#point-set-topology | Mathematics | Page 192

gritty widget Dec 26, 2020, 5:26 PM

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and the stuff i posted from lee was self reading

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catThhhh

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semester's over catshrug

dusk heron Dec 27, 2020, 2:01 AM

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Given a one-form $\xi$ on an oriented Riemannian $4$-manifold, what is the intuitive meaning of the one-form $\star(\dd{\xi}\wedge\xi)$, where $\dd{}$ is the exterior derivative and $\star$ is the Hodge star?

gentle ospreyBOT Dec 27, 2020, 2:03 AM

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gustavn64

📎 608412719451865089.png

dusk heron Dec 27, 2020, 2:04 AM

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This is apparently something called the "twist form".

gentle ospreyBOT Dec 27, 2020, 3:19 AM

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slimvesus

📎 98936686641156096.png

gritty widget Dec 27, 2020, 3:32 AM

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urysohn lemma?

river granite Dec 27, 2020, 3:33 AM

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yeah sounds quite a bit like that or some separation axiom

gritty widget Dec 27, 2020, 3:33 AM

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ya i think u can urysohn this

river granite Dec 27, 2020, 3:33 AM

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was just looking at it lmao

📎 Captura_de_pantalla_2020-12-27_00-33-00.png

gritty widget Dec 27, 2020, 3:34 AM

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compact hausdorff spaces satisfy the hypotheses of urysohn's lemma (normal space)

river granite Dec 27, 2020, 3:34 AM

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sounds like you want some kind of bump function in C?

gritty widget Dec 27, 2020, 3:37 AM

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just take {u} to be the closed set opencry

tough imp Dec 27, 2020, 4:10 AM

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Are irreducible components disjoint?

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I felt like they were, but then when I think about minimal primes I don’t think so anymore lol

tight agate Dec 27, 2020, 4:22 AM

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no

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k[x,y]/(xy)

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the irred components are the x and y axes

red garden Dec 27, 2020, 8:34 AM

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if X is a T_1 space with atleast two points , show that an open base which contains X as a member remains an open base if X is dropped

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my attempt ( i know this is wrong and its bad but i wasnna know why ) :

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suppose {B_1,B_2,....B_n,X,.......} is an open base

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we wish to show that {B_1,B_2,....B_n,.....} is an open base

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so for the sake of contradiction suppose it is not which means there exists x and y such that x and y are not inside B_1,B_2....B_n (it fails to cover it)

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but {B_1,B_2,....B_n,X} is an open base

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so for both x and y

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we know since X is T_1 there exists neighbourhoods for x and y such that one does not contain the other , call them G_x G_y respectively

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{B_1,B_2,....B_n,X} is an open base --> x is in B_i for some i which is a subset of G_x and also likewise for y

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but x is not in B_i for any i and likewise for y

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so B_i must equal X

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hence G_x is a subset of G_y or G_y is a subset of G_x

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as X is a subset of G_x

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contradiction

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this sounds like HELLA wrong

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but i dont see why

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i just feel why

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no?

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should i?

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idk

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no

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i dont

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any open set must be like a subset of X ig so it turns out the neighbourhoods are exactly X hence contradiction

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i am sure this is wrong but i just want to know why

gritty widget Dec 27, 2020, 8:42 AM

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ah oops ignore what i said

red garden Dec 27, 2020, 8:42 AM

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its okay

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so what do u think

tough imp Dec 27, 2020, 8:53 AM

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mo2men

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You started off with a finite base

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Your first step isn’t correct bro

red garden Dec 27, 2020, 8:59 AM

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yea

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okay but what does this change

red garden Dec 27, 2020, 9:27 AM

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@tough imp

vocal wharf Dec 27, 2020, 9:47 AM

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how did you define base and what gives you two points that are not covered by the B_i @red garden

red garden Dec 27, 2020, 10:28 AM

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idk

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i guess i only have one point

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base is defined as like

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for any x in an open set G there exists a base element B such that x is in B subset of G

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@vocal wharf

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fuck is this supposed to be hard

vocal wharf Dec 27, 2020, 10:36 AM

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well, you can characterize a base like: covers whole space and for every point in the intersection there is another base element that includes that point and is contained in the intersection

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seemed like you were going for this

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then the second thing about the intersection thing is easy

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so you have to check that your new base covers X

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assume it doesnt, get a point that is and one that isnt covered

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apply T_1 to solve

red garden Dec 27, 2020, 10:44 AM

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ohhhhhhhhh

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one that is covered and one not

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yea clever

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lmao

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ty

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got it @vocal wharf

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if i assume it doesnt then im only allowd to assume

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1 point that is not covered

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right? that was the wrong thing

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that i assumed existence of two pointst tht are not covered

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right?

vocal wharf Dec 27, 2020, 10:45 AM

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i think so

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at least i don't see how that follows

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and what chmonkey said is correct as well

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you should index your base not with natural numbers

red garden Dec 27, 2020, 10:52 AM

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damn

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i thought it was hard

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okay tysm

slow tree Dec 27, 2020, 8:35 PM

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Ok I asked in another channel but can someone help me with this problem:

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Here's the problem I'm facing: in the Poincare disk, given a point P, an angle a, and a hyperbolic distance D, I want to find the point Q such that the angle between the line from P to Q and the horizontal through P is a and the hyperbolic distance between P and Q is D. See https://drive.google.com/file/d/1uUGQbjNDh04GKZ5PnoCiD2jZR6PnjIcN/view?usp=sharing for an illustration of what I mean. I've been working on this problem for a couple of days but I can't seem to get it right.

One idea I had for doing it is calculating the line first, which will be a circle in euclidean geometry, then the circle of points of distance D around P, which will be another circle in euclidean geometry, and finally calculating the intersections between these two circles in cartesian coordinates. Unfortunately, I can't seem to figure out how to calculate either of these, and I can't find too many accessible resources for this. Can anybody help?

Google Docs

screenshot.png

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In short, given P, the angle, and the distance in the Poincare disk, I want to find Q

gritty widget Dec 28, 2020, 3:41 AM

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just to make sure I'm not crazy, it's always true that $Y \subseteq Z(T)$ if and only if $T \subseteq I(Y)$ right

gentle ospreyBOT Dec 28, 2020, 3:42 AM

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zd

📎 209443096990187520.png

sleek thicket Dec 28, 2020, 3:54 AM

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Yeah so $Y \subseteq Z(T)$ iff $I(Z(T)) \subseteq I(T)$. Any element of $T$ vanishes on all of $Z(T)$, so $T \subseteq I(Z(T))$

gentle ospreyBOT Dec 28, 2020, 3:55 AM

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Schamrock

📎 228562950703546368.png

gritty widget Dec 28, 2020, 4:07 AM

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right

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ok I am stumped

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📎 unknown.png

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why does this follow

tight agate Dec 28, 2020, 4:13 AM

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are Y1 and Y2 irreducible

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oh wait nvm

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this shouldnt be true

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take p = 0

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and uh

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hmm

gritty widget Dec 28, 2020, 4:16 AM

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sorry p is a prime ideal

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but I don't see how I can find all elements being products of elements from I(Y1) and I(Y2) in that intersection

tight agate Dec 28, 2020, 4:17 AM

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what are Y1 and Y2

gritty widget Dec 28, 2020, 4:18 AM

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Closed sets in affine space

tight agate Dec 28, 2020, 4:19 AM

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assume p is not equal to I(Y1)

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take some element r in I(Y1) that is not in p

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for any element s in I(Y2)

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rs is in P (as it is in the intersection)

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so s is in P (as r is not in P)

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so I(Y2) \subset P

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and we're done

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oh fuck

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nvm i screwed up

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nah

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it works

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right @gritty widget ?

tight agate Dec 28, 2020, 4:24 AM

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gritty widget but I don't see how I can find all elements being products of elements from I(Y1...

the product I(Y1)I(Y2) is contained in the intersection

gritty widget Dec 28, 2020, 4:24 AM

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That is clever but I am just a little confused why rs is in P

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Oh is it? ok

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hmm lemme think why

tight agate Dec 28, 2020, 4:25 AM

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rs is in I(Y1)

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it is also in I(Y2)

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I(Y1) is an ideal, so r in I(Y1) implies rs is in I(Y1) for all s

gritty widget Dec 28, 2020, 4:28 AM

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bruh moment

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ty

tight agate Dec 28, 2020, 7:00 AM

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I'm stuck on a part of Hartshorne problem 3.2.6

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Let X be a noetherian topological space.

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Let $U \subset X$ be open. Let $\mathscr{R} \subset i_{!}\mathbf{Z}_U$ be a subsheaf. Then the exercise asks us to prove that $\mathscr{R}$ is finitely generated.

gentle ospreyBOT Dec 28, 2020, 7:02 AM

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Brofibration

📎 351398829238648832.png

tight agate Dec 28, 2020, 7:02 AM

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where i: U ---> X is the inclusion map

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I'm struggling to show that even $i_{!}Z_{U}$ is finitely generated

gentle ospreyBOT Dec 28, 2020, 7:04 AM

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Brofibration

📎 351398829238648832.png

tight agate Dec 28, 2020, 7:16 AM

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hmm I might have something

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alright I think I'm being stupid

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Z_U is just generated by 1 \in Z_U(U) right?

meager python Dec 28, 2020, 8:03 AM

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What does it mean that a subsheaf is fg?

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It is not true that Z_U is finitely generated as an abelian group

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Or am I missing something?

uncut surge Dec 28, 2020, 10:12 AM

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https://math.stackexchange.com/questions/587916/what-is-a-finitely-generated-sheaf this problem has apparently been had and the exercise is not formulated well

Mathematics Stack Exchange

What is a finitely generated sheaf?

I do exercise asking to show on Noetherian space $X$, any subsheaf $\mathscr{R} \subseteq \Bbb{Z}_U$ is finitely generated. $\Bbb{Z}U$ is sheaf $i{!}(\Bbb{Z}|_U)$ for $U \subseteq X$ open. What ...

long hornet Dec 28, 2020, 1:19 PM

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If we have two disks and attach one to the other by sending a boundary point (x, y) to (-x, -y), how will the resulting manifold look like?

sweet wing Dec 28, 2020, 1:21 PM

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long hornet If we have two disks and attach one to the other by sending a boundary point (x,...

a S^2?

long hornet Dec 28, 2020, 1:22 PM

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I thought so at first, but I can't visualize it..

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Generally, if we have a manifolds-with-boundary, does it matter which homeomorphism of boundary we use?

uncut surge Dec 28, 2020, 1:27 PM

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This sounds like you're just attaching the two disks in the "obvious" way, but one of them is rotated by 180 degrees, since mapping a point x on the boundary to -x is the same as rotating it by 180 degrees around the origin, right?

long hornet Dec 28, 2020, 1:28 PM

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uncut surge This sounds like you're just attaching the two disks in the "obvious" way, but o...

Oh yeah, I saw it by thinking about two intervals..

uncut surge Dec 28, 2020, 1:28 PM

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So in terms of what manifold you get, it doesn't matter whether you glue (x,y) to (x,y) or (x,y) to (-x,-y), it's just a rotation of one of the disks

long hornet Dec 28, 2020, 1:29 PM

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I get it now

uncut surge Dec 28, 2020, 1:29 PM

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naisu

long hornet Dec 28, 2020, 1:29 PM

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long hornet Generally, if we have a manifolds-with-boundary, does it matter which homeomorph...

What about this?

uncut surge Dec 28, 2020, 7:32 PM

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@long hornet took me a while to get back to this, but yes, in general, the choice of homeomorphism does matter

📎 220px-Circle-surgery.png

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The above picture shows you that if you glue two compact intervals to two compact intervals, you can either get two circles or one, depending on how you map the boundary points of the intervals

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But a homeomorphism of the circle, I believe, can always be continuously "straightened out" to be either identity or -identity, so with two disks you should never run into problems

long hornet Dec 28, 2020, 7:36 PM

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uncut surge But a homeomorphism of the circle, I believe, can always be continuously "straig...

Yeah. I think in general if a homeomorphism of the boundary can be extended to the whole manifold (always the case for balls), then it doesn't matter. I am not quite sure about my reasoning, however.

uncut surge Dec 28, 2020, 7:38 PM

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Hm, I'm also not sure about that. I believe in the above example with the compact intervals, you can extend the homeomorphisms on the boundaries to the full manifolds, right?

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If the picture makes sense at all lol

long hornet Dec 28, 2020, 7:38 PM

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The arrows are confusing honestly

uncut surge Dec 28, 2020, 7:39 PM

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Hahaha, it's from the wikipedia article about surgery on manifolds which is somewhat related but not quite

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But basically, take the manifold with boundary given by the disjoint union of two intervals [0,1] and [0,1]. We want to glue this manifold to another copy of itself. So one copy of this manifold is the two black lines, the other copy is the two red lines

long hornet Dec 28, 2020, 7:40 PM

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Got it

uncut surge Dec 28, 2020, 7:40 PM

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Now, the boundary of this manifold is equal to the four boundary points of the intervals, 0, 0, 1, 1, and depending on how we map the boundary to itself, we can end up with two circles or one circle

long hornet Dec 28, 2020, 7:40 PM

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The boundary in each case is a four-point set

uncut surge Dec 28, 2020, 7:41 PM

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Yup!

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Oh, but it's true, you can't always extend the maps

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From the boundaries to the manifolds, since that might require you to create a path between a zero of the first interval to the zero of the second interval, for example

long hornet Dec 28, 2020, 7:43 PM

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uncut surge Oh, but it's true, you can't always extend the maps

Why? I thought you can use 1-x

long hornet Dec 28, 2020, 7:44 PM

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uncut surge From the boundaries to the manifolds, since that might require you to create a p...

Ohhh..

uncut surge Dec 28, 2020, 7:45 PM

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Denoting the boundary points by $0_A, 1_A, 0_B, 1_B$ depending on wheter we're in interval A or B, your boundary homeomorphism maps $0_A \mapsto 0_A$ and $1_A \mapsto 0_B$, then there is no way to extend this to the whole manifold, since there is no path between $0_A$ and $0_B$

gentle ospreyBOT Dec 28, 2020, 7:45 PM

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Lartomato

📎 151058908755001344.png

uncut surge Dec 28, 2020, 7:45 PM

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So some homeomorphisms of the boundary can be extended, but not all

long hornet Dec 28, 2020, 7:47 PM

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Is it reasonable to expect to know all the cases when this is possible?

uncut surge Dec 28, 2020, 7:49 PM

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Hmm, no idea really. In this simple example, it seems like whenever you end up gluing in a way that gets you two disjoint circles, it's possible, but in general it all may be more tricky

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This seems pretty close to this whole surgery theory business, so that may be the right area to look, but I'm an amateur when it comes to this gluing stuff

marsh forge Dec 28, 2020, 8:35 PM

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could someone summarize the question

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having trouble following the discussion lol

long hornet Dec 28, 2020, 8:42 PM

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Given a manifold-with-boundary M and boundary B, you can glue two copies of M to get a new manifold (without boundary), by identifying the boundaries of the two copies through a homeomorphism h : B ---> B. The question is whether the choice of h matters.

marsh forge Dec 28, 2020, 8:43 PM

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but to address larto's remark, a big goal of surgery theory is to talk about how you can do surgery to the interior of a manifold with boundary without damaging the boundary and also preserving other structure

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i think if Aut(B)/htpy is nontrivial there should be issues

long hornet Dec 28, 2020, 8:45 PM

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marsh forge but to address larto's remark, a big goal of surgery theory is to talk about how...

Other structure like what?

marsh forge Dec 28, 2020, 8:45 PM

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It might even be an issue if Aut(B) is nontrivial

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Smooth structure for example

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Other times you're trying to specifically kill generators in H* or pi*

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well, no, this is not necessarily the case actually

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Aut(S^1)/htpy is Z/2

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but the choice of homeo doesn't matter as long as topology is all you care about

long hornet Dec 28, 2020, 8:48 PM

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marsh forge It might even be an issue if Aut(B) is nontrivial

Well, sometimes we can always extend an element of Aut B to one of Aut M. This is the case for balls.

marsh forge Dec 28, 2020, 8:48 PM

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otoh if we have some way of differentiating sides of the disk it does matter, like if we are talking about orientation, only one of these two choices should preserve it

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like take a disk and give it a clockwise vector field

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one choice of gluing will match up the two vector fields

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and the other won't

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If you take a filled in torus I think it matters because Aut(T^2) has a lot of funky stuff

long hornet Dec 28, 2020, 8:51 PM

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marsh forge but the choice of homeo doesn't matter as long as topology is all you care about

Well, Lartomato gave an example where it does matter.

marsh forge Dec 28, 2020, 8:52 PM

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not for a disk

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which is what i was talking about there specifically

long hornet Dec 28, 2020, 8:52 PM

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marsh forge otoh if we have some way of differentiating sides of the disk it *does* matter, ...

I don't care about it (yet) since I don't even know its definition 😂

marsh forge Dec 28, 2020, 8:53 PM

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I think my guess would be that if Aut(B) is trivial up to orientation it doesn't matter

long hornet Dec 28, 2020, 8:53 PM

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marsh forge If you take a filled in torus I think it matters because Aut(T^2) has a lot of f...

Can you describe Aut(T^2)

marsh forge Dec 28, 2020, 8:53 PM

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GL(2,Z)

long hornet Dec 28, 2020, 8:57 PM

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marsh forge I think my guess would be that if Aut(B) is trivial up to orientation it doesn't...

Hmmmm..

marsh forge Dec 28, 2020, 8:57 PM

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Sorry I was being uncautious

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there are way more automorphisms of a torus

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these are the automorphisms of the torus as a lie group

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(they are the nice automorphisms of the torus that are linear on the universal cover)

long hornet Dec 28, 2020, 8:59 PM

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Actually I can't even see why GL(2, Z) is a subgroup of Aut(T^2). Is it related to Möbius transformations somehow?

marsh forge Dec 28, 2020, 9:00 PM

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Are you aware of the description of the torus as R^2/Z^2

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GL(2,Z) is basically the automorphisms of R^2 preserving Z^2

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so that if L is in GL(2,Z), so that L: R^2->R^2

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then L descends to a map R^2/Z^2

long hornet Dec 28, 2020, 9:01 PM

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marsh forge Are you aware of the description of the torus as R^2/Z^2

Yeah

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Ohh, I see.

delicate hollow Dec 28, 2020, 10:34 PM

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If asked: Determine all topologies on X = {a,b}

Is the answer?:
T_1 = {∅,{a,b}} , T_2 = {∅,{a,b},{a}} , T_3 = {∅,{a,b},{b}} , T_4 = {∅,{a,b},{a},{b}}

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If asked: On the three-point set X = {a, b, c}, For each of n = 3
7, either find a topology on X consisting of n open sets or prove that no such topology exists.

Is the answer:
Proof:
For n=3
T = {∅,{a,b,c},{a}}
For n=4
T = {∅,{a,b,c},{a,b},{a}}
For n=5
T = {∅,{a,b,c},{a,b},{b,c},{b}}
For n =6
The only collections consisting of 6 sets are
T = {∅,{a,b,c},{a,b},{a,c},{a},{b}}
T = {∅,{a,b,c},{a,b},{a,c},{b},{c}}
T = {∅,{a,b,c},{a,b},{a,c},{a},{c}}
T = {∅,{a,b,c},{a,b},{b,c},{a},{b}}
T = {∅,{a,b,c},{a,b},{b,c},{b},{c}}
T = {∅,{a,b,c},{a,b},{b,c},{a},{c}}
T = {∅,{a,b,c},{a,c},{b,c},{a},{b}}
T = {∅,{a,b,c},{a,c},{b,c},{b},{c}}
T = {∅,{a,b,c},{a,c},{b,c},{a},{c}}
However none of these are valid topologies
For n=7
A collection with 7 sets is,
T = {∅,{a,b,c},{a,b},{b,c},{a},{b},{c}}
However {a} U {c} is not in T.
Or
T = {∅,{a,b,c},{a,b},{b,c},{a,c},{a},{b}}
However {b,c} ∩ {a,c} is not in T.
Without loss of generality it can be shown that there is no collection with 7 sets that satisfy all properties of a topology
QED

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Should n=6 use a without loss of generality argument?

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I could ask 9 more

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but im pretty sure no one wants to check 9 exercises.

frosty sundial Dec 28, 2020, 11:08 PM

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I don't agree with your statement "the only collections consisting of 6 sets are..."

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unless you're using some kind of "no loss of generality"?

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also your "no loss of generality" argument for n = 7 isn't really correct

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in this case, "no loss of generality" means you can do things like switch a and b

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not just "I'm only going to try one or two things and observe they fail, therefore all possibilities will fail"

delicate hollow Dec 28, 2020, 11:10 PM

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whats the best way to go about arguing 7 without writing a bunch that is hard to follow

frosty sundial Dec 28, 2020, 11:10 PM

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well first I think your argument for 6 is wrong because I found a topology on {a,b,c} with 6 elements

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namely the very first one you wrote down

delicate hollow Dec 28, 2020, 11:11 PM

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ok

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i see that

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didnt at first idk why

frosty sundial Dec 28, 2020, 11:13 PM

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for 7 there are a couple of approaches

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first, notice that there are only 8 subsets of {a, b, c}

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so any collection of 7 subsets we could think of as just "exclude one of the 8"

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we can't exclude {} or {a,b,c} and that leaves 6 options

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you could just write out all 6 of those and observe that none are topologies

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or you could use a WLOG argument -- by changing the labeling as necessary we only have to consider the cases of "remove {a}" and "remove {a,b}"

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because removing {b} is really just the same as removing {a} -- just swap the labels on a and b

delicate hollow Dec 28, 2020, 11:14 PM

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or {c}

frosty sundial Dec 28, 2020, 11:14 PM

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sure

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the point is you can just consider "remove a 1-element subset" and "remove a 2-element subset" so you only ahve two cases

delicate hollow Dec 28, 2020, 11:14 PM

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oh ok

frosty sundial Dec 28, 2020, 11:14 PM

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and you can see directly that neither of those are topologies

delicate hollow Dec 28, 2020, 11:15 PM

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i was trying to go with something like that

frosty sundial Dec 28, 2020, 11:15 PM

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it looks like you were on the right track, but you need to be careful about your wording. You can't just say "WLOG" and move on, you should explain why you're not losing any generality

delicate hollow Dec 28, 2020, 11:15 PM

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not gonna lie but for the n=6 one I couldnt imagine something off the top of my head so I copy and pasted a bunch

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oh ok

frosty sundial Dec 28, 2020, 11:15 PM

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haha

delicate hollow Dec 28, 2020, 11:16 PM

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i thought you could just state wlog as a magic word and people might follow what ur getting at

cloud owl Dec 28, 2020, 11:16 PM

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lol

frosty sundial Dec 28, 2020, 11:16 PM

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well, eventually you can, but you have to learn how to use it before you can just wave it around

cloud owl Dec 28, 2020, 11:16 PM

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it's not quite a synonym for 'trivially'

frosty sundial Dec 28, 2020, 11:16 PM

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I mean it's not at all a synonym for "trivially"

cloud owl Dec 28, 2020, 11:16 PM

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it means you can do a swappity

frosty sundial Dec 28, 2020, 11:17 PM

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"trivially" means "this is easy". "WLOG" means "I'm only going to consider a specific case, but don't worry, the general case is equivalent to this"

cloud owl Dec 28, 2020, 11:17 PM

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yeah, that's my point

frosty sundial Dec 28, 2020, 11:17 PM

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in this problem, technically we would have to consider each of the cases "remove {a}", "remove {b}", and "remove {c}" separately since technically each of the resulting collections of 7 sets are not hte same

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however, we realize that we don't really have to do that since removing {b} is the same as removing {a} -- we just swap the labels on {b} and {a} so they become the same

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so, what we're telling the reader is

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"I know that I'm only writing down a specific case, not the most general case, however, I'm not losing any generality because this specific case really does cover all the possible cases, which I know because I can just swap labels on elements as necessary"

delicate hollow Dec 28, 2020, 11:19 PM

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even though im only 1.5 years in to writing proofs and never had a class to talk about them, is it ok to just use words in a precise way instead of copying how i see proofs in textbooks?
An example being if I wanted to break down the two cases where I first remove an element of size 2 and then an element of size 1

frosty sundial Dec 28, 2020, 11:20 PM

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in my experience, proof writing is hard because there's not always a formula for how to write a proof. There's no "magic word" or magical order you can write words in to get a correct proof

delicate hollow Dec 28, 2020, 11:20 PM

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Could I state explicit, In this case we are removing any element of size 2 ... and then in this case we remove element of size 1

frosty sundial Dec 28, 2020, 11:20 PM

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yeah definitely

delicate hollow Dec 28, 2020, 11:21 PM

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it just has to be precise/meticulous english?

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or whichever langauge

frosty sundial Dec 28, 2020, 11:21 PM

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"There are two cases to consider: Removing a subset of size 1 and removing a subset of size 2. Without loss of generality, we can just consider removing {a} and {a, b} because [reason].
Case 1: Removing {a}. This does not result in a topology because...
Case 2: Removing {a, b}. This also does not result in a topology because..."

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yeah

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the point of a proof is communication

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you are trying to communicate what you are doing to a reader who isn't inside your head

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you're writing your thought process down

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so if you think of this problem in cases -- write that down!

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as long as your thought process is correct, your proof will be correct

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(well, as long as your thought process is correct and as long as you've written all of the steps down)

delicate hollow Dec 28, 2020, 11:28 PM

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so i can set proofs up in a way where it seems like im talking to the person

frosty sundial Dec 28, 2020, 11:28 PM

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yep

delicate hollow Dec 28, 2020, 11:28 PM

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so if i say "Lets think of the proof in cases"

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ok

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thats cool now that i know

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always thought it had to have some type of 3rd person perspective

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where we dont refer to the reader

frosty sundial Dec 28, 2020, 11:29 PM

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On the contrary, I think most proofs are written either in the first person plural or the second person imperative

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so like "first we do this, then we do this"

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or just "first do this, then do that"

delicate hollow Dec 28, 2020, 11:30 PM

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i never say we or you

frosty sundial Dec 28, 2020, 11:30 PM

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I would never say "you"

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I would just use the imperative

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like

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Consider this case. Now, look at this.

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(there's an implied "you", the reader, who I am asking to do the considering or the looking)

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but I use "we" all the time in proofs

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https://math.stackexchange.com/questions/1305775/why-do-single-author-math-papers-use-we-instead-of-i

Mathematics Stack Exchange

Why do single-author math papers use "we" instead of "I"?

Even when math papers are authored by one person, you will see stuff like "We now prove so and so" or "In the following section, we show that...".

How come papers with one author use "we" instead...

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there are lots of other related questions about this as well

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and I hope you'll get the idea that using "we" is very common, and even encouraged, over other options

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It shouldn't be in every sentence, you don't want to write "we do this. now we do this. now we do that. now we're done"

delicate hollow Dec 29, 2020, 12:40 AM

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this is so weird

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why do we do this

river granite Dec 29, 2020, 1:12 AM

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frosty sundial (there's an implied "you", the reader, who I am asking to do the considering or ...

it's not weird at all if we you consider this

sleek thicket Dec 29, 2020, 2:27 AM

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Sometimes I use "I" on my homework for metamathematical notes

rotund thicket Dec 29, 2020, 3:06 AM

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I've gotten used to the we

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I sort of like it

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I mean it's done like that in science as well

gritty widget Dec 29, 2020, 4:43 AM

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I always use I cause I'm not a team stabbyDuck

rugged swan Dec 29, 2020, 10:29 AM

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Btw you it can be a heritage from other language. French use "we" and it's not weird at all

uncut surge Dec 29, 2020, 12:09 PM

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french is very weird

shut moat Dec 29, 2020, 2:37 PM

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Not sure if this is the right place for this, but I'm starting a chapter of Hubbard that introduces integration on manifolds. What it does is it defines the k-volume of a k-parallelogram in R^n spanned by v_1, v_2, ... v_k as sqrt(det(T^T T)) where T = [v_1, ...v_k]. To justify this, it basically just tests it out for a few cases in R, R^2, and R^3, and says we should probably define it this way for arbitrary k. Is there any more concrete justification/"proof" for why we choose this definition?

gritty widget Dec 29, 2020, 2:42 PM

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I think you want to expand on the question your are asking, you don’t ‘prove’ a defintion

tough imp Dec 29, 2020, 2:42 PM

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I think it's perfectly well-stated as is

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They're just asking why you define it that way, the "proof" given that it's the right definition is a calculation for R^n for n = 1,2,3

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you certainly can "prove" you've chosen the right definition for something

gritty widget Dec 29, 2020, 2:47 PM

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I don’t think it’s clear, I also think if they try ask the question again they will nearly answer the question for themselves

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The question could either be: why are we allowed to define generalizations, or how can I see that this generalization agrees with some other definition of volume that I have. If they realise it’s the first question they want answered, they’ll get a better answer. If they realise it the second, they’ll realise they don’t have some other defintion they are trying to compare it to.

shut moat Dec 29, 2020, 2:52 PM

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Why's that last line implied by the first?

📎 Screen_Shot_2020-12-29_at_9.51.55_AM.png

shut moat Dec 29, 2020, 2:52 PM

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gritty widget The question could either be: why are we allowed to define generalizations, or h...

yeah i know there's no prior definition, but i feel that there should be at least a geometrically intuitive justification for why this is the right choice

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even though there isn't a rigorous prior definition, intuitively if you have a 2-parallelogram in the plane and just shift it so it doesn't fit in the plane, it shouldn't change the 2-volume

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and in R^n, the volume of an arbitrary n-parallelogram is already defined as |det[v_1, v_2, ...]|

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so I guess one potential "justification" would be to move the k-parallelogram to a hyperplane and then use |det| but I'm not sure how to do that

shut moat Dec 29, 2020, 4:30 PM

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sry had to go for a sec. I think I figured it out pandaHugg
I think I disagree that "it could practically be called a theorem", but what they did is exactly what I was spelling out above lol. Change of basis, then identify e_1, ... e_k (in R^n) with the standard basis of R^k by a map that kills off the "lower components", then take the volume of that

wanton marsh Dec 29, 2020, 10:32 PM

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what's a GW invariant ?

tight agate Dec 29, 2020, 10:32 PM

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gromov-witten I assume

gritty widget Dec 29, 2020, 10:34 PM

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Yes. Someone who does enumerative geometry would be the best fit. I am willing to pay because I hear it could take some work

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Would be very helpful to me

golden gust Dec 29, 2020, 11:40 PM

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when people say that all metrics X^2 to R are continuous, that's only with respect to the standard topology on R, right?

elder yew Dec 29, 2020, 11:43 PM

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Don't metrics induce a topology

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and it would be with respect to that topology

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But they all (if I remember correctly) end up more or less the same

golden gust Dec 29, 2020, 11:44 PM

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i thought the induced topology is on X, not R

marsh forge Dec 30, 2020, 12:05 AM

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"when people say that all metrics X^2 to R are continuous, that's only with respect to the standard topology on R, right?" I don't know how to parse this

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My guess is that you mean given a metric the metric is continuous in the topology induced by the metric

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which is true

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but given some metric X^2->R it is not necessarily true that this metric is continuous with respect to arbitrary topologies on X

golden gust Dec 30, 2020, 12:08 AM

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we can use the continuity definition of "preimages of open sets are open". then it seems to me like it matters which toplogy R is equipped with

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like say if R has the discrete topology so all sets are open

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then the preimage of (an open set in the standard topology) is open, but this doesn't necessarily hold for the preimage of (an open set in the discrete toplogy)

golden gust Dec 30, 2020, 12:35 AM

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ah thanks. can you elaborate/send resources on "This is by construction"? I don't fully get this

obtuse meteor Dec 30, 2020, 2:02 AM

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yeah this is basically it

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like the topology induced by a metric is precisely like the coarsest topology on X which makes the distance function X x X -> [0, infty) continuous where [0, infty) is given the standard topology

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I smell a universal property lol

gritty widget Dec 30, 2020, 5:15 PM

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Can anyone see why this is true? Let G be an abelian topological group, let H be the intersection of all open sets containing the identity. Show that if G is hausdorff then H consists of only the identity

uncut surge Dec 30, 2020, 5:19 PM

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This is not too tricky I think, is there a specific part that you're stuck on?

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Since Hausdorffness is pretty directly linked to how different points can be separated by different open sets

gritty widget Dec 30, 2020, 5:25 PM

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I don't think it should be two tricky but I'm not seeing how to piece together the two things. The haussdorf lets us separate any x and y by open sets.

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If we knew H was open then we could assume its not just the identity and get a contradiction

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oh

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damn we don't need to know H is open to seperate the points

uncut surge Dec 30, 2020, 5:26 PM

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Nope, we don't, we just need the openness of the sets that we intersect to get H 😄

gritty widget Dec 30, 2020, 5:26 PM

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But now it seems I haven't used any properties of being topological group

uncut surge Dec 30, 2020, 5:26 PM

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So this direction of the implication should follow once you've made that realization

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You're going to use it in the other direction!

gritty widget Dec 30, 2020, 5:27 PM

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Is it true in general for a hausdorf space, the intersection of every open set contain x is x

uncut surge Dec 30, 2020, 5:27 PM

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Yup! And that is how you prove it

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The special thing about topological groups is now that it's enough to check that this holds for a single point x to check that we have a Hausdorff space

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(for example the identity point, which is what we're looking at in this exercise)

woeful oasis Dec 30, 2020, 5:29 PM

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Pretty much anything local you have to say, you can say it at the identity

uncut surge Dec 30, 2020, 5:31 PM

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Ye that's the magic of having continuous multiplication and inversion

gritty widget Dec 30, 2020, 5:33 PM

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ty

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i have the other direction

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weird that I never noticed this is true for general hausdorff top spaces

woeful oasis Dec 30, 2020, 5:34 PM

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I have a question involving topological groups. The fundamental group of any topological group is abelian, which comes from the facts that multiplication is continuous and the unit square is simply connected.
Does any of this require group inverses or connectedness? It seems like only continuity of multiplication is required.

uncut surge Dec 30, 2020, 5:34 PM

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Now you'll never forget at least 😛

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wow i hate that emoticon picture

gritty widget Dec 30, 2020, 5:35 PM

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I know in the context of lie groups, smooth multiplication implies smooth inverse

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Does this not also hold for continous?

woeful oasis Dec 30, 2020, 5:36 PM

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I mean, if it's not a group to begin with

uncut surge Dec 30, 2020, 5:36 PM

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wiki says dis

📎 unknown.png

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So sounds like you're right, only some continuous multiplication map is required

woeful oasis Dec 30, 2020, 5:37 PM

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that's some mighty big tiny words

marsh forge Dec 30, 2020, 5:37 PM

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You can weaken the requirements to a htpy magma

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Referred to as an H Space

uncut surge Dec 30, 2020, 5:37 PM

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sawry my screen 2 big

marsh forge Dec 30, 2020, 5:38 PM

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I’m not sure if this is related to your question

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Not all spaces w cont multiplication have cont inverses though

gritty widget Dec 30, 2020, 5:38 PM

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Do you have a good example?

woeful oasis Dec 30, 2020, 5:38 PM

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So the "main homotopy" happening is still within the domain of parametrization, and continuity just passes this info to the product

marsh forge Dec 30, 2020, 5:39 PM

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I don’t have a good example

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I don’t know what that means apop

gritty widget Dec 30, 2020, 5:39 PM

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I know we could take a monoid with the discrete topology

woeful oasis Dec 30, 2020, 5:39 PM

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addition in (R^+)^2

marsh forge Dec 30, 2020, 5:40 PM

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https://math.stackexchange.com/questions/151878/why-metrizable-group-requires-continuity-of-inverse

Mathematics Stack Exchange

Why metrizable group requires continuity of inverse?

A metrizable group is a metric space $(G,d)$ with a binary operation $\cdot$ such $(G,(\cdot))$ is a group and maps $(\cdot):G\times G\to G$ and $f:G\to G$ given by $(\cdot)(x,y)=xy$ and $f(x)=x^{-...

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This should work

woeful oasis Dec 30, 2020, 5:40 PM

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My first thought was "but what about the Cayley graph of a group" but duh that's simply connected

marsh forge Dec 30, 2020, 5:40 PM

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Even if the underlying set is a group

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Taking inverses can be discontinuous

marsh forge Dec 30, 2020, 5:41 PM

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woeful oasis My first thought was "but what about the Cayley graph of a group" but duh that's...

I don’t think this is true

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Lots of Cayley graphs have cycles, no?

woeful oasis Dec 30, 2020, 5:42 PM

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err of a lie group

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If a discrete group is considered a subgroup of a lie group, then the cayley graph embeds in the universal cover of the lie group

gritty widget Dec 30, 2020, 5:43 PM

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hmm, what goes "right" in the case of lie groups

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is it just being a smooth map on a smooth manifold is so much stronger?

marsh forge Dec 30, 2020, 5:43 PM

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I would guess that if you look at the proof

gritty widget Dec 30, 2020, 5:44 PM

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yeah

marsh forge Dec 30, 2020, 5:44 PM

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It would be clear

gritty widget Dec 30, 2020, 5:44 PM

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i forget how to show it

woeful oasis Dec 30, 2020, 5:45 PM

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inverse function theorem

gritty widget Dec 30, 2020, 5:49 PM

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I wonder then can you get k-times differentiable multiplication implies k'-times differentiable inverse

woeful oasis Dec 30, 2020, 5:52 PM

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yes, you can do inverse function theorem with respect to order of differentiability.

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whatever order you're considering, usually with differentiable it's implicit that you're dealing with a manifold, so weird examples would still involve manifolds

gritty widget Dec 30, 2020, 6:05 PM

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by manifold here we mean smooth manifold

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i suppose this is all a bit point less since for lie groups C^1 implies C^infinity

woeful oasis Dec 30, 2020, 6:09 PM

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there's weird stuff like infinite dimensional C^k manifolds... But once you have the inverse function theorem for k at least one, then you have an inverse of equal order differentiability

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so as far as semigroup vs group, all you need is order 1

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The weirdness of the MSE example I think was in the topology underlying the multiplication. Once you restrict to manifolds...

gritty widget Dec 30, 2020, 6:12 PM

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interestante

woeful oasis Dec 30, 2020, 6:16 PM

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muy

gritty widget Dec 30, 2020, 6:21 PM

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Every monoid object in the category of C^k manifolds is automatically a group object

woeful oasis Dec 30, 2020, 6:27 PM

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well, if it was already already a group, and multiplication is C^k, then inverse is C^k

gritty widget Dec 30, 2020, 6:32 PM

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Oh

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So we can still have a C^k manifold with a C^k multiplication that does not have an inverse

woeful oasis Dec 30, 2020, 6:50 PM

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Or you might have an inverse, but the inverse might not be an element of the monoid you started with

marsh forge Dec 30, 2020, 6:51 PM

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I think lie monoids/semi groups can exist without being equal to their group competitions

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Or without having inverses

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That’s what mse suggests anyway

gritty widget Dec 30, 2020, 7:47 PM

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We have some top space X, and A subset B. We have quotient maps pi_a from X to X/A and similarly pi_b.

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We we also have a map from X/B to X/A

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Is it obvious that this is also a quotient map

woeful oasis Dec 30, 2020, 8:46 PM

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I think it would be X/A to X/B, squishing less to squishing more

marsh forge Dec 30, 2020, 9:07 PM

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for cont you need some extra point set stuff i think

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maybe that all things in consideration are like good pairs or whatever

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maybe less

gritty widget Dec 30, 2020, 9:11 PM

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sorry i got distracted and never asked the question

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the question was why is this a quoitent map

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and yes the A and B should be swapped

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but it comes from the universal property of the quoitent map pi_b

marsh forge Dec 30, 2020, 9:14 PM

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i dont know if it is in general but yeah just think about equivalence classes here

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as long as the identification is well defined everything works on the level of sets

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and then you just need point set considerations for things to work out topologically

obtuse meteor Dec 30, 2020, 9:33 PM

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you can just use universal prop of quotient set right?

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Like to show the map X / A -> X / B is a quotient map it suffices to show that if you have a thing X / A -> Y that's constant on B then that gives you a thing X / B -> Y that makes the diagram commute

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X / A -> Y that's constant on B is the same as X -> Y constant on B by universal prop which gives you your X / B -> Y by universal prop

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paste some diagrams together QED

gritty widget Dec 30, 2020, 9:36 PM

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yeah

marsh forge Dec 30, 2020, 9:41 PM

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I don’t think this needs that level of machinery

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But first the first map has to exist in Top

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And second the is obvious by construction without reference to category theory

obtuse meteor Dec 30, 2020, 9:52 PM

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I can only think in terms of pasting and universal props lol

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It is obvious in terms of “if this isn’t true the theory is fucked” and by the construction tho

gritty widget Dec 30, 2020, 9:57 PM

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Its definitely good to be able to do it out explicitly from the definitions. But I do find that the abstract nonsense helps to better organise it in the mind

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Having said that In hindsight its not too hard to check using the definitions. The map is clearly a surjection, and there isn't to much checking that the open sets work nicely

elder yew Dec 30, 2020, 10:01 PM

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Every time I try to learn algebraic topology I just find it unsatisfying or uninteresting

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I don't know why

gritty widget Dec 30, 2020, 10:02 PM

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what do you find interesting

elder yew Dec 30, 2020, 10:02 PM

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I like Low dim'l topology

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Knots, Links, 3 manifolds

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I'm currently reading about mapping class groups, which relies a lot on Alg. Top. results

gritty widget Dec 30, 2020, 10:02 PM

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fair

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I've never learned much low dim top

elder yew Dec 30, 2020, 10:03 PM

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It's a very niche area, very few people learn anything about it

marsh forge Dec 30, 2020, 10:03 PM

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Is that true

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Low dim top is like

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A huge research area

elder yew Dec 30, 2020, 10:03 PM

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It's huge in research, but when you look at schools where you can go do research in it

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Most schools are hard pressed to even have one low dim'l topologist

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Of course there are the heavy hitters like Austin, Davis, Princeton, etc.

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Santa Barbara, but like Berkeley has 2

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UW Seattle doesn't even have one

marsh forge Dec 30, 2020, 10:05 PM

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I think having 2 is a lot

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Half the schools in applying to have like one or two topologists

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Studying my stuff

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Almost every school has at least one low dim person

elder yew Dec 30, 2020, 10:05 PM

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What specifically are you applying for max?

gritty widget Dec 30, 2020, 10:05 PM

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are you also into low dim top?

marsh forge Dec 30, 2020, 10:05 PM

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No lol

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Algebraic Topology and homotopy theory

elder yew Dec 30, 2020, 10:06 PM

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That's more popular/has more than ldt

marsh forge Dec 30, 2020, 10:06 PM

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I don’t think that’s true

elder yew Dec 30, 2020, 10:06 PM

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Do you wanna stay at Chicago?

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Or are you gonna move around?

marsh forge Dec 30, 2020, 10:06 PM

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I mean I’d be happy to

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But only because it’s a good school

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I don’t have any desire to leave/stay outside of that

gritty widget Dec 30, 2020, 10:07 PM

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How low counts as low dim top?

marsh forge Dec 30, 2020, 10:07 PM

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It might be better for me to leave I think

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Sub 5

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Usually

elder yew Dec 30, 2020, 10:07 PM

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2, 3, 4

gritty widget Dec 30, 2020, 10:07 PM

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exotic spheres are out

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(possibly)

elder yew Dec 30, 2020, 10:07 PM

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Right now I only think about 2 and 3. 4 scares me

marsh forge Dec 30, 2020, 10:07 PM

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Exotic spheres aren’t out lol

elder yew Dec 30, 2020, 10:08 PM

#

have you taken a class from farb, max?

marsh forge Dec 30, 2020, 10:08 PM

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Two

elder yew Dec 30, 2020, 10:09 PM

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I'm reading one of his books, it's quite excellent

gritty widget Dec 30, 2020, 10:09 PM

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Do people expect dim 4 to have exotic spheres

marsh forge Dec 30, 2020, 10:09 PM

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i dont really know if people have strong opinions

gritty widget Dec 30, 2020, 10:09 PM

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or is there much consensus either way

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gg

marsh forge Dec 30, 2020, 10:09 PM

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im not a low dim person though

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i dont really care about manifolds or anything

gritty widget Dec 30, 2020, 10:09 PM

#

what kind of stuff are you into

elder yew Dec 30, 2020, 10:09 PM

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he answered that, homotopies!

marsh forge Dec 30, 2020, 10:10 PM

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i like generalized cohomology and stuff like that

gritty widget Dec 30, 2020, 10:10 PM

#

gg

elder yew Dec 30, 2020, 10:10 PM

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outside of topology, I like harmonic analysis and PDEs

marsh forge Dec 30, 2020, 10:10 PM

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obviously manifolds are often interesting examples for these things

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I'm slightly interested in the intersection of dynamics and AT

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although its gone out of fashion

gritty widget Dec 30, 2020, 10:11 PM

#

kind of stuff do you look at their

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just super general only obeying the eilneberg steenrod axioms

marsh forge Dec 30, 2020, 10:11 PM

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a good example is Schub's entropy conjecture (which was false as originally stated but people think something similar is true)

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Well normally I care about specific generalized cohomology theories

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like MU/KU etc

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I also care about equivariant stuff

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and am getting into CHT

marsh forge Dec 30, 2020, 10:13 PM

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marsh forge a good example is Schub's entropy conjecture (which was false as originally stat...

this is dynamics not coho

gritty widget Dec 30, 2020, 10:14 PM

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I have no idea what MU/KU and CHT are

tough imp Dec 30, 2020, 10:14 PM

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wtf is an exotic sphere

gritty widget Dec 30, 2020, 10:14 PM

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topologically the same as a sphere but not the same in terms of manifold structure

marsh forge Dec 30, 2020, 10:15 PM

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Smooth manifold homeomorphic to but not diffeomorphic to the standard sphere

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it is the same in terms of manifold structure

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just not smooth manifold structure

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most dimensions don't have them

tough imp Dec 30, 2020, 10:15 PM

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ohhh

#

I see

gritty widget Dec 30, 2020, 10:16 PM

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They form a group

tough imp Dec 30, 2020, 10:16 PM

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under?

gritty widget Dec 30, 2020, 10:16 PM

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connected sum

marsh forge Dec 30, 2020, 10:16 PM

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homotopy spheres are a group under connect sum

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with unit the standard sphere

gritty widget Dec 30, 2020, 10:16 PM

#

sorry yeah you need to include the standard sphere

tough imp Dec 30, 2020, 10:17 PM

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ah gotcha

gritty widget Dec 30, 2020, 10:17 PM

#

milnor must have felt like some king

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aftering figuring this out

tough imp Dec 30, 2020, 10:17 PM

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Like...

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J Milnor?

gritty widget Dec 30, 2020, 10:18 PM

#

yeap

tough imp Dec 30, 2020, 10:18 PM

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Oh wait

#

I'm thinking of J Milne

marsh forge Dec 30, 2020, 10:18 PM

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yes

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j milnor

gritty widget Dec 30, 2020, 10:18 PM

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as far as I know, the history story is

tough imp Dec 30, 2020, 10:18 PM

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Turns out that it's also J Milnor

#

lmfao

gritty widget Dec 30, 2020, 10:18 PM

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he was doing some stuff working under the assumption that homeomorphic to standard sphere implies diffeomorphic

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calculating some betti numbers or something

#

wasn't looking right

#

wham bam thank you maam

#

the thing must have not been diffeomorphic to a standard sphere

marsh forge Dec 30, 2020, 10:19 PM

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the Kervaire Milnor paper Groups of Homotopy Spheres is the first piece of the puzzle

#

Well the first piece of the general classification

#

Smale, Milnor, and someone else found some examples early on

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iirc

#

the history is kinda hard to remember exactly

#

the fun part is that these days the problem is basically a problem in stable homotopy theory

#

outside of dim 4

gritty widget Dec 30, 2020, 10:21 PM

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If you know about characteristic classes the papers aren't inpreitable

#

what is the suss with stable homotopy

marsh forge Dec 30, 2020, 10:22 PM

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the kervaire invariant one problem was recently solved for all dimensions except 126(?) using methods from stable homotopy theory

elder yew Dec 30, 2020, 10:23 PM

#

Smale's interviews are fun to watch

gritty widget Dec 30, 2020, 10:23 PM

#

Whoever was saying they like low dim top

elder yew Dec 30, 2020, 10:23 PM

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yA das me

gritty widget Dec 30, 2020, 10:23 PM

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Is this continous family of exotic R^4's hard to read about

marsh forge Dec 30, 2020, 10:23 PM

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Yeah okay counting diffeo classes of spheres has been reduced to computing the cokernel of the image of the J homomorphism up to the kervaire invariant problem

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the latter is almost solved

elder yew Dec 30, 2020, 10:23 PM

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I don't think about 4

marsh forge Dec 30, 2020, 10:24 PM

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and coker J is hard

elder yew Dec 30, 2020, 10:24 PM

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I think about 1, 2, and 3

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That's it

marsh forge Dec 30, 2020, 10:24 PM

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i think the sweet spot is to study cobordisms of 3 manifolds

#

that way you get just a hint of dim 4

obtuse meteor Dec 30, 2020, 10:25 PM

#

It’s astounding how hard low dimensional shit is

elder yew Dec 30, 2020, 10:25 PM

#

Yeah, I think the knot table goes up to crossing number 18 now

#

After that it's currently computationally infeasible

#

(Well maybe if some hardcore CS people dedicated more time to computing knots they could get a lot more, but it's probably not worth it, they have ads to run)

gritty widget Dec 30, 2020, 10:26 PM

#

Is it harder or can you ask harder questions?

marsh forge Dec 30, 2020, 10:26 PM

#

harder

gritty widget Dec 30, 2020, 10:27 PM

#

Is there something in particular about knots in 3 dimesion

marsh forge Dec 30, 2020, 10:27 PM

#

lots of 'high dimensional' results get killed in one sweep

elder yew Dec 30, 2020, 10:27 PM

#

There's lots of unreasonably hard questions in knot theory

marsh forge Dec 30, 2020, 10:27 PM

#

so like, iirc Poincare conjecture was solved for n>4 in one paper

#

n=2 in one paper and n=3 in a separate

obtuse meteor Dec 30, 2020, 10:27 PM

#

gritty widget Is there something in particular about knots in 3 dimesion

Philosophically there’s enough to tangle but not enough to like move around things

gritty widget Dec 30, 2020, 10:27 PM

#

yeah this is a good point

marsh forge Dec 30, 2020, 10:27 PM

#

well n=2 is trivial from the classification

#

but n=4 was perelman's famous work

obtuse meteor Dec 30, 2020, 10:28 PM

#

n = 4 was solved with a stronger conjecture right iirc?

#

That like every manifold can be cut up into pieces with some specified list of geometries

marsh forge Dec 30, 2020, 10:28 PM

#

oh i have no idea

obtuse meteor Dec 30, 2020, 10:28 PM

#

Something Thurston thought about

marsh forge Dec 30, 2020, 10:28 PM

#

the details are not something i know much about

elder yew Dec 30, 2020, 10:28 PM

#

The classification wasn't exactly trivial. n = 4 is still unsolved in certain cases

tight agate Dec 30, 2020, 10:28 PM

#

thurston geometrization?

obtuse meteor Dec 30, 2020, 10:28 PM

#

Ye

#

Geometrization

marsh forge Dec 30, 2020, 10:29 PM

#

the classification for n=2 was not trivial but poincare follows trivially

tight agate Dec 30, 2020, 10:29 PM

#

ye that's what perelman proved

obtuse meteor Dec 30, 2020, 10:29 PM

#

Perelman’s proof of n = 4 point are either directly used or like is simple to modify to get geometrization

gritty widget Dec 30, 2020, 10:29 PM

#

obtuse meteor Philosophically there’s enough to tangle but not enough to like move around thin...

So I know any knot be untangled if you give your self 4-dimensions to play with

tight agate Dec 30, 2020, 10:29 PM

#

geometrization implies poincare

obtuse meteor Dec 30, 2020, 10:29 PM

#

Ye

#

That’s what I thought

#

Thurston is a god Jesus

gritty widget Dec 30, 2020, 10:30 PM

#

but can you generalise knot to some co-dimesion 1 manifold in general and get interesting stuff

elder yew Dec 30, 2020, 10:30 PM

#

Even knot connect sum how it behaves with crossing number

#

Even the definition of crossing number is a bit trouble some

#

Since it relies on a projection

#

There's no intrinsic definition of crossing number, it's always done in relation to a regular projection

marsh forge Dec 30, 2020, 10:32 PM

#

All I want to do is compute $H^V(*)$

gentle ospreyBOT Dec 30, 2020, 10:32 PM

#

Ultimate TA Emailer

📎 158305429615673345.png

elder yew Dec 30, 2020, 10:32 PM

#

A topological understanding of the Alexander-Jones polynomial is still unknown

obtuse meteor Dec 30, 2020, 10:32 PM

#

elder yew A topological understanding of the Alexander-Jones polynomial is still unknown

Huge L

elder yew Dec 30, 2020, 10:32 PM

#

Yeah yeah

marsh forge Dec 30, 2020, 10:32 PM

#

Lmao

elder yew Dec 30, 2020, 10:32 PM

#

I made a funny mistake

#

I thought it was one guy

#

Yeah yeah hilarious

#

Anyway, I just read the statement of baum-connes

#

I don't understand it

#

It's very easy to ask a very hard question in knot theory

#

My prof. once had to construct a knot that had more crossings than there are atoms in the observable universe to find a counter example

#

to

#

w(K#J) = w(K) + w(J) - 2?

marsh forge Dec 30, 2020, 10:36 PM

#

is that still the best known counterexample

elder yew Dec 30, 2020, 10:36 PM

#

No, it was reduced down to a few thousand crossings now

#

He said if he gets bored one afternoon he will draw it

#

https://arxiv.org/pdf/1005.1359.pdf

#

Here's the paper if anyone is interested

tight agate Dec 30, 2020, 10:39 PM

#

colorful pictures

elder yew Dec 30, 2020, 10:40 PM

#

his recent papers have much better diagrams

little hemlock Dec 30, 2020, 10:57 PM

#

does anyone know what the f_* map refers to in rotman's AT?

📎 unknown.png

gritty widget Dec 30, 2020, 11:00 PM

#

In general

#

your X and Y are top spaces

#

H_0(X) and H_0(Y) are groups

#

f_* is the group homo induced by f

little hemlock Dec 30, 2020, 11:02 PM

#

you mean the image of f under the H_n functor?

gritty widget Dec 30, 2020, 11:02 PM

#

Yes

little hemlock Dec 30, 2020, 11:03 PM

#

ahh thanks!

gritty widget Dec 30, 2020, 11:03 PM

#

If you want to know what it is explicitly

#

Elements in H(X) are equivalence classes of paths

#

If p is a path in X, and you wan to make a path in Y the only thing you can really do is compose p and f

#

So f_* sends [p] to [p \circ f]

little hemlock Dec 30, 2020, 11:07 PM

#

yup, makes sense!

gritty widget Dec 31, 2020, 11:50 PM

#

Was it expected that the a proof of the geometreization conjecture would come before a proof of poincaire

elder yew Dec 31, 2020, 11:55 PM

#

I'm not sure, but it seems like an area that was making a lot of progress for a while

#

My professor was a grad student in LDT when perelman's proof was being verified, basically everyone that was interested in it knew nothing about ricci flow

gritty widget Jan 1, 2021, 12:02 AM

#

Where ldt?

elder yew Jan 1, 2021, 12:02 AM

#

low dimensional topology

#

He was at Santa Barbara

delicate hollow Jan 1, 2021, 12:24 AM

#

Question given:
Show that the set T containing ∅,ℝ, and all intervals (-∞,p) for p∈ℝ, is a topology on ℝ.

#

The first condition is satisfied because for X = ℝ, T contains ∅ and X=ℝ.

#

For the second condition, a finite intersection of members of T is also in T,
Let p∈ℝ and q = p+1 then intersecting (-∞,p) and (-∞,q) gives us (p,q)

#

but (p,q) isnt in T?

tight agate Jan 1, 2021, 12:28 AM

#

that is not the intersection

delicate hollow Jan 1, 2021, 12:28 AM

#

oh

#

its (-∞,p)

tight agate Jan 1, 2021, 12:28 AM

#

yes

delicate hollow Jan 1, 2021, 12:28 AM

#

so if im trying to show 2nd condition

#

I need to show for q > p and q < p

#

and that should be sufficient?

tight agate Jan 1, 2021, 12:30 AM

#

you need to show that for any finite collection (-infty, p_1), ..., (-infty, p_n), the intersection of those is of the form (-infty, x) for some real number x

#

or the empty set

#

or the whole thing

delicate hollow Jan 1, 2021, 12:30 AM

#

oh ok

#

its trivial if the finite collection is empty or the whole thing

#

so the main part of the proof is showing that for a finite collection (-infty, p_1), ..., (-infty, p_n),

#

and showing that its intersection is an open set in the topology

#

am I right?

tight agate Jan 1, 2021, 12:31 AM

#

yes

delicate hollow Jan 1, 2021, 12:32 AM

#

thanks bro

#

Is it proof enough to say that Given a set {p_n|p_i are reals} $\bigcap\limits_1^n (-\infty,p_i) = (-\infty,sup({p_n}))$

#

lol

#

Is it proof enough to say that Given a set {p_n|p_i are reals} $\bigcap\limits_1^n (-\infty,p_i) = (-\infty,sup({p_n}))$

gentle ospreyBOT Jan 1, 2021, 12:44 AM

#

Marlin Flier
Compile Error! Click the errors reaction for more information.
(You may edit your message to recompile.)

📎 770350381435322418.png

delicate hollow Jan 1, 2021, 12:45 AM

#

wait not again

sleek thicket Jan 1, 2021, 12:45 AM

#

Nope

delicate hollow Jan 1, 2021, 12:45 AM

#

lub

#

wait

#

inf

sleek thicket Jan 1, 2021, 12:45 AM

#

You can't just work with sequences

#

wait hang on

#

Sorry I thought you were looking at the union

delicate hollow Jan 1, 2021, 12:45 AM

#

yeah my bad

#

i dont know if my message came across clearly

sleek thicket Jan 1, 2021, 12:46 AM

#

I think it's simpler to just work with two sets in the topology

#

Instead of an arbitrary finite number of them

#

So $(-\infty, p) \cap (-\infty, q) $

delicate hollow Jan 1, 2021, 12:47 AM

#

$(-\infty, p) \cap (-\infty, q) = (-\infty,min(p,q))$

gentle ospreyBOT Jan 1, 2021, 12:47 AM

#

Marlin Flier

📎 770350381435322418.png

sleek thicket Jan 1, 2021, 12:47 AM

#

Right

delicate hollow Jan 1, 2021, 12:47 AM

#

then to argue that to arbitrary finite union what would I say>?

#

or I dont argue that at all and asume it to be somewhat trivial?

#

Would the flow of argument be to argue that:

#

$(-\infty, p) \cap (-\infty, q) = (-\infty,min(p,q))$ and $(-\infty, p) \cap \mathbb{R} = (-\infty,p)$ and $(-\infty, p) \cap \emptyset = (-\infty, p)$ so an arbitrary number of intersections of any of these elements will create some sort of repeating cycle that gets elements already in the set?

gentle ospreyBOT Jan 1, 2021, 12:51 AM

#

Marlin Flier

📎 770350381435322418.png

delicate hollow Jan 1, 2021, 12:52 AM

#

last one meant to be emptyset

#

not sure how to elegantly put the

so an arbitrary number of intersections of any of these elements will create some sort of repeating cycle that gets elements already in the set?

part. @sleek thicket

sleek thicket Jan 1, 2021, 12:55 AM

#

umm basically if you have open sets $U_1,\ldots, U_n$ then you can write their intersections as $U_1 \cap (U_2 \cap (\ldots \cap (U_{n-1} \cap U_n)\ldots)$ and say that $U_{n-1} \cap U_n$ is in your topology because it's a binary intersection, $U_{n-2} \cap (U_{n-1} \cap U_n)$ is for the same reason, and so on. Formally you'd phrase this as an inductive proof

gentle ospreyBOT Jan 1, 2021, 12:55 AM

#

Schamrock

📎 228562950703546368.png

delicate hollow Jan 1, 2021, 12:57 AM

#

is this not sorta obvious?

#

do i need to give this argument

#

or is it enough to just show

📎 unknown.png

meager python Jan 1, 2021, 9:59 AM

#

Eh very basic question. Assume X is a projective hypersurface that’s reducible. Does it follow that X = V(f) = V(g) \cup V(h) for some g,h? (We can assume algebraically closed if it’s needed)

gritty widget Jan 1, 2021, 11:36 AM

#

I don’t think you are even have X=V(f)?

#

Plenty of varieties can’t be written as the zero set of just one equation

meager python Jan 1, 2021, 2:02 PM

#

An algebraic hypersurface can be

#

Depending on the definition. But assume that X is defined by a principal ideal

high monolith Jan 1, 2021, 2:45 PM

#

I’m pretty sure V(f) is reducible iff f is reducible, so it should be true given the assumption

summer jolt Jan 1, 2021, 3:34 PM

#

Could anyone explain why this identity is true?

📎 unknown.png

#

I'm having trouble understanding the notation on the RHS as "composition of pullbacks"

#

Ok my confusion is about the fact that we regard psi_2* as a function.

#

Definition of pullback that was given to me was psi*f = f composed psi

#

but psi is a function but not psi* right?

#

@gritty widget yes, my confusion was indeed that I didn't think of the pullback as a mapping but as some kind of weird operator

pastel linden Jan 1, 2021, 6:48 PM

#

this is a stupid question

#

I'm trying to prove that the local matrix of the differential induced by a smooth map between manifolds is independent of coordinate charts

#

But I'm failing to see how to prove this fact

#

wait I missed an obvious fact never mind

gritty widget Jan 1, 2021, 7:31 PM

#

is it? thonk

pastel linden Jan 1, 2021, 7:32 PM

#

yeah it's not I had a brain fart

gritty widget Jan 1, 2021, 7:32 PM

#

i know it's rank ~~and, if the manifolds have the same dimension, it's determinant,~~ is well defined

pastel linden Jan 1, 2021, 7:33 PM

#

that is the true part

#

I misinterpreted independence of the differential as an object from coordinate charts as "local matrices are the same" which is just obviously false by the word "local"

pastel linden Jan 1, 2021, 7:35 PM

#

gritty widget i know it's rank ~~and, if the manifolds have the same dimension, it's determina...

is this because the differential becomes an isomorphism of the tangent spaces that the determinant is well defined?

gritty widget Jan 1, 2021, 7:36 PM

#

wait i might be having a smooth brain moment

#

you need a linear map to go from a vector space to itself to have a determinant, sorry

#

i was thinking of invertibility (which is subsumed by the rank condition)

pastel linden Jan 1, 2021, 7:37 PM

#

makes sense

gritty widget Jan 1, 2021, 7:38 PM

#

hmmm

sleek thicket Jan 2, 2021, 3:12 AM

#

ttera has done too much with manifolds

#

His brain is now smooth

river granite Jan 2, 2021, 1:45 PM

#

C^\infty brain moment

plain raven Jan 2, 2021, 5:47 PM

#

My brain is literally a perfect sphere. Scientists don't understand how such a flawless geometric figure could have even arisen through biological processes

river granite Jan 2, 2021, 5:48 PM

#

exotic sphere brain hmmm

vast estuary Jan 3, 2021, 9:14 AM

#

Hello, can someone help me with the following theorem reformulation please? It's been bothering me for a while now. It's related to convex geometry and I hope this is the right place to post:

#

https://math.stackexchange.com/questions/3970267/reformulation-of-a-theorem-about-the-lower-bound-on-no-of-facets-of-a-symmetric

Mathematics Stack Exchange

Reformulation of a theorem about the lower bound on no. of facets o...

Preliminaries

The distance $d(K, L)$ between symmetric convex bodies $K$ and $L$ is the least positive $d$ for which there is a linear image $\tilde L$ of $L$ such that $\tilde L ⊂ K ⊂ d\tilde L$....

willow spear Jan 4, 2021, 3:04 AM

#

how would one prove that an open interval is homeomorphic to the real number line?

#

like what would be teh exact steps?

#

im sort of confused

gritty widget Jan 4, 2021, 3:05 AM

#

define a homeomorphism from that open interval to R

#

as in

willow spear Jan 4, 2021, 3:05 AM

#

how does that work, all I know is teh defintion of a homeomorphism

#

this sio my first time applying it

#

*is

gritty widget Jan 4, 2021, 3:06 AM

#

come up with a function from your open interval to R which is
-continuous,
-bijective,
-and has a continuous inverse

willow spear Jan 4, 2021, 3:07 AM

#

is that it?

gritty widget Jan 4, 2021, 3:07 AM

#

yes

willow spear Jan 4, 2021, 3:07 AM

#

I dont need to prove anythign right?

gritty widget Jan 4, 2021, 3:07 AM

#

sully well you should prove that your function is actually a homeomorphism

willow spear Jan 4, 2021, 3:07 AM

#

and how are you just supposed to come up with a function

gritty widget Jan 4, 2021, 3:08 AM

#

well you can start by looking at some examples of functions that take open intervals to all of R

willow spear Jan 4, 2021, 3:09 AM

#

gritty widget <:sully:651816820122189834> well you should prove that your function is actually...

by this you just mean going through the checklist right? Not like a formal proof?

gritty widget Jan 4, 2021, 3:09 AM

#

that's what a "formal proof" is

willow spear Jan 4, 2021, 3:09 AM

#

lol ok

gritty widget Jan 4, 2021, 3:10 AM

#

like it might be easier to start by finding one that works for something simpler like (0, 1), and then it should be pretty clear how to make one for any open interval of the form (a, b)

willow spear Jan 4, 2021, 3:10 AM

#

The inteval must have values right liek (2,3) or (5, 7) right? It can't just be a and b

gritty widget Jan 4, 2021, 3:10 AM

#

(of course tht's in the case of a bounded one)

rotund thicket Jan 4, 2021, 3:10 AM

#

make sure to include "this follows trivially" and then it is a formal proof

gritty widget Jan 4, 2021, 3:10 AM

#

willow spear The inteval must have values right liek (2,3) or (5, 7) right? It can't just be ...

post the full question

#

as in

#

the source

willow spear Jan 4, 2021, 3:11 AM

#

k

#

From allen hatcher's text:

#

📎 unknown.png

rotund thicket Jan 4, 2021, 3:14 AM

#

i'd just start with a simple open interval like (0,1) or (-1,1)

willow spear Jan 4, 2021, 3:14 AM

#

ok so then would I just magically write down a formula lol?

rotund thicket Jan 4, 2021, 3:14 AM

#

yeah pretty much

willow spear Jan 4, 2021, 3:14 AM

#

bruh

rotund thicket Jan 4, 2021, 3:15 AM

#

i mean you're doing like an existence proof

willow spear Jan 4, 2021, 3:15 AM

#

i meant function, not formula my bad

rotund thicket Jan 4, 2021, 3:15 AM

#

just go into a graphing calculator and think of some cool functions that work

#

idk if you're familiar with the sigmoid function from ML

#

that does what you want

#

but backwards

willow spear Jan 4, 2021, 3:15 AM

#

so like would y = x^2 work?

rotund thicket Jan 4, 2021, 3:16 AM

#

well what open interval would map to the entirety of R?

#

(a,b) -> (a^2, b^2)

#

probably some rational function would work

willow spear Jan 4, 2021, 3:17 AM

#

oh ok

#

so once i find a function, do i need ot generalize it in terms of a and b?

rotund thicket Jan 4, 2021, 3:18 AM

#

yeah probably

willow spear Jan 4, 2021, 3:19 AM

#

Woudl u mind guiding me through this problem, this is my first experience in Topology and it'd be nice for a walkthrough

rotund thicket Jan 4, 2021, 3:19 AM

#

📎 unknown.png

#

this might be just some random idea

#

see how the middle part of the function maps (-1, 1) to (-1, -infinity)

#

it isn't bijective but

willow spear Jan 4, 2021, 3:20 AM

#

yee

rotund thicket Jan 4, 2021, 3:20 AM

#

like if you could get something that looks like x^3

willow spear Jan 4, 2021, 3:21 AM

#

so essentially any interval that is taken on the function that you find shoudl map to (-infinity, infinty)?

#

or is it only teh interval u are looking for?

rotund thicket Jan 4, 2021, 3:22 AM

#

well a homeomorphism is a function

willow spear Jan 4, 2021, 3:22 AM

#

yes

rotund thicket Jan 4, 2021, 3:22 AM

#

so just for exploratory purposes just think of a simple open interval

#

and find the function which satisfies the properties of a homeomorphism

#

hint: maybe something trigonometric might be good

#

@willow spear

willow spear Jan 4, 2021, 3:32 AM

#

alright gotcha

little hemlock Jan 4, 2021, 7:04 AM

#

📎 unknown.png

#

dumb question, but how does this definition ensure that $$\partial'n(S_n') \subset S{n-1}'?$$

gentle ospreyBOT Jan 4, 2021, 7:05 AM

#

kxrider

📎 195237017049759744.png

tough imp Jan 4, 2021, 7:38 AM

#

It’s defined as a map into S’_n-1

#

Like... the fact you draw an arrow from S’_n -> S’_n-1 means it’s a map between those two

#

And in addition you require that that map be obtained by restriction

little hemlock Jan 4, 2021, 7:39 AM

#

right, but it just comes from restricting the original map to the domain of the new subgroup

tough imp Jan 4, 2021, 7:40 AM

#

Right, but I think the fact that it’s a sub_complex_ implies it’s still a complex

#

Aka that the map actually maps into the right thing

#

You just require in addition

#

That the maps be obtained via restriction

#

Altho I guess it’s not clear via that

#

That that’s the case which is kinda pepega...

#

I think it’s implicit in that that a subcomplex is still a complex but it doesn’t really state it

little hemlock Jan 4, 2021, 7:43 AM

#

i mean, lets say i have a part of a complex S_n -> S_{n-1} -> ......
and d_n : S_n to S_{n-1} is a nonzero map. There would be no way to make a subcomplex where S_{n-1}' = 0 and S_n' = S_n.
So you cant just choose any sequence of subgroups to make a subcomplex. Is that correct?

tough imp Jan 4, 2021, 7:45 AM

#

Right it’s not a complex then

#

I think the description should be like

#

A subcomplex is a complex where each S’_n is a subgroup of S_n and the maps are obtained by restriction

#

Then by requiring it be a complex we know that the codomains work as needed

little hemlock Jan 4, 2021, 7:47 AM

#

ah okay, that makes sense to me. thanks

obtuse meteor Jan 4, 2021, 6:42 PM

#

little hemlock

does this reduce to it just having a monic from S' to S

#

it looks like it does

#

and that's a lot easier to explain / read imo

marsh forge Jan 4, 2021, 6:47 PM

#

yeah coho

#

a subcomplex is just a complex w a monic map of complexes

#

where the commutativity data is encoded in the second half of that

willow spear Jan 5, 2021, 12:26 AM

#

what does a polart rectabgle look like?

marsh forge Jan 5, 2021, 12:26 AM

#

?

willow spear Jan 5, 2021, 12:27 AM

#

polar rectangles

#

?

cloud owl Jan 5, 2021, 12:27 AM

#

define rectangle

willow spear Jan 5, 2021, 12:27 AM

#

here ill send u a screenshot

#

📎 unknown.png

cloud owl Jan 5, 2021, 12:29 AM

#

ah, so this will just be like the region where x =< theta =< y and a =< r =< b

willow spear Jan 5, 2021, 12:29 AM

#

so does it pictorially look like a rectangle or something?

cloud owl Jan 5, 2021, 12:29 AM

#

because here they mean a rectangle in the sense of two variables varying

#

it's like, a rectangle at the origin, x could be going from -1 to 1, y could be going from -3 to 3

#

cartesian, that is

#

but these 'rectangles', they're varying over r and theta

willow spear Jan 5, 2021, 12:30 AM

#

excluding 0 right?

cloud owl Jan 5, 2021, 12:31 AM

#

uhhh including 0, for cartesian

willow spear Jan 5, 2021, 12:31 AM

#

oh

cloud owl Jan 5, 2021, 12:31 AM

#

for the polar ones, r =/= 0, but they've covered that because the 'rectangles' are open

willow spear Jan 5, 2021, 12:31 AM

#

kk

#

bruh topology is hard to understand lol

marsh forge Jan 5, 2021, 1:06 AM

#

📎 wrapping_paper.png

ivory dragon Jan 5, 2021, 1:16 AM

#

this is why my present boxes are nonorientable

obtuse meteor Jan 5, 2021, 1:42 AM

#

willow spear so does it pictorially look like a rectangle or something?

it pictorially looks like a slice of pie

#

a section of some circle

#

well

#

more like an annulus

heady grove Jan 5, 2021, 1:42 AM

#

i love you guys

obtuse meteor Jan 5, 2021, 1:42 AM

#

like, first you chop a slice of pie out of the plane

#

then you cut it off with circular cutty bois

#

Like the gray part of this picture

📎 iu.png

sleek thicket Jan 5, 2021, 4:12 AM

#

heady grove i love you guys

i love me too

tacit stratus Jan 5, 2021, 8:59 AM

#

Why is this true? It feels like it should be pretty clear, but I just can’t see it.
Here, M is a riemann surface, K is its canonical line bundle, and P is a principle G-bundle over M
Shouldn’t it give a map ad P*^d —> /mathbb{C}?

📎 Screen_Shot_2021-01-05_at_3.49.19_AM.png

sonic hill Jan 5, 2021, 2:37 PM

#

Looking at fibers, a homogenous polynomial of degree $d$ maps $\mathfrak g$ to $\bC$. The effect of tensoring by $K$ makes the right hand side tensor by $K^{\otimes d}$

gentle ospreyBOT Jan 5, 2021, 2:38 PM

#

Icy001

📎 143174781561339904.png

tacit stratus Jan 5, 2021, 6:50 PM

#

ahhh okay thank you

bright acorn Jan 5, 2021, 7:52 PM

#

I was thinking about this problem, what are some necessary conditions, the least the best, one should impose on your space in other for it to always have a basis?

meager python Jan 5, 2021, 8:07 PM

#

Eh, how does one check practically that certain points are in "general position" in a projective space?

tight agate Jan 5, 2021, 8:26 PM

#

in affine space you can do it by calculating the rank of some matrix

#

so restricting to some affine open and checking might work

meager python Jan 5, 2021, 8:30 PM

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Is it really the rank of a matrix? I barely remember what it means in linear algebra

tight agate Jan 5, 2021, 8:31 PM

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ig you need to translate them to turn it into a matrix calculation

sleek thicket Jan 5, 2021, 8:32 PM

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Tolaria, iirc general position is a statement about linear or affine independence

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At least when you're talking about a set of points being in general position

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so the fact that it comes down to computing the rank of a matrix shouldn't be too surprising

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since that's exactly telling you that things are linearly independent

meager python Jan 5, 2021, 8:32 PM

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But it says something like any m points are linearly independent out of say n

tight agate Jan 5, 2021, 8:33 PM

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Say we have 3 points in R2

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and we want to check if they're colinear

meager python Jan 5, 2021, 8:33 PM

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Rank of a matrix = dimension of the image?

sleek thicket Jan 5, 2021, 8:33 PM

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Yes, and also the number of linearly independent columns (the equivalence of this with what you said should make sense)

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Or the number of linearly independent rows

meager python Jan 5, 2021, 8:34 PM

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Ok so how does this definition translate into projective space?

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Would I have to check on all affine cover elements?

sleek thicket Jan 5, 2021, 8:35 PM

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I'm not sure, but that's sort of my intuition. I think you could also translate it into a statement about whether these n lines lie on a common plane in A^(n+1)

meager python Jan 5, 2021, 8:35 PM

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By the affine cone or something?

sleek thicket Jan 5, 2021, 8:35 PM

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Right

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Like, general position is a statement about how many of the points lie on a given line, right? And lines in P^n are planes through the origin one dimension up

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unless I'm getting mixed up lol

meager python Jan 5, 2021, 8:36 PM

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Yeah

sleek thicket Jan 5, 2021, 8:37 PM

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I'm not really sure how to do this in projective space, sorry. I was just trying to say earlier why it makes sense that it would come down to computing the rank of a matrix

meager python Jan 5, 2021, 8:37 PM

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Yeah thanks, I never did classic linear algebra

tight agate Jan 5, 2021, 8:38 PM

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What exactly is your definition of general position?

meager python Jan 5, 2021, 8:38 PM

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I have 6 points in P^2 and want to check if they're in general position. So would it be like checking if any 3 of them are linearly independent?

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That's the thing, I don't know. Lol

sleek thicket Jan 5, 2021, 8:38 PM

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so it's not exactly linear independence

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Like three points in R^2 can be in general position

tight agate Jan 5, 2021, 8:39 PM

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because there isnt a single definition

sleek thicket Jan 5, 2021, 8:39 PM

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If we stick to A^n it's about affine independence

tight agate Jan 5, 2021, 8:39 PM

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subtract one of those points from the other two to turn it into a linear independence question

meager python Jan 5, 2021, 8:39 PM

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📎 Screenshot_2021-01-05_at_21.39.34.png

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This is from Gathmann and the only place where it mentions general position

sleek thicket Jan 5, 2021, 8:40 PM

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So x0, x1,...,xn are affinely independent if {x1-x0,...,xn-x0} are linearly independent, equivalently no single line in A^k goes through more than 2 of them

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note that we are not restricting to lines through the origin

meager python Jan 5, 2021, 8:40 PM

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I hate how classic algebraic geometry is just linear algebra, like literally linear algebra

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Ah

tight agate Jan 5, 2021, 8:41 PM

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meager python I have 6 points in P^2 and want to check if they're in general position. So woul...

sometimes in this context, general position can mean that those 6 points do not lie on a conic

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it really depends on what they mean by general position

tight agate Jan 5, 2021, 8:44 PM

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meager python

in this case it probably means that they intersect at 5 distinct points

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and the uniqueness follows from Bezout

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oh wait tangent to all of them nvm

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for any five points there is at most one conic

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why is there only one choice of 5 points hmmm

meager python Jan 5, 2021, 8:52 PM

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"In a projective space of dimension d (projective dimension), a set of points is usually considered to be in general position if no d+1 are contained in a hyperplane."

I found this on mathstack. So what does it mean in terms of say rank of a matrix?

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For say d = 2, number of points = 6

tight agate Jan 5, 2021, 8:53 PM

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so we want no 3 of them to lie on a line

meager python Jan 5, 2021, 8:53 PM

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I literally feel like a calculus student right now thonkstein

tight agate Jan 5, 2021, 8:54 PM

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there is a unique line going through two of them

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so you just need to check if the third one is on the line in this case

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but if we want a method that works in general

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find a line that misses all 3

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the complement is an affine space

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subtract one of the points from the other two

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now we have 2 vectors in C^2

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check if the matrix has nonzero determinant

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that should do it

meager python Jan 5, 2021, 8:58 PM

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So you have to do that for all combinations? Or will those determinants be some minors of a larger matrix

tight agate Jan 5, 2021, 8:59 PM

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umm yeah, probably just do it for all collections of 3 points among those 6

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did you do the conic tangent to 5 lines problem?

meager python Jan 5, 2021, 9:01 PM

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Nah

tight agate Jan 5, 2021, 9:02 PM

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I wonder if there's an easy way to turn it into a problem about bundles

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we're asking for a unique section of O(2) tangent to 5 lines hmmm

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or perhaps some easy dimension computation

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conics in P2 are parameterized by P5

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as we have x^2, y^2, z^2, xy, yz. xz

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for a conic to be tangent to the line at some point p, ig we want the conic to vanish to at least order 2

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so maybe each thing corresponds to a hypersurface in P5

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so roughly we're looking at the intersection of 5 hypersurfaces in P5

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which is a point?

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which corresponds to a conic

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something like that I assume

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so there's just one conic tangent to five lines

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but 3264 conics tangent to 5 conics

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weird

meager python Jan 5, 2021, 9:14 PM

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I like that book, 3264 and all that

tight agate Jan 5, 2021, 9:15 PM

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it looks super interesting

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I might read it sometime next year

meager python Jan 5, 2021, 9:17 PM

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I tried to read small parts to get motivation for the hilbert scheme

tight agate Jan 5, 2021, 9:17 PM

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did it work?

meager python Jan 5, 2021, 9:18 PM

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Didn't really have the time, but it was very "geometric" so I might try it again at some point

tight agate Jan 5, 2021, 9:18 PM

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the hilbert scheme shouldnt be hard to motivate anyway

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it's just that the construction is super technical

meager python Jan 5, 2021, 9:19 PM

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Well motivation for the technical definition. I tried to work some easy cases explicitly

tight agate Jan 5, 2021, 9:20 PM

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noice

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do they construct it in 3264 and all that?

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looks like they just sketched it

meager python Jan 5, 2021, 9:24 PM

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Only for projective spaces afair

tight agate Jan 5, 2021, 9:26 PM

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ye the general construction requires a whole bunch of vanishing theorems

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I think Kollar does it in his book

meager python Jan 5, 2021, 9:29 PM

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Ah, which one?

tight agate Jan 5, 2021, 9:30 PM

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rational curves on algebraic varieties

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or something like that

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lectures on curves on an algebraic surface also does it

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both hilbert and picard schemes

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and proves that they're smooth at points corresponding to 1-regular curves

shut moat Jan 5, 2021, 10:10 PM

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Can I get a hint on how to approach this?

📎 Screen_Shot_2021-01-05_at_5.10.47_PM.png

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for context, definition 5.2.3 is:

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📎 Screen_Shot_2021-01-05_at_5.11.20_PM.png

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now it's fairly easy to show that it's a smooth 1-manifold because it's a union of opens subsets of R, but I couldn't think of any way to make a [relaxed] parametrization

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possibly related to my inability to visualize this manifold lol
(and the identity map doesn't work here because vol_1(∂U) != 0)

gritty widget Jan 7, 2021, 9:17 PM

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I don't see why the identity map doesn't work? The boundary of U is some discrete collection of points? So long as there is no continuous strip this has zero volume\

shut moat Jan 7, 2021, 9:25 PM

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📎 Screen_Shot_2021-01-07_at_4.25.20_PM.png

sleek thicket Jan 8, 2021, 1:22 AM

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Let $G$ be a Lie group and $\rho : G \to \mathrm{GL}_n(\R)$ a continuous representation. Is $\rho$ necessarily smooth? If not, can someone provide a counterexample?

gentle ospreyBOT Jan 8, 2021, 1:22 AM

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shamrocK^n(X)

📎 228562950703546368.png

sleek thicket Jan 8, 2021, 1:22 AM

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Sorry ~S^1, I thought about your problem and nothing popped out at me

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@gritty widget you've become Lie pilled right

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over break

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solve

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now

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🔫

gritty widget Jan 8, 2021, 1:39 AM

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i do not know how to answer this

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sorry

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i am only lee irm chapter 3 pilled

tight agate Jan 8, 2021, 3:10 AM

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sleek thicket Let $G$ be a Lie group and $\rho : G \to \mathrm{GL}_n(\R)$ a continuous represe...

The wikipedia page on Lie groups claims that any continuous homomorphism between lie groups is real analytic

sleek thicket Jan 8, 2021, 3:10 AM

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oooohhhhhh

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Very interesting

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I did not know that

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This is also not helpful for my actual question I care about

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huh, any topological group which is a topological manifold has a unique analytic lie group structure

obtuse meteor Jan 8, 2021, 3:12 AM

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that's weird and cool

sleek thicket Jan 8, 2021, 3:12 AM

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Yeah haha

obtuse meteor Jan 8, 2021, 3:12 AM

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feels like a generalization of what happens for complex geometry stuff

sleek thicket Jan 8, 2021, 3:12 AM

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Faye you might like the question that inspired this

marsh forge Jan 8, 2021, 3:12 AM

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wait sham

obtuse meteor Jan 8, 2021, 3:12 AM

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me
liking Lie groups

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bold claim

marsh forge Jan 8, 2021, 3:13 AM

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actually nvm

obtuse meteor Jan 8, 2021, 3:13 AM

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but I'll hear it out

marsh forge Jan 8, 2021, 3:13 AM

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my question was dumb

sleek thicket Jan 8, 2021, 3:13 AM

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wtf lie groups are based

marsh forge Jan 8, 2021, 3:13 AM

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lie groups are lame

obtuse meteor Jan 8, 2021, 3:13 AM

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they're cool

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I just don't know anything about them

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I am too dummy

sleek thicket Jan 8, 2021, 3:13 AM

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but also the only lie group I'm talking about is GL(n, R) and GL(n, C)

marsh forge Jan 8, 2021, 3:13 AM

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compact Hausdorff Groups are cool and good

obtuse meteor Jan 8, 2021, 3:13 AM

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H-groups only

sleek thicket Jan 8, 2021, 3:13 AM

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Let $\mathsf{Vect}$ denote the category of finite dimensional vector spaces over $\R$ or $\C$

gentle ospreyBOT Jan 8, 2021, 3:13 AM

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shamrocK^n(X)

📎 228562950703546368.png

obtuse meteor Jan 8, 2021, 3:13 AM

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I reject differentiability, continuity is my friend now

sleek thicket Jan 8, 2021, 3:13 AM

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equivalently the category of matrices

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this is enriched in top and diff

obtuse meteor Jan 8, 2021, 3:14 AM

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mhm

sleek thicket Jan 8, 2021, 3:14 AM

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So we have a notion of a continuous or smooth endofunctor

obtuse meteor Jan 8, 2021, 3:14 AM

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yes

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all of them are that non?

sleek thicket Jan 8, 2021, 3:14 AM

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Note that continuous does not mean preserves categorical limits

obtuse meteor Jan 8, 2021, 3:14 AM

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oh endofunctor

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nvm gotcha

sleek thicket Jan 8, 2021, 3:14 AM

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yeah, or between R Vect and C Vect or whatever

obtuse meteor Jan 8, 2021, 3:14 AM

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you mean a diff-enriched endofunctor

sleek thicket Jan 8, 2021, 3:15 AM

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Yeah, or Top-enriched

obtuse meteor Jan 8, 2021, 3:15 AM

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mhm

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got it

sleek thicket Jan 8, 2021, 3:15 AM

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is a continuous functor automatically smooth?

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Just for endofunctors of Vect

obtuse meteor Jan 8, 2021, 3:15 AM

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if not I doubt the viability of modern mathematics