wanna look at <T(x+y), T(x+y)>?

cold needle Jul 2, 2021, 7:00 AM

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no this is gonna suck

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yeah

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ok let me write this up from beginning

neat lintel Jul 2, 2021, 7:01 AM

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that's a norm so you can use your hypotheses

cold needle Jul 2, 2021, 7:01 AM

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we will use symmetry of inner product later

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same stuff will cancel when i get ther

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but

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yes

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so then |T(x+y)| = sqrt(<T(x+y),T(x+y)>)

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and so Rhs also equal sqrt(<x+y,x+y>)

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square both sides

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then apply bilinearity and symmetry

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cancel terms

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pane

neat lintel Jul 2, 2021, 7:03 AM

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pane

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i see

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cool

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i was thinking of using the polarization identity

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since it expresses <v, w> in terms of norms

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here's a similar/harder exercise

cold needle Jul 2, 2021, 7:04 AM

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wait

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no

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mine is wrong

neat lintel Jul 2, 2021, 7:05 AM

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stare

cold needle Jul 2, 2021, 7:05 AM

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pain

neat lintel Jul 2, 2021, 7:05 AM

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is it?

cold needle Jul 2, 2021, 7:05 AM

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er

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no it isn't

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i combine my naive attempt with this improved one

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yes

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its correct

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cus the naive attempt is part of how i can cancel some terms out after applying bilinearity

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so i guess not so naive

neat lintel Jul 2, 2021, 7:06 AM

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you have to be a little careful on why you can cancel those

cold needle Jul 2, 2021, 7:06 AM

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yeah

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see you can start with |T(x)| = |x|

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and then do the same squaring game

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and thats why we can cancel

neat lintel Jul 2, 2021, 7:06 AM

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yes

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good

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ok here's something similar

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let V be a real inner product space, and let f: V -> V be a function with f(0) = 0 and |f(x) - f(y)| = |x - y| for all x, y in V. show that f is a linear map such that <f(x), f(y)> = <x, y> for all x, y in V

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note that f is not assumed linear

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!

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as a hardmode, is the same true of maps between complex inner product spaces? (i don't actually know off the top of my head)

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stare

cold needle Jul 2, 2021, 7:09 AM

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oof

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in complex we have to conjugate a lot of things

neat lintel Jul 2, 2021, 7:09 AM

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right im thinking that might mess something up

cold needle Jul 2, 2021, 7:09 AM

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yeah

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Perhops...

neat lintel Jul 2, 2021, 7:09 AM

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anyways the thing i have in mind uses ||polarization||

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note that real inner product spaces and complex inner product spaces have different ||polarization identities||

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spoilers for problem

cold needle Jul 2, 2021, 7:10 AM

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ok wont click yet

neat lintel Jul 2, 2021, 7:10 AM

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this one might get a little computational

cold needle Jul 2, 2021, 7:10 AM

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oug

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ic

neat lintel Jul 2, 2021, 7:10 AM

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oug

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you dont have to do it now

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just something to ponder

cold needle Jul 2, 2021, 7:11 AM

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ok

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ill take a stab tonight and probably go sleep soon

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then tomorrow is another day

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this is nice

forest jackal Jul 2, 2021, 7:11 AM

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https://tenor.com/view/cat-cat-sleep-sleepy-sleppy-cat-cat-roll-gif-21923694

Tenor

neat lintel Jul 2, 2021, 7:12 AM

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linear algebra is good for you

cold needle Jul 2, 2021, 7:12 AM

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talking about math here is very good for me :3

neat lintel Jul 2, 2021, 7:12 AM

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neat lintel let V be a real inner product space, and let f: V -> V be a function with f(0) =...

lee says this comes up in the proof of the myers steenrod theorem so i guess you're doing some sophisticated geometry opencry

cold needle Jul 2, 2021, 7:12 AM

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idek what that is but very cool :3

blazing pawn Jul 2, 2021, 7:13 AM

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thats ok tterra doesnt either

neat lintel Jul 2, 2021, 7:13 AM

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first fundamental form turns a surface into a metric space. theorem: isometry in the metric-space sense is the same as a DG isometry

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roughly

blazing pawn Jul 2, 2021, 7:13 AM

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he is just possessed by the sleep paralysis geometry demons when he posts here

neat lintel Jul 2, 2021, 7:14 AM

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very rough theorem sttement

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i dont want to use the word riemannian

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also u need connected

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sad

blazing pawn Jul 2, 2021, 7:15 AM

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i finally found non connected spaces that matter

neat lintel Jul 2, 2021, 7:15 AM

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stare

cold needle Jul 2, 2021, 7:15 AM

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lmao

blazing pawn Jul 2, 2021, 7:15 AM

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tensor product of field extensions over the field of meromorphic functions on a riemann surface get you a non connected cover usually

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and the connected components correspond to compositum

neat lintel Jul 2, 2021, 7:16 AM

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these are words

blazing pawn Jul 2, 2021, 7:16 AM

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i think this result is intuitive actually

neat lintel Jul 2, 2021, 7:16 AM

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remind me of the thing about meromorphic functions you tlod me the other day

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or maybe it was pty

blazing pawn Jul 2, 2021, 7:17 AM

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ok so you know how covering theory looks similar to galois theory

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but with the directions reversed

neat lintel Jul 2, 2021, 7:17 AM

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i know 0 galois but i know this is supposed to be true

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i was planning on going into galois theory with this pov but then we just

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didnt

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opencry

blazing pawn Jul 2, 2021, 7:17 AM

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lmfao

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basically u can reformulate galois theory as talking about the category of finite etale algebras being equivalent to actions of the absolute galois group

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which sounds complicated

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but its not really

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if we have a field F then we can think about finite separable extensions right

velvet dagger Jul 2, 2021, 7:18 AM

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No we can't

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But continue

blazing pawn Jul 2, 2021, 7:18 AM

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Dami moment

vivid halo Jul 2, 2021, 7:18 AM

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hello

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nozoomi

neat lintel Jul 2, 2021, 7:18 AM

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hi

velvet dagger Jul 2, 2021, 7:19 AM

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Whaddup

blazing pawn Jul 2, 2021, 7:19 AM

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Shut up ng im trying to have my moment in the spotlight where i pretend to be smart by regurgitating information i read in a book 5 to 7 business days ago

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dont take this from me 😡

velvet dagger Jul 2, 2021, 7:19 AM

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I'm actually curious lol so go on

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My annoying chime ins are signs that I'm listening

neat lintel Jul 2, 2021, 7:19 AM

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moth

vivid halo Jul 2, 2021, 7:20 AM

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I did some affine Weyl group computations by literally folding a right triangle out of paper and writing 0,1,2 on the edges and flipping it around in the air

Screen_Shot_2021-07-02_at_12.19.12_AM.png

neat lintel Jul 2, 2021, 7:20 AM

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literally any mathematical discussion is welcomed

velvet dagger Jul 2, 2021, 7:20 AM

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Not diffgeo

neat lintel Jul 2, 2021, 7:20 AM

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cope

velvet dagger Jul 2, 2021, 7:20 AM

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Well diffgeo under limited circumstances

blazing pawn Jul 2, 2021, 7:20 AM

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ok so yeah basically I dont want to get into the details because they are kind of elaborate but u can formulate galois theory by talking about

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if i fix a field k

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and a separable closure k_s of k

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we can take the absolute galois group Gal(k_s, k) = Gal(k)

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so now basically if we have a finite separable extension L of k then we get an action of the Gal(k) automorphisms on Hom(L, k_s) right

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literally by just composing

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every g in Gal(k) is an automorphism k_s -> k_s

velvet dagger Jul 2, 2021, 7:21 AM

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Fax

blazing pawn Jul 2, 2021, 7:22 AM

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so it defines an action on hom sets by sending f to f circ g

velvet dagger Jul 2, 2021, 7:22 AM

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Okay I'm starting to see where this is going I think

blazing pawn Jul 2, 2021, 7:23 AM

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so without going into too much detail a finite etale algebra over k is a finite dimensional algebra A over k isomorphic to a finite direct product of separable extensions

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and we have the same action of Gal(k) on Hom(A, k_s)

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so its not hard to see that we get a functor from finite etale algebras over k into sets with Gal(k) actions on them

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and specifically separable extensions will give you transitive Gal(k) actions

velvet dagger Jul 2, 2021, 7:24 AM

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:0

blazing pawn Jul 2, 2021, 7:24 AM

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and Galois extensions give you finite quotients

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nozoomi

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And this is very similar to the covering space situation

velvet dagger Jul 2, 2021, 7:24 AM

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That's nifty

blazing pawn Jul 2, 2021, 7:25 AM

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if i take the universal cover p: X' -> X over a (semilocally simply connected blah blah blah) space X

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then covers Y -> X give rise to actions of pi_1(X, x)

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connected covers correspond to transitive pi-1 actions

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and galois covers correspond to coset spaces of normal subgroups of pi_1

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so you have the analogy here

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finite etale algebras over a field look like covers over a space

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separable extensions look like connected covers

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galois extensions look like galois covers

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riemann surfaces and ramification are the (or at least a) setting where this equivalence is realized

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specifically compact connected riemann surfaces

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where covers Y -> X turn M(Y) into a finite etale algebra over M(X)

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where M(X) and M(Y) are the ring/fields of meromorphic functions

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M(Y) is a field and thus separable bc characteristic 0 since we are working over C precisely when Y is connected

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and similarly M(Y) is galois over M(X) precisely when Y -> X as a ramified cover restricts to a galois cover of X

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in fact in a way that preserves degree

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nozoomi

velvet dagger Jul 2, 2021, 7:30 AM

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Okay I'm back sorry

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But yeah it seems this is secretly the fact that transitive G-sets are all G/H

blazing pawn Jul 2, 2021, 7:31 AM

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which part

velvet dagger Jul 2, 2021, 7:31 AM

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The whole thing kinda, like once you have a Galois correspondence

uncut creek Jul 2, 2021, 7:32 AM

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sounds like u guys are unraveling the secrets of the universe

velvet dagger Jul 2, 2021, 7:32 AM

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Then it's also an equivalence of categories between a category of G-sets and whatever you're Galois corresponding to

blazing pawn Jul 2, 2021, 7:32 AM

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im not sure which part u mean specifically but like yea the fact that transitive G-sets look like G/H is important for the topology stuff

velvet dagger Jul 2, 2021, 7:33 AM

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I guess there's some technical detail regarding the fact that you're taking finite etale algebras rather than just separable field extensions but I imagine it's because of Hom business

blazing pawn Jul 2, 2021, 7:33 AM

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or like its important for establishing the kinks of what the functor from fin etale algebras -> Gal(k) sets preserves

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and similarly for covers -> pi_1(X, x) sets

vivid halo Jul 2, 2021, 7:36 AM

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nozoomi

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one way to look at this is the following

blazing pawn Jul 2, 2021, 7:37 AM

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integral

vivid halo Jul 2, 2021, 7:38 AM

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when X is normal (don't worry about precisely what this means) we can talk about the function field K of X and we have π^et_1(X)=Gal(K^ur/K) where K^ur is a maximal unramified extension of K

cold needle Jul 2, 2021, 7:38 AM

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normal

vivid halo Jul 2, 2021, 7:38 AM

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yea I hate this terminology

cold needle Jul 2, 2021, 7:38 AM

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lovely

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this is what

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the 88th time the word normal has been used

vivid halo Jul 2, 2021, 7:38 AM

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anyways this applies in the case of compact Riemann surfaces for instance

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GAGA will tell you these things are algebraic

cold needle Jul 2, 2021, 7:38 AM

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for something even more specific

vivid halo Jul 2, 2021, 7:39 AM

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the case of something like compact Riemann surfaces has this extra step of relating the complex analytic picture to the algebraic picture

blazing pawn Jul 2, 2021, 7:39 AM

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mathematicians just call everything they like "normal" in the hopes of coping so hard mathematics itself bends to their will

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and all the examples just look like the nice case

vivid halo Jul 2, 2021, 7:39 AM

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so you have an algebraic function field and a meromorphic function field, and algebraic (finite) covers versus analytic (possibly infinite) covers

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normalize singularities

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holy shit

neat lintel Jul 2, 2021, 7:45 AM

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definition: a normal singularity...

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Who's feynman of Mathematics?

devout nacelle Jul 2, 2021, 7:56 AM

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Nikita Shlyapustin.

velvet dagger Jul 2, 2021, 8:00 AM

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~~Who?~~

devout nacelle Jul 2, 2021, 8:01 AM

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toxic schooner Jul 2, 2021, 8:01 AM

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velvet dagger Jul 2, 2021, 8:01 AM

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Uh

neat lintel Jul 2, 2021, 8:01 AM

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nikita is fine

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if you're gonna shit on someone shit on dami for SLANDERING dg

devout nacelle Jul 2, 2021, 8:02 AM

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kekw

neat lintel Jul 2, 2021, 8:02 AM

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opencry

velvet dagger Jul 2, 2021, 8:02 AM

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DG slanders itself lmfao

neat lintel Jul 2, 2021, 8:04 AM

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spivak calls the ricci calculus chapter "a debauch of indices" in the chapter title opencry

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a bout of excessive indulgence in sensual pleasures, especially eating and drinking.

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ah yes

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the sensual pleasure of ricci calculus

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stare

devout nacelle Jul 2, 2021, 8:04 AM

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stare

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What is Ricci calculus

neat lintel Jul 2, 2021, 8:04 AM

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https://en.wikipedia.org/wiki/Ricci_calculus

Ricci calculus

In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be called the absolute differential calculus (the foundation of tensor calculus), developed by Gregorio Ric...

devout nacelle Jul 2, 2021, 8:04 AM

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opencry

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Were you anticipating this question

neat lintel Jul 2, 2021, 8:04 AM

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manipulation of tensor fields in coordinates

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yes

devout nacelle Jul 2, 2021, 8:05 AM

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Ahh

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So it is calculus in the sense of symbolic calculi

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Is this related to that Ricci flow stuff stare

neat lintel Jul 2, 2021, 8:06 AM

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maybe?

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idk anything about ricci flow lol

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oh it's a pde

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i guess you'd run into some ricci calculus while studying ricci flow

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like if you wanted to write this out in coordinates and play with it

devout nacelle Jul 2, 2021, 8:08 AM

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cros

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What is g_t?

neat lintel Jul 2, 2021, 8:08 AM

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a riemannian metric

devout nacelle Jul 2, 2021, 8:08 AM

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catThink

neat lintel Jul 2, 2021, 8:08 AM

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(you may also have heard "first fundamental form" before)

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it's a smoothly varying inner product on each tangent space

devout nacelle Jul 2, 2021, 8:09 AM

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Just the word, yes, I don't know what it means.

toxic schooner Jul 2, 2021, 8:09 AM

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stare

devout nacelle Jul 2, 2021, 8:09 AM

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neat lintel it's a smoothly varying inner product on each tangent space

I see

neat lintel Jul 2, 2021, 8:09 AM

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like

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g is a riemannian metric if for every p, g(p) is an inner product on the tangent space to your surface at p, and if this is "smooth" in some sense

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something like

devout nacelle Jul 2, 2021, 8:10 AM

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catThink

neat lintel Jul 2, 2021, 8:11 AM

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tangent space could be thought of as the best linear approximation to a surface at a point

devout nacelle Jul 2, 2021, 8:11 AM

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Okay, makes sense

neat lintel Jul 2, 2021, 8:11 AM

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inner product on the tangent space thus measures geometric stuff on your manifold near that point up to first order

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roughly vaguely

devout nacelle Jul 2, 2021, 8:11 AM

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From what I vaguely remember from an interaction between Dami and Metal, each of these tangent spaces is endowed with an inner product?

neat lintel Jul 2, 2021, 8:11 AM

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yup

devout nacelle Jul 2, 2021, 8:12 AM

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neat lintel inner product on the tangent space thus measures geometric stuff on your manifol...

I see

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Why do we care about the inner product?

neat lintel Jul 2, 2021, 8:12 AM

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you can measure geometric information with it

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angle, length, etc.

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think of all the things you can do with the dot product

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geometry

devout nacelle Jul 2, 2021, 8:13 AM

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I see

neat lintel Jul 2, 2021, 8:13 AM

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actually that's an example of a riemannian metric

devout nacelle Jul 2, 2021, 8:13 AM

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neat lintel like if you wanted to write this out in coordinates and play with it

Does this have some physical interpretation catThink

neat lintel Jul 2, 2021, 8:13 AM

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neat lintel actually that's an example of a riemannian metric

if you have a surface in euclidean space, just restrict the dot product to each tangent space and you get a riemannian metric

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stareFlushed

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(this one's more commonly called "first fundamental form")

neat lintel Jul 2, 2021, 8:14 AM

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devout nacelle Does this have some physical interpretation <:catThink:526983414226747393>

idk

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opencry

devout nacelle Jul 2, 2021, 8:14 AM

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It feels like a manifold's structure is evolving with time, and this equation tells you how this affects the metric, maybe

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I may or may not be making sense

devout nacelle Jul 2, 2021, 8:14 AM

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neat lintel if you have a surface in euclidean space, just restrict the dot product to each ...

cocatThink

neat lintel Jul 2, 2021, 8:15 AM

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i guess it's saying something like

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hm

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yeah nah you basically said what i was thinking

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a riemannian metric is "extra structure" on a manifold, right

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and you're looking at how a family of these change over time

chilly smelt Jul 2, 2021, 8:17 AM

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neat lintel Jul 2, 2021, 8:17 AM

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if you wrote it out in coordinates the ricci flow equation would be

devout nacelle Jul 2, 2021, 8:17 AM

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Okay yeah t need not have any meaningful assignment in terms of time here

neat lintel Jul 2, 2021, 8:17 AM

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rather complicated

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yeah im just used to calling it time lol

devout nacelle Jul 2, 2021, 8:18 AM

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Okay physicist

neat lintel Jul 2, 2021, 8:18 AM

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if you wrote it out in coordinates the ricci flow equation would be a rather complicated PDE monkaS

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what would you call it

devout nacelle Jul 2, 2021, 8:19 AM

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PDEs are already monkagiga

neat lintel Jul 2, 2021, 8:19 AM

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system of n^2 + 1 partial differential equations in n + 1 variables?

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that might be wrong

devout nacelle Jul 2, 2021, 8:19 AM

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cros

neat lintel Jul 2, 2021, 8:20 AM

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no n^2 and n + 1

devout nacelle Jul 2, 2021, 8:20 AM

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TTerra geometric analysis arc

chilly smelt Jul 2, 2021, 8:20 AM

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That’s easy then

neat lintel Jul 2, 2021, 8:20 AM

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since each g_t has n^2 components

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but symmetry cuts this down quite a bit

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n(n+1)/2 and n + 1??

chilly smelt Jul 2, 2021, 8:20 AM

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n(n+1)/2

neat lintel Jul 2, 2021, 8:21 AM

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sniped xd

neat lintel Jul 2, 2021, 8:21 AM

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devout nacelle TTerra geometric analysis arc

yes

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one of these days

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my uni is offering two geometric analysis courses in the upcoming semesters

devout nacelle Jul 2, 2021, 8:22 AM

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hype hype

neat lintel Jul 2, 2021, 8:22 AM

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i absolutely have the differential geometry background

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but maybe not the analysis background

devout nacelle Jul 2, 2021, 8:22 AM

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Do they need a course in PDEs?

neat lintel Jul 2, 2021, 8:23 AM

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idk

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one of them has the grad analysis as a prereq (coreq)

devout nacelle Jul 2, 2021, 8:23 AM

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cros

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What all do typical grad analysis classes cover?

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Measure theory, functional analysis?

neat lintel Jul 2, 2021, 8:24 AM

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all? no clue lol

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probably

devout nacelle Jul 2, 2021, 8:24 AM

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I see

neat lintel Jul 2, 2021, 8:24 AM

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Measure Theory: Lebesgue measure and integration, convergence theorems, Fubini's theorem, Lebesgue differentiation theorem, abstract measures, Caratheodory theorem, Radon-Nikodym theorem.

Functional Analysis: Hilbert spaces, orthonormal bases, Riesz representation theorem, compact operators, L^p-spaces, Holder and Minkowski inequalities.

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Fourier Analysis: Fourier series and transforms, Fourier inversion and Plancherel formula, estimates and convergence results, topological vector spaces, Schwartz space, distributions.

Functional Analysis: The main topic here will be the spectral theorem for bounded self-adjoint operators, possibly together with its extensions to unbounded and differential operators.

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1 and 2

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pretty close lol

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you got 2/3 of em

devout nacelle Jul 2, 2021, 8:25 AM

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OwO niceeee

neat lintel Jul 2, 2021, 8:25 AM

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depending on the rest of my courses i might take this

devout nacelle Jul 2, 2021, 8:25 AM

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This seems rather overkill for a semester though monkaS

neat lintel Jul 2, 2021, 8:25 AM

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this is over two semesters

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lol

devout nacelle Jul 2, 2021, 8:25 AM

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Ah, that is more reasonable

neat lintel Jul 2, 2021, 8:26 AM

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topology comes first, but the grad admissions here really like to see these two analysis courses

devout nacelle Jul 2, 2021, 8:27 AM

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catThink

neat lintel Jul 2, 2021, 8:27 AM

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they're the qualifying exam courses

devout nacelle Jul 2, 2021, 8:27 AM

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Ohh

neat lintel Jul 2, 2021, 8:27 AM

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grad school likes those

devout nacelle Jul 2, 2021, 8:27 AM

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So if you take these classes, you can skip quals in grad school?

neat lintel Jul 2, 2021, 8:27 AM

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if you're a graduate student already and you do sufficiently well, yes

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but if you're an undergrad

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idk

devout nacelle Jul 2, 2021, 8:28 AM

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Yeah, I meant an undergrad taking them before starting grad school

neat lintel Jul 2, 2021, 8:28 AM

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i did well enough in complex analysis to not have to write the qual if i were a graduate student, but i'll probably have to write the complex qual at some point anyways (if i am accepted here)

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sadness!

devout nacelle Jul 2, 2021, 8:29 AM

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sad

neat lintel Jul 2, 2021, 8:29 AM

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pure agony

velvet dagger Jul 2, 2021, 8:30 AM

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Hmm?

neat lintel Jul 2, 2021, 8:30 AM

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yo

velvet dagger Jul 2, 2021, 8:30 AM

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I feel like you could probably just go to the admin and be like

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Hey look at my grade

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It took it while an undergrad but I still have satisfied the requirement of "Take this class or pass this qual"

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So I feel like when I did it shouldn't matter

neat lintel Jul 2, 2021, 8:30 AM

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"don't make me write this qual or i'll toss you off the roof bitch"

velvet dagger Jul 2, 2021, 8:30 AM

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Honestly yeah

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Well

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Kind word and a gun works better than a kind word

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And probably than a mean word

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So instead of doing that just accidentally drop your concealed carry license

neat lintel Jul 2, 2021, 8:31 AM

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speaking from experience?

velvet dagger Jul 2, 2021, 8:31 AM

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While pleading to their ability to reason critically

neat lintel Jul 2, 2021, 8:31 AM

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stare

velvet dagger Jul 2, 2021, 8:31 AM

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Probably from someone's experience

neat lintel Jul 2, 2021, 8:32 AM

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the people in the math department here are nice

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i would not want to ~~have to~~ hurt anyone

devout nacelle Jul 2, 2021, 8:36 AM

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What is your about me section saying TTerra KEK

neat lintel Jul 2, 2021, 8:37 AM

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a long time ago shamrock told me a really stupid proof that $\binom{n}{k} = \binom{n}{n - k}$ using the cohomology of the $n$-torus

fathom swallowBOT Jul 2, 2021, 8:37 AM

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TTerra

neat lintel Jul 2, 2021, 8:37 AM

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only the other day did i actually understand it

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(de rham cohomology)

mellow moon Jul 2, 2021, 8:38 AM

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Best proof, heavy technical stuff perhaps

neat lintel Jul 2, 2021, 8:38 AM

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yes

velvet dagger Jul 2, 2021, 8:38 AM

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Hah

neat lintel Jul 2, 2021, 8:38 AM

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it uses some very non trivial algebraic topology KEK

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show this to cs students in their combinatorics classes

velvet dagger Jul 2, 2021, 8:39 AM

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Fwiw along these lines there are some actual combinatorial things which are non-trivial and can be done topologially

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If you take complex Grassmannians

neat lintel Jul 2, 2021, 8:39 AM

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reminds me of "subgroups of free groups are free" proven using AT

velvet dagger Jul 2, 2021, 8:40 AM

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You can reference stuff like the weak and hard Lefschetz theorems to know that the betti numbers are unimodal

neat lintel Jul 2, 2021, 8:40 AM

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hmm

velvet dagger Jul 2, 2021, 8:40 AM

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And symmetric

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But the betti numbers are combinatorial

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b_{2i} = number of partitions of i into at most d parts with largest part at most size n-d

neat lintel Jul 2, 2021, 8:41 AM

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stare

velvet dagger Jul 2, 2021, 8:42 AM

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Meaning you're counting partitions of i whose young diagrams fit into a d x (n-d) sized box

neat lintel Jul 2, 2021, 8:43 AM

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if only i had not dropped combinatorics

velvet dagger Jul 2, 2021, 8:43 AM

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Combinatorics is fun with the right prof

neat lintel Jul 2, 2021, 8:43 AM

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my class was something

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the prof was very good

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but he was rather insistent on in class exercises where you discuss with the person next to you

velvet dagger Jul 2, 2021, 8:44 AM

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Oh but "next to you" is tricky with Zoom lmfao

neat lintel Jul 2, 2021, 8:44 AM

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opencry

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this was before covid

velvet dagger Jul 2, 2021, 8:44 AM

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Oh

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Then what was wrong?

neat lintel Jul 2, 2021, 8:45 AM

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opencry

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how do i say this

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no ok the actual reason i dropped was because i wasn't comfortable taking 3 math courses at once at the time

velvet dagger Jul 2, 2021, 8:46 AM

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Ah that's fair

neat lintel Jul 2, 2021, 8:46 AM

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i was going to make a meme about it but i realized it was basically an isolated incident

velvet dagger Jul 2, 2021, 8:46 AM

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What were the other two?

neat lintel Jul 2, 2021, 8:47 AM

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topology and "advanced calc" (spivak's calc on manifolds)

velvet dagger Jul 2, 2021, 8:47 AM

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If either of them were less interesting ima judge your taste in math even harder than I already do

neat lintel Jul 2, 2021, 8:47 AM

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the latter course was veyr hard

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but both were very good

velvet dagger Jul 2, 2021, 8:47 AM

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Point-set? Hmm it's not quite as interesting but it's kinda important so I can respect it

neat lintel Jul 2, 2021, 8:47 AM

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well

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we did AT for like

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half of it

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lol

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baby AT though

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munkres speedrun any%

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these classes may or may not have played major roles in my current mathematical interests stare

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both classes were very well run and had amazing professors

velvet dagger Jul 2, 2021, 8:49 AM

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Hmm... normally those are fine to take but if they ended up with you doing diffgeo then perhaps you should've done combo instead

neat lintel Jul 2, 2021, 8:49 AM

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right

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oh well

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i can just take it in the fall if i want

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or pick it up here and there

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why take a course in something when you can get people on math discord to explain it to you?

velvet dagger Jul 2, 2021, 8:50 AM

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Just stop doing diffgeo and do combinatorics instead tbh

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Like as your research area

neat lintel Jul 2, 2021, 8:51 AM

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differential-geometric combinatorics

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stareFlushed

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prove combinatorical identities using

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stokes

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opencry

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i can see it tbh

velvet dagger Jul 2, 2021, 8:55 AM

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Honestly I need to ask my advisor for more deets about the quantum chaos on graphs business

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That could legit be my style

neat lintel Jul 2, 2021, 8:58 AM

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catThonk

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sounds like something ultra would know about

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ask them

robust stratus Jul 2, 2021, 9:37 AM

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hi, i'm looking for some advice. math was my strong point in high school, I liked it, probably my favorite subject, but it was never passionate. Today I want to discern if I was only interested in mathematics or if it is something I can dedicate my life to. What do you think can help me find an answer to this? (anything can help)

leaden torrent Jul 2, 2021, 9:43 AM

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do you have any exposure to proof-based mathematics? (not high school geometry "proofs", actual proofs in definition-theorem-proof-corollary style)

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if not, try self-studying some.

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its more reflective of what mathematics "actually" feels like at a high level

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even applied math is frequently proofsy

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certainly way more proofsy than high school mathematics, at least.

robust stratus Jul 2, 2021, 9:47 AM

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leaden torrent do you have any exposure to proof-based mathematics? (not high school geometry "...

mathematical induction counts?

leaden torrent Jul 2, 2021, 9:47 AM

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induction is one proof technique

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but theres a bit of a philosophical difference between a course structured around proofs and rigour, and one that just happens to cover basic induction at one point

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would you be able to prove, for example, that f(x) = x^3 is a bijection (one-to-one & onto function) from ℝ to ℝ?

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whether directly from the definition or proving continuity and applying the monotone continuous function theorem

torn willow Jul 2, 2021, 9:50 AM

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Induction is the proof version of "My intuition is clearly correct"

chilly smelt Jul 2, 2021, 9:55 AM

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The guy is deciding if he can find passion in math and you start talking about boring theorems

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If that’s what someone told me math was I wouldn’t want to devote my life to it

leaden torrent Jul 2, 2021, 9:56 AM

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im just trying to scope out whether theyve taken a proof based course before.

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im not math's PR department

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though either way, id certainly consider this more interesting than rote induction drills.

neat lintel Jul 2, 2021, 9:57 AM

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quick start taking about your favorite k theory result

leaden torrent Jul 2, 2021, 9:58 AM

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every K-theory result sucks

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sorry

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dont even know K_12(ℤ)

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pathetic

neat lintel Jul 2, 2021, 9:59 AM

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frogN

leaden torrent Jul 2, 2021, 9:59 AM

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(aside: i think people should go into math will full knowledge that a lot of it will be boring rote work)

robust stratus Jul 2, 2021, 10:00 AM

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leaden torrent would you be able to prove, for example, that f(x) = x^3 is a bijection (one-to-...

maybe I know where the solution can be (x ^ 3 retains the sign of x, and x ^ 1/3 is real, independent if x is positive or negative??), but can I prove it formally? I do not think so.

leaden torrent Jul 2, 2021, 10:00 AM

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(as you get good, you can skip a lot of that rote work, but its still worth setting reasonable expectations)

robust stratus Jul 2, 2021, 10:00 AM

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sorry for the late reply, i barely speak english

leaden torrent Jul 2, 2021, 10:00 AM

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hmmm you have the inklings of the right idea actually

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proving that x^3 is odd and that x^1/3 = 0 has no solutions for x > 0 is enough

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but it requires a bit more work

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either way, it seems you havent taken a "proper" proofsy course - which is fine

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makes that probably a good place to start though

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to see how much it "clicks"

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unfortunately im unfamiliar with foreign-language recommendations, though.

robust stratus Jul 2, 2021, 10:02 AM

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chilly smelt The guy is deciding if he can find passion in math and you start talking about b...

exactly that worries me

leaden torrent Jul 2, 2021, 10:03 AM

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i did give you a very boring problem.

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perhaps a more interesting one would be worthwhile, but I don't know how much you know

robust stratus Jul 2, 2021, 10:04 AM

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leaden torrent unfortunately im unfamiliar with foreign-language recommendations, though.

ah don't worry about that, I can read, but when I write I try to do it correctly

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and that takes time

leaden torrent Jul 2, 2021, 10:04 AM

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I see.

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here's perhaps a more interesting question:

it's well-known that a positive integer is divisible by 3 precisely when the sum of its digits is. for example, 4 + 7 + 1 = 12 and 1 + 2 = 3, so 471 is divisible by 3. meanwhile, 1 + 0 + 0 + 7 = 8, so 1007 is not divisible by 3. a similar thing is true for division by 9.

(a) why is this true? can you prove it? (it helps if you know modular arithmetic, but that isn't necessary)
(b) if youre familiar with bases other than ten: for what divisors is this true in a given base? (hint: how are 9 and 10 related? how are 3 and 9 related?)

robust stratus Jul 2, 2021, 10:09 AM

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robust stratus exactly that worries me

But the point is that math is not very exciting for me. I mean, when a new topic is presented to me, I like to try to understand the "logic behind it", try to connect it with previous topics. but i never looked for stuff myself (except a few numberphile videos xd).

leaden torrent Jul 2, 2021, 10:11 AM

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dont worry about answering that to me btw

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im just giving it as an example of a problem that one might be posed in a proofsy mathematics course

robust stratus Jul 2, 2021, 10:11 AM

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i will try it, but not now probably (it's very late here).

leaden torrent Jul 2, 2021, 10:11 AM

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would require some thought and breaking down

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the honest truth is that a career in just mathematics (even applie) is fairly niche

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and higher math is very different from high school/early university

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so you cant really figure out if it "sticks" unless you just try it

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its boring at times, frustrating at others, and not the most financially lucrative career

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but its also really interesting IMO

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problem solving is fun

robust stratus Jul 2, 2021, 10:14 AM

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leaden torrent do you have any exposure to proof-based mathematics? (not high school geometry "...

actually any recommendation to learn / practice that? maybe in a "boring" way it could help me decide better

leaden torrent Jul 2, 2021, 10:15 AM

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know any calculus? spivak's calculus is a good textbook with a more proof-oriented style than a typical calculus course.

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though a warning: the first few chapters are very slow and boring

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necessary but boring

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takes a while to get into actual calc

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apostol is a bit faster-paced but doesnt teach proofs as well IMO

robust stratus Jul 2, 2021, 10:16 AM

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leaden torrent know any calculus? spivak's _calculus_ is a good textbook with a more proof-orie...

i don't know exactly what calculus is so i don't know, but i will check it

leaden torrent Jul 2, 2021, 10:16 AM

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what language do you speak?

robust stratus Jul 2, 2021, 10:17 AM

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leaden torrent what language do you speak?

spanish, but don't worry about the language barrier

leaden torrent Jul 2, 2021, 10:18 AM

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límites, integrales y derivadas

robust stratus Jul 2, 2021, 10:18 AM

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robust stratus spanish, but don't worry about the language barrier

it's not my first time reading a english book

leaden torrent Jul 2, 2021, 10:18 AM

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sound familiar?

robust stratus Jul 2, 2021, 10:19 AM

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yes, but i know almost zero about the topic

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anyway, probably anything I'll study will have math in it

leaden torrent Jul 2, 2021, 10:20 AM

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hm, spivak would be theoretically doable, but may be difficult without a teacher giving you intuition for the calculus

robust stratus Jul 2, 2021, 10:21 AM

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maybe the hard way is the ideal way to decide better

leaden torrent Jul 2, 2021, 10:23 AM

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well proofs are unnaturally disorienting to first-time learners

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so im a bit wary

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dedicated proof textbooks exist but i think they all kinda suck

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and its better (+ way more interesting) to learn proofs "naturally" in the context of actual mathematics

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rather than contrived "random easy topics from elementary number theory"

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(though my earlier divisibility-by-3 problem is arguably an example of that...)

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but its tough since youre learning 2 things at once that both kind of depend on each other

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and because you lack instructor feedback while self-studying, it can be hard to tell how well you're actually doing.

robust stratus Jul 2, 2021, 10:27 AM

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leaden torrent and its better (+ way more interesting) to learn proofs "naturally" in the conte...

I guess it makes you more skilled too?

leaden torrent Jul 2, 2021, 10:27 AM

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(though some people post their proofs in #proofs-and-logic and ask for feedback which helps)

robust stratus Jul 2, 2021, 10:27 AM

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This problem solving thing is to find out if I like science oriented math, right?

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because if we are talking about engineering oriented mathematics, should I look at other aspects (like real life applications)?

leaden torrent Jul 2, 2021, 10:30 AM

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for engineering, look at engineering coursework

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my advice is for "math majors"

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like your uni degree says "mathematics" on it

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maybe "pure mathematics" or "applied mathematics", but still math

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other fields dont have nearly as much emphasis on proofs

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(physics and computer science have some but not as much)

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(same with statistics)

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(other fields dont really have proofs at all)

robust stratus Jul 2, 2021, 10:32 AM

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I probably like more science-oriented math, but the point is, I'll have to find a job too. And where I live, science is not something that people are willing to finance.

leaden torrent Jul 2, 2021, 10:34 AM

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yeah, employment is an issue, no doubt.

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engineering or computer science are far "safer" paths relatively

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even statistics if you still want a very mathy flair

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(and in fact, i'd say the bulk of statistics "feels more like" high school math than mathematics does)

robust stratus Jul 2, 2021, 10:35 AM

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Maybe it's a stupid question, but where will I find more science-oriented math? an engineering in mathematics? or things like physics or computer science?

leaden torrent Jul 2, 2021, 10:36 AM

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pure math is mostly just academia (professorial research + teaching), but applied math is widespread as financial analysts in industry positions

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and statisticians & whatnot

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you can often get a job in some sort of software development with any sort of mathematical degree

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so you can pursue a passion in school then settle into a relatively safe programming job

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still, im not gonna pretend the job market has an amazing wealth of options

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the vast majority of those who pursue pure math have a mentality of

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i'll try for a job in academia, and if i cant get one (which is the most likely scenario), i'll find a job programming or teaching high school

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if you value job security highly, mathematics sadly isnt the best field to pursue

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though it's still better than a typical arts degree

robust stratus Jul 2, 2021, 10:41 AM

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Something that worries me is that maybe I like mathematics, but an engineering in mathematics maybe not, I have the impression that it takes away the "fun" part. But again, I know very little about engineering in general.

leaden torrent Jul 2, 2021, 10:41 AM

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for whatever that's worth.

robust stratus Jul 2, 2021, 10:42 AM

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is computer science a much safer bet?

leaden torrent Jul 2, 2021, 10:42 AM

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yes

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like i could sugar-coat it

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but yes

robust stratus Jul 2, 2021, 10:43 AM

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but maybe just a little safer?

leaden torrent Jul 2, 2021, 10:43 AM

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computer science is so lucrative that universities face a shortage of cs professors

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since a typical cs phd enters industry instantly

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in mathematics, academia is "the dream job"

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in cs, academia begs to employ you

robust stratus Jul 2, 2021, 10:44 AM

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robust stratus Something that worries me is that maybe I like mathematics, but an engineering i...

sorry if I'm bothering you. but your opinion in this?

leaden torrent Jul 2, 2021, 10:44 AM

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huge difference in employability

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honestly i dont have a great answer

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i have no clue how youd figure that out

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sorry ¯_(ツ)_/¯

robust stratus Jul 2, 2021, 10:46 AM

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no problem

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You have already been very helpful, thank you

sleek wing Jul 2, 2021, 10:47 AM

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Engineering bad

robust stratus Jul 2, 2021, 10:48 AM

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if you don't mind, a brief why?

thorn brook Jul 2, 2021, 10:48 AM

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Abelian grape good

bronze pelican Jul 2, 2021, 10:50 AM

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kot

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I have a friend doing an electrical engineering PhD and he regularly takes graduate level math courses

summer nest Jul 2, 2021, 12:17 PM

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leaden torrent computer science is so lucrative that universities face a shortage of cs profess...

Now I regret not taking CS

sleek wing Jul 2, 2021, 12:46 PM

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thorn brook Abelian grape good

sotrue

toxic schooner Jul 2, 2021, 12:49 PM

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thorn brook Abelian grape good

stare

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what the hell

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is an abelian group?

summer nest Jul 2, 2021, 12:51 PM

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Some group

toxic schooner Jul 2, 2021, 12:55 PM

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i mean abelian grape

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ik abelian group, lol

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a group with commutativity

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right?

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hmmCat

inner finch Jul 2, 2021, 12:55 PM

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ulti ho gai sari tadbeerein
Koi dawa naa kam kari

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Mujhe is ke baad yaad naheen hai
Meri soch se bahir hui

summer nest Jul 2, 2021, 12:56 PM

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inner finch ulti ho gai sari tadbeerein Koi dawa naa kam kari

Try Jamal gota riehlshit3

inner finch Jul 2, 2021, 12:56 PM

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riehlshit

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it's a reference to some mir taki mir sher

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or ghazal rather

neat lintel Jul 2, 2021, 12:57 PM

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inner finch Jul 2, 2021, 12:57 PM

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https://www.rekhta.org/ghazals/ultii-ho-gaiin-sab-tadbiiren-kuchh-na-davaa-ne-kaam-kiyaa-meer-taqi-meer-ghazals

Rekhta

Read full ghazal by Meer Taqi Meer

ulTi ho gain sab tadbiren kuchh na dawa ne kaam kiya | Complete Ghazal with lyrics and translation, along with audio and video, at Rekhta

sleek wing Jul 2, 2021, 1:01 PM

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toxic schooner i mean abelian grape

What’s purple...... and..... commutes......

inner finch Jul 2, 2021, 1:02 PM

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A purple person in a car

summer nest Jul 2, 2021, 1:02 PM

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Tele tubby

inner finch Jul 2, 2021, 1:04 PM

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lmao

toxic schooner Jul 2, 2021, 1:05 PM

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sleek wing What’s purple...... and..... commutes......

wow

toxic schooner Jul 2, 2021, 1:05 PM

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summer nest Tele tubby

toxic schooner Jul 2, 2021, 1:05 PM

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sleek wing What’s purple...... and..... commutes......

Algebra has gotten onto ur head kid

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u need to become analysis pilled

toxic schooner Jul 2, 2021, 1:06 PM

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inner finch ulti ho gai sari tadbeerein Koi dawa naa kam kari

This was nice

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hmmCat

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so my nitro runs out

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on the 4th of july

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sleek wing Jul 2, 2021, 1:08 PM

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toxic schooner u need to become analysis pilled

I’ll do algebraic topology cause that’s basically analysis

toxic schooner Jul 2, 2021, 1:09 PM

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No

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i mean pure analysis

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nothing else

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No AT for u kid

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pick up Tao

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and start it

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also is AT basically analysis?

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stare

surreal sapphire Jul 2, 2021, 1:10 PM

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do p-adic analysis instead

toxic schooner Jul 2, 2021, 1:11 PM

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stare

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i cant do normal analysis

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how will i do p-adic

surreal sapphire Jul 2, 2021, 1:12 PM

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its easier

toxic schooner Jul 2, 2021, 1:12 PM

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it is?

surreal sapphire Jul 2, 2021, 1:12 PM

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in some sense, yes

toxic schooner Jul 2, 2021, 1:12 PM

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and i suppose it requires some analysis knowledge?

surreal sapphire Jul 2, 2021, 1:13 PM

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well, maybe

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the p-adic absolute value is an ultrametric, which makes it somewhat 'easier'

toxic schooner Jul 2, 2021, 1:14 PM

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ConfusedPsyduck

surreal sapphire Jul 2, 2021, 1:14 PM

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\sum a_n converges iff a_n is null sequence as it should be

devout nacelle Jul 2, 2021, 1:15 PM

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realshit

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Wait so

toxic schooner Jul 2, 2021, 1:15 PM

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i didnt get far enough to know what a null sequence is sed

devout nacelle Jul 2, 2021, 1:15 PM

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Convergence in p-adics means the sequence is eventually all 0s?

toxic schooner Jul 2, 2021, 1:16 PM

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stare

sleek wing Jul 2, 2021, 1:16 PM

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Tbf ive never heard that term either but I assume it means this ^

surreal sapphire Jul 2, 2021, 1:16 PM

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no, but

toxic schooner Jul 2, 2021, 1:16 PM

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is that what a null sequence is?

surreal sapphire Jul 2, 2021, 1:16 PM

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you know how \sum 1/n diverges even though 1/n is null

devout nacelle Jul 2, 2021, 1:16 PM

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Yes

sleek wing Jul 2, 2021, 1:16 PM

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Yur

devout nacelle Jul 2, 2021, 1:16 PM

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OH

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I missed sum

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Okay

surreal sapphire Jul 2, 2021, 1:16 PM

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stuff like that does not happen in the p-adics

devout nacelle Jul 2, 2021, 1:16 PM

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That's neat

surreal sapphire Jul 2, 2021, 1:16 PM

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or in any ultrametric space

toxic schooner Jul 2, 2021, 1:16 PM

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ConfusedPsyduck

sleek wing Jul 2, 2021, 1:16 PM

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devout nacelle I missed sum

I did as well 🥺

surreal sapphire Jul 2, 2021, 1:17 PM

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ultrametric space just means you have a triangle equality but stronger

devout nacelle Jul 2, 2021, 1:17 PM

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So the sum corresponding to every sequence which converges to 0 converges as well

sleek wing Jul 2, 2021, 1:17 PM

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That is very nice

toxic schooner Jul 2, 2021, 1:17 PM

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sleek wing I did as well 🥺

i missed the entire point

surreal sapphire Jul 2, 2021, 1:17 PM

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devout nacelle So the sum corresponding to every sequence which converges to 0 converges as wel...

yes

sleek wing Jul 2, 2021, 1:18 PM

#

Friendship with real numbers over, p-adics are my best friend now (I like 101-adics the best but don’t tell the others)

devout nacelle Jul 2, 2021, 1:18 PM

#

KEK

#

p-adics good

surreal sapphire Jul 2, 2021, 1:18 PM

#

|a + b| <= max(|a|, |b|)

toxic schooner Jul 2, 2021, 1:18 PM

#

surreal sapphire Jul 2, 2021, 1:18 PM

#

and this gives you lots of strong results

devout nacelle Jul 2, 2021, 1:18 PM

#

catThimc

surreal sapphire Jul 2, 2021, 1:18 PM

#

so its "easier" in that sense

#

but harder in the sense that you have 0 geometric intuition

toxic schooner Jul 2, 2021, 1:19 PM

#

surreal sapphire but harder in the sense that you have 0 geometric intuition

stare

devout nacelle Jul 2, 2021, 1:19 PM

#

Yeah, I don't see any meaningful geometric intuition here

#

But then I barely understand ultrametrics

#

Or even metrics KEK

sleek wing Jul 2, 2021, 1:19 PM

#

Just distance innit

#

Like metres and kilograms are those metric things

devout nacelle Jul 2, 2021, 1:20 PM

#

thonkzoom

#

I do know the formal definition, I'm just familiar with very basic examples though

sleek wing Jul 2, 2021, 1:20 PM

#

Same

devout nacelle Jul 2, 2021, 1:20 PM

#

Like I don't have any intuition outside of Euclidean metric on R^n

sleek wing Jul 2, 2021, 1:21 PM

#

Taxicab is pretty easy as well

devout nacelle Jul 2, 2021, 1:21 PM

#

Maybe taxicab

#

Yeah

toxic schooner Jul 2, 2021, 1:21 PM

#

hmm

sleek wing Jul 2, 2021, 1:21 PM

#

Jungle river as well

toxic schooner Jul 2, 2021, 1:21 PM

#

I like your funny words magic men

toxic schooner Jul 2, 2021, 1:21 PM

#

sleek wing Jungle river as well

dont tell me thats an actual metric

#

stare

sleek wing Jul 2, 2021, 1:21 PM

#

Metrics are baby to understand the definition of

#

It is

devout nacelle Jul 2, 2021, 1:21 PM

#

sleek wing Jungle river as well

Which metric is this?

toxic schooner Jul 2, 2021, 1:21 PM

#

i mean ik what a metric is

#

but the rest idk anything about

sleek wing Jul 2, 2021, 1:22 PM

#

You uhhh draw a line until it hits another pre-set line perpendicular to the first one and the distance is the sum of those two iirc

surreal sapphire Jul 2, 2021, 1:22 PM

#

general metrics are kinda fine, since i just think of blobs in space

#

but p-adics are really weird, since it deals with objects you know like integers but assigns them weird absolute values

sleek wing Jul 2, 2021, 1:23 PM

#

Yeah it’s a lot easier to visualise everything as “Euclidean but a bit dodgy”

toxic schooner Jul 2, 2021, 1:23 PM

#

hmmCat

sleek wing Jul 2, 2021, 1:23 PM

#

As opposed to something like non-metric topological spaces which I cannot visualse for the life of me

toxic schooner Jul 2, 2021, 1:24 PM

#

p-adics seem kinda interesting

sleek wing Jul 2, 2021, 1:24 PM

#

Number theory 🚨

surreal sapphire Jul 2, 2021, 1:24 PM

#

i think https://www.youtube.com/watch?v=XFDM1ip5HdU introduces the 2-adics

YouTube

3Blue1Brown

What does it feel like to invent math?

An exploration of infinite sums, from convergent to divergent, including a brief introduction to the 2-adic metric, all themed on that cycle between discovery and invention in math.
Home page: https://www.3blue1brown.com/

Music: Legions (Reverie) by Zoe Keating

3blue1brown is a channel about animating math, in all senses of ...

▶ Play video

sleek wing Jul 2, 2021, 1:24 PM

#

Yeah that’s a good intro, I need to rewatch it tbh

#

I would like to see the actual formal definition of the p-adics tho

bronze pelican Jul 2, 2021, 1:25 PM

#

There are several equivalent definitions

devout nacelle Jul 2, 2021, 1:25 PM

#

What is the motivation for p-adics?

sharp mulch Jul 2, 2021, 1:25 PM

#

Number theory

sleek wing Jul 2, 2021, 1:25 PM

#

They are... cool...

surreal sapphire Jul 2, 2021, 1:26 PM

#

i think hensel considered formal laurent series

devout nacelle Jul 2, 2021, 1:26 PM

#

Debatable, although I do like the sum convergence fact

toxic schooner Jul 2, 2021, 1:26 PM

#

surreal sapphire i think https://www.youtube.com/watch?v=XFDM1ip5HdU introduces the 2-adics

i saw it some time ago

surreal sapphire Jul 2, 2021, 1:26 PM

#

the formal definition is either as formal laurent series or you define the p-adic metric on Q and complete it "in the usual way"

toxic schooner Jul 2, 2021, 1:26 PM

#

sleek wing They are... cool...

in my case everything is cool

#

sed

surreal sapphire Jul 2, 2021, 1:26 PM

#

or just as inverse limit of some category i guess

devout nacelle Jul 2, 2021, 1:27 PM

#

Ew

#

I momentarily forgot p-adic metric is described on Q catThink

#

Wait so

#

Does every Cauchy sequence in this metric converge?

surreal sapphire Jul 2, 2021, 1:28 PM

#

if you complete, yes

#

that's how completions work

devout nacelle Jul 2, 2021, 1:28 PM

#

What is the completion here? R?

bronze pelican Jul 2, 2021, 1:28 PM

#

Its Q_p

devout nacelle Jul 2, 2021, 1:28 PM

#

catThink

surreal sapphire Jul 2, 2021, 1:28 PM

#

you take all sequences in Q

#

so thats Q^N

devout nacelle Jul 2, 2021, 1:29 PM

#

Yes

surreal sapphire Jul 2, 2021, 1:29 PM

#

then you take the subring R of Cauchy sequences with respect to some metric

devout nacelle Jul 2, 2021, 1:29 PM

#

Ring hyperhonk

bronze pelican Jul 2, 2021, 1:29 PM

#

Q_p is the completion of Q with respect to the p-adic metric.
R is the completion of Q with respect to the usual metric.

surreal sapphire Jul 2, 2021, 1:29 PM

#

(confirm this actually is a subring as you do in analysis, sums and product of Cauchy are Cauchy etc)

devout nacelle Jul 2, 2021, 1:29 PM

#

Aah

#

Okay

surreal sapphire Jul 2, 2021, 1:29 PM

#

then confirm that the subset of null sequences N is an ideal

#

(this is not easy)

#

it's in fact a maximal ideal

#

by algebra R/N is a field

devout nacelle Jul 2, 2021, 1:30 PM

#

Haven't learnt algebra moment sad

surreal sapphire Jul 2, 2021, 1:30 PM

#

and depending on what metric you started with you get the p-adics Q_p or the real numbers

bronze pelican Jul 2, 2021, 1:30 PM

#

kot

surreal sapphire Jul 2, 2021, 1:30 PM

#

(this is a theorem by ostrowski)

#

well, if you have a ring R and a maximal ideal m in R, then R/m is a field

bronze pelican Jul 2, 2021, 1:31 PM

#

surreal sapphire Jul 2, 2021, 1:31 PM

#

i think its easier to just confirm manually that R/N in this case is a field

#

by computing explicit inverses

toxic schooner Jul 2, 2021, 1:31 PM

#

sh_Confused_Dog

surreal sapphire Jul 2, 2021, 1:32 PM

#

also it's not that important, but it "just works" the same way the construction of R works

devout nacelle Jul 2, 2021, 1:32 PM

#

I see catThink

#

Where do p-adics show up in number theory, what kind of questions do they help in answering?

bronze pelican Jul 2, 2021, 1:33 PM

#

"Local global priciples"

devout nacelle Jul 2, 2021, 1:33 PM

#

catThink

surreal sapphire Jul 2, 2021, 1:34 PM

#

the actual way you think of those objects is via formal laurent series or as the inverse limit $\varprojlim_n\bZ/p^n\bZ$

fathom swallowBOT Jul 2, 2021, 1:34 PM

#

Lochverstärker

devout nacelle Jul 2, 2021, 1:34 PM

#

What is inverse limit?

#

Is this some analytic notion

#

Or algebraic

surreal sapphire Jul 2, 2021, 1:35 PM

#

its category theory memes

#

but you can think of it pretty easily

devout nacelle Jul 2, 2021, 1:35 PM

#

bronze pelican "Local global priciples"

I googled this and found Hasse principle, seems to answer my question. catthumbsup

surreal sapphire Jul 2, 2021, 1:37 PM

#

so you have the product of set $\bZ/p\bZ \times \bZ/p^2\bZ \times \dots$

fathom swallowBOT Jul 2, 2021, 1:37 PM

#

Lochverstärker

devout nacelle Jul 2, 2021, 1:38 PM

#

Okay

surreal sapphire Jul 2, 2021, 1:39 PM

#

and then a sequence $(x_1, x_2, x_3, \dots)$ is a $p$-adic number if $\lambda_n(x_{n+1})= x_{n}$, where $\lambda_n\colon \bZ/p^{n+1}\bZ\to \bZ/p^{n}\bZ$ is the canonical projection

fathom swallowBOT Jul 2, 2021, 1:39 PM

#

Lochverstärker

bronze pelican Jul 2, 2021, 1:39 PM

#

So this defines Z_p as the inverse limit of Z/p^nZ

devout nacelle Jul 2, 2021, 1:40 PM

#

catThink

surreal sapphire Jul 2, 2021, 1:40 PM

#

oh yeah

bronze pelican Jul 2, 2021, 1:40 PM

#

Q_p is the fraction field of Z_p

toxic schooner Jul 2, 2021, 1:40 PM

#

hmmCat

#

i suppose i will do algebra before p-adics

bronze pelican Jul 2, 2021, 1:40 PM

#

There are several equivalent definitions of p-adics

toxic schooner Jul 2, 2021, 1:41 PM

#

as of rn

#

all seem too ahead of me

#

hmmm

chilly smelt Jul 2, 2021, 1:41 PM

#

It’s just numbers that go brrr to the left

toxic schooner Jul 2, 2021, 1:42 PM

#

Ooo

#

im gonna believe that blindly

surreal sapphire Jul 2, 2021, 1:43 PM

#

ye, you can probably do that rigorously

cinder zephyr Jul 2, 2021, 1:43 PM

#

Oof p-adics are wack

#

There's a research project I'm interested in and I emailed the prof

surreal sapphire Jul 2, 2021, 1:43 PM

#

also yes, be decent in algebra before

cinder zephyr Jul 2, 2021, 1:43 PM

#

and she was like "yea start reading up on P-adic numbers"

#

and I'm like ok

#

and I did

#

💀

surreal sapphire Jul 2, 2021, 1:44 PM

#

because you want to do galois theory over p-adics

cinder zephyr Jul 2, 2021, 1:44 PM

#

I have not taken Algebra and will not take it until next fall cause of how the honors sequence works here at UIUC

#

soooooo

#

I'm a bit in over my head rn but we're working on it

surreal sapphire Jul 2, 2021, 1:44 PM

#

or like, deal with p-adic completions of number fields to do number theory but the right way

cinder zephyr Jul 2, 2021, 1:44 PM

#

those are words

chilly smelt Jul 2, 2021, 1:45 PM

#

I feel you

surreal sapphire Jul 2, 2021, 1:45 PM

#

ye, i think you can do some stuff with p-adics 'easily' as an undergrad

#

but you can also easily get lost in stuff that is really hard

#

(introducing p-adics and doing some small thing with them is a popular bachelor thesis topic)

cinder zephyr Jul 2, 2021, 1:46 PM

#

I see

bronze pelican Jul 2, 2021, 1:46 PM

#

Gouvea's book is written for an undergrad audience

#

cinder zephyr Jul 2, 2021, 1:47 PM

#

this is the project I'm looking at and the prof (she was my honors abstract lin alg prof last semester) sent me two textbooks as well >_>

cinder zephyr Jul 2, 2021, 1:48 PM

#

bronze pelican

dope I'll look at it

rose dock Jul 2, 2021, 1:48 PM

#

I've always wondered about the difference between "universitext" and ugtm/gtm

bronze pelican Jul 2, 2021, 1:49 PM

#

That's cool

#

I have no idea how homotopy theory interacts with number theory

cinder zephyr Jul 2, 2021, 1:50 PM

#

I have no idea what homotopy theory is lol

#

I just saw undergrad research project led by a prof whose class I did well in

modest rune Jul 2, 2021, 1:50 PM

#

homotopy is a simple and natural notion that appears all over

#

homotopy theory usually refers to like

#

a much more specific thing

cinder zephyr Jul 2, 2021, 1:50 PM

#

what is homotopy

modest rune Jul 2, 2021, 1:50 PM

#

its like

vast surge Jul 2, 2021, 1:50 PM

#

bronze pelican

I have a copy of this and I'm planning on reading it soon

modest rune Jul 2, 2021, 1:50 PM

#

continuously deforming one function into another

cinder zephyr Jul 2, 2021, 1:51 PM

#

interesting

bronze pelican Jul 2, 2021, 1:51 PM

#

vast surge I have a copy of this and I'm planning on reading it soon

Cool, you should give lectures on what you learn in VC

#

hyperhonk

cinder zephyr Jul 2, 2021, 1:51 PM

#

I presume not every function can be deformed into every other function?

modest rune Jul 2, 2021, 1:51 PM

#

that is a good assumption

cinder zephyr Jul 2, 2021, 1:51 PM

#

Is it a correct one?

modest rune Jul 2, 2021, 1:51 PM

#

yes

cinder zephyr Jul 2, 2021, 1:51 PM

#

ay nice

azure kettle Jul 2, 2021, 1:53 PM

#

bronze pelican

gouvea is nice

cinder zephyr Jul 2, 2021, 1:54 PM

#

is there an easy to understand example?

#

just something I can look up and read in a bit when I'm more focused

modest rune Jul 2, 2021, 1:55 PM

#

Path homotopies are the simplest examples

#

well no that requires basepoint stuff

#

there are simple examples but you do need to learn the proper defns and stuff

cinder zephyr Jul 2, 2021, 1:55 PM

#

I see

#

figured

toxic schooner Jul 2, 2021, 1:58 PM

#

modest rune there are simple examples but you do need to learn the proper defns and stuff

max, will ur talk be on zoom? or discord only?

#

hmmCat

modest rune Jul 2, 2021, 1:58 PM

#

zoom

toxic schooner Jul 2, 2021, 1:58 PM

#

oooh

#

ok

#

Link_Thumbs_Up

hallow wasp Jul 2, 2021, 2:03 PM

#

proprietary shill

modest rune Jul 2, 2021, 2:04 PM

#

lmao zoom is free

bronze pelican Jul 2, 2021, 2:04 PM

#

?

toxic schooner Jul 2, 2021, 2:04 PM

#

?

modest rune Jul 2, 2021, 2:04 PM

#

i assume i was being called a proprietary shill

bronze pelican Jul 2, 2021, 2:05 PM

#

hallow wasp proprietary shill

My ? is directed at @hallow wasp

toxic schooner Jul 2, 2021, 2:05 PM

#

bronze pelican My ? is directed at <@770350381435322418>

same

#

and looks like the talk thats coming is at 3 pm for me

#

hype

#

i can attend it properly,
~~tho ofc i wont understand basically all of it~~

thorn brook Jul 2, 2021, 2:06 PM

#

the talk for me will be at 21:00 so it's fine for me hype

toxic schooner Jul 2, 2021, 2:06 PM

#

this is the first talk

#

where the time is not late at night for me

#

lol

hallow wasp Jul 2, 2021, 2:07 PM

#

bronze pelican My ? is directed at <@770350381435322418>

imagine not using Zfone

modest rune Jul 2, 2021, 2:08 PM

#

...what

#

no one knows what that is

toxic schooner Jul 2, 2021, 2:08 PM

#

hallow wasp imagine not using Zfone

whats a zfone

#

is that some meme?

hallow wasp Jul 2, 2021, 2:08 PM

#

meh

modest rune Jul 2, 2021, 2:08 PM

#

it looks like some garbage open source thing

hallow wasp Jul 2, 2021, 2:08 PM

#

untechnical baboons

#

go back to your windows/ macos envs

toxic schooner Jul 2, 2021, 2:09 PM

#

hallow wasp untechnical baboons

Dont disrespect baboons by comparing me to them

#

hmmCat

modest rune Jul 2, 2021, 2:09 PM

#

lmfao

thorn brook Jul 2, 2021, 2:09 PM

#

Z catFone

brave hollow Jul 2, 2021, 2:09 PM

#

I bet you took the moderna vaccine

toxic schooner Jul 2, 2021, 2:09 PM

#

hallow wasp go back to your windows/ macos envs

ah, we have a linux elitist here

#

lol

modest rune Jul 2, 2021, 2:09 PM

#

you specifically please don't come to my talk lmfao

toxic schooner Jul 2, 2021, 2:09 PM

#

brave hollow I bet you took the moderna vaccine

idk why thats a joke

#

but its funny

obsidian parrot Jul 2, 2021, 2:09 PM

#

Hey guys cos(theta)*cos(theta) = cos^2theta right?

toxic schooner Jul 2, 2021, 2:09 PM

#

modest rune you specifically please *don't* come to my talk lmfao

sh_kek_3

hallow wasp Jul 2, 2021, 2:10 PM

#

toxic schooner ah, we have a linux elitist here

if its not gentoo see ya

obsidian parrot Jul 2, 2021, 2:10 PM

#

obsidian parrot Hey guys cos(theta)*cos(theta) = cos^2theta right?

i forgot

modest rune Jul 2, 2021, 2:10 PM

#

#❓how-to-get-help

toxic schooner Jul 2, 2021, 2:10 PM

#

obsidian parrot Hey guys cos(theta)*cos(theta) = cos^2theta right?

first of all, #❓how-to-get-help
secondly.. go to a help channel i will answer there

modest rune Jul 2, 2021, 2:11 PM

#

i am so mad that civ5 doesn't run properly on mac

#

i'd never buy another pc

inner finch Jul 2, 2021, 2:11 PM

#

jhund mara

sharp mulch Jul 2, 2021, 2:11 PM

#

Macs are quite nice

hallow wasp Jul 2, 2021, 2:11 PM

#

dont buy another pc, it will be ported soon

modest rune Jul 2, 2021, 2:11 PM

#

its a bit late for civ 5 to get ported properly lol

hallow wasp Jul 2, 2021, 2:11 PM

#

not true

#

it happens with old games often

modest rune Jul 2, 2021, 2:12 PM

#

i think you are missing the point

#

it is ported

#

but the port dev is abandoned

hallow wasp Jul 2, 2021, 2:12 PM

#

fixed*

modest rune Jul 2, 2021, 2:12 PM

#

and the modding is impossible

#

and aspyr has openly said

#

they have no intention to fix it

hallow wasp Jul 2, 2021, 2:12 PM

#

baseddev

#

idk how ppl have balls to announce that type of thing

modest rune Jul 2, 2021, 2:13 PM

#

maybe i will get lucky and civ7 won't suck

hallow wasp Jul 2, 2021, 2:13 PM

#

maybe you get a free windows machine gifted to you

rose dock Jul 2, 2021, 2:13 PM

#

Can you use boot camp or smth?

sharp mulch Jul 2, 2021, 2:13 PM

#

Max are you going to play Foundation?

hallow wasp Jul 2, 2021, 2:14 PM

#

rose dock Can you use boot camp or smth?

lol i was joking

modest rune Jul 2, 2021, 2:14 PM

#

boot camp has always been a pain

#

whats foundation

hallow wasp Jul 2, 2021, 2:14 PM

#

zoom is better than discord for streaming

sharp mulch Jul 2, 2021, 2:14 PM

#

New game

#

4X

#

Looks exciting

modest rune Jul 2, 2021, 2:15 PM

#

do you have a link i cant find anything

hallow wasp Jul 2, 2021, 2:15 PM

#

that looks like a fun game

sharp mulch Jul 2, 2021, 2:15 PM

#

Wait oops not Foundation

#

Foundation is city builder

modest rune Jul 2, 2021, 2:15 PM

#

oh a game called Old World just dropped

#

maybe i should try it

hallow wasp Jul 2, 2021, 2:16 PM

#

oh wow

sharp mulch Jul 2, 2021, 2:16 PM

#

Humankind

hallow wasp Jul 2, 2021, 2:16 PM

#

old world looks like archipelgo board game

sharp mulch Jul 2, 2021, 2:16 PM

#

There you go

modest rune Jul 2, 2021, 2:16 PM

#

oh!

#

I was like

#

into the humankind stuff

#

but then everyone i know who plays 4x didnt buy it

#

so i also didnt

#

idk if its good or not or anything

sharp mulch Jul 2, 2021, 2:16 PM

#

Oh

#

I'll let you know if it's good

#

I bought it

modest rune Jul 2, 2021, 2:17 PM

#

I'll play with you if you like it, it looks neat and civ is getting stale for me

crystal stone Jul 2, 2021, 3:16 PM

#

@grim token

#

You awake u had a question

grim token Jul 2, 2021, 3:34 PM

#

Ph yeah

#

I made a college list

#

But i had some trouble with my safeties

#

What is the best community college in california

#

so I can get an easy transfer into a good UC school

#

@crystal stone

crystal stone Jul 2, 2021, 3:36 PM

#

IDK about the best one

#

But I know a few good ones, depends on the area you want to live in

#

For Math, the strongest one is probably Orange Coast College

grim token Jul 2, 2021, 3:36 PM

#

Ok

#

Do you think I will have a good chance at trasnferring to berkeley

crystal stone Jul 2, 2021, 3:36 PM

#

If you're into Engineering there's probably better ones

grim token Jul 2, 2021, 3:36 PM

#

or ucla

#

if I go to Orange Coast

crystal stone Jul 2, 2021, 3:37 PM

#

If you go to orange coast and do well

#

Then yeah

grim token Jul 2, 2021, 3:37 PM

#

How well are we talking

crystal stone Jul 2, 2021, 3:37 PM

#

They have a lot of Honors classes, you can look up transfer rates by your major

#

but if you do Honors math sequence

#

a very good chance of getting in

grim token Jul 2, 2021, 3:37 PM

#

I tried making sense of that website

#

But they don't give trasnfer % by CC

crystal stone Jul 2, 2021, 3:37 PM

#

Ah the UC transfer admission rates, no they don't

#

But they give you an idea of how competitive each school is and for what major

#

OCC has a very strong math/physics/computer-science/chemistry department

grim token Jul 2, 2021, 3:38 PM

#

Do you think OCC and UCSD for example

#

are on the same leve/

#

level?

crystal stone Jul 2, 2021, 3:38 PM

#

For lower division teaching, OCC is better than most UCs

grim token Jul 2, 2021, 3:39 PM

#

Really??

#

That's crazy

crystal stone Jul 2, 2021, 3:39 PM

#

You can look this up, Prof. Arnold Guerra the Third teaches full time at OCC and part-time at UC Irvine

grim token Jul 2, 2021, 3:39 PM

#

Was that ur teacher?

crystal stone Jul 2, 2021, 3:39 PM

#

He was offered his own lab at UC Irvine, and turned it down because the pay was too little

#

OCC pays their instructors more than UCs do for basic level courses

#

I took Physics 280/285 from Guerra

grim token Jul 2, 2021, 3:40 PM

#

When ur teacher is famou

#

s

#

do you ever get a chance to talk to him after class?

#

Like a normal teacher?

crystal stone Jul 2, 2021, 3:40 PM

#

What do you mean?

#

That depends on the prof

grim token Jul 2, 2021, 3:40 PM

#

Like can you ask questions in class

crystal stone Jul 2, 2021, 3:40 PM

#

Some love to talk to students

grim token Jul 2, 2021, 3:40 PM

#

And talk to them in office hours

crystal stone Jul 2, 2021, 3:40 PM

#

Some don't like that

grim token Jul 2, 2021, 3:40 PM

#

That sucks

crystal stone Jul 2, 2021, 3:41 PM

#

I dunno, it's more a personality thing than it is

#

A famous thing

grim token Jul 2, 2021, 3:41 PM

#

That's lame

crystal stone Jul 2, 2021, 3:41 PM

#

🤷‍♂️

sharp mulch Jul 2, 2021, 3:41 PM

#

Berkeley has a lot of transfers from some Bay Area CCs like De Anza