Mathematics

5:00 PM

you're welcome!

@BalarkaSen OK how about this question

I have a geometric line segment in $\Bbb R^3$, and I'm embedding a disc so that the line segment is a part of the boundary

How bad can that embedding be near the segment?

Balarka Sen

Extremely wild I think

Imagine a Cantor set many Alexander horns popping out as you approach the boundary

Akiva Weinberger

I guess I wanna $\epsilon$-approximate it with something nice

Those Alexander horns gotta shrink

Balarka Sen

My worry is that after approximating you only get some immersed smooth S^1 x I

Akiva Weinberger

I can triangular domainland so that the output of each section has tiny diameter or something

Oh yeah I'm assuming the boundaries of S^2 x I are already smooth so this won't change them too much. Otherwise the ribboned F-A arc minus a disc fucks us

Balarka Sen

5:09 PM

Yeah got your assumptions

Ted Shifrin

Howdy, DogAteMy and a @Balarka.

Akiva Weinberger

So what's your worry

Heyo

Oh immersed

Sorry misread

Yeah 100%

Lukas Heger

hi @Ted

Ted Shifrin

Hi @Lukas!

Lukas Heger

I got a job where I design exercise sheets for a masters algebraic number theory course, really fun stuff

Akiva Weinberger

5:13 PM

Can I always triangulate stuff so that no triangle is in any other triangle's convex hull or something @BalarkaSen

Ted Shifrin

Writing (and grading) interesting homeworks seems to be something professors hate to do. I always enjoyed it, especially the former. Of course, if there's a text with excellent exercises, that makes it easy, but often there isn't.

When I taught out of Griffiths/Harris, I had to write lots of problems. Some were routine verifications, but some were interesting :)

Akiva Weinberger

Hm actually I don't think that's useful

Balarka Sen

@AkivaWeinberger Isnt it true that if you have an immersed disk bounding a knot in S^3 with an embedded collar then its actually an unknot.

a la Papakyriakapoulous

leslie townes

one of my least favorite jobs was grading homework for a prof who assigned small amounts of homework every monday, wednesday, and friday. he put no effort into it and it was obvious. students experienced whiplash. the scores provided no information.

Balarka Sen

Maybe something like this is true for an annulus as well

Lukas Heger

5:15 PM

@Ted I'm kind of sad that when we'll get to discrete valuations, I can't make exercises about how a vanishing order/pole order at a point on a Riemann surface gives you a discrete valuation

I find that really helpful to explain the terminology "ramification" as well

Ted Shifrin

Agreed, that is the example that motivates everything.

You can give the algebraic set-up and put a remark about the complex curve case.

Lukas Heger

yeah maybe I'll do something like that

the thing is those students haven't necessarily taken complex analysis or algebraic geometry

Ted Shifrin

Or maybe an optional problem ... "For those of you who've seen Riemann surfaces ..."

Yeah, but you don't need much. It's just a local holomorphic coordinate. You can just use that as an Ansatz.

You certainly don't need algebraic geometry for this. And everyone knows what a smooth manifold is since their first year in Germany ...

fin

hi teddd

Ted Shifrin

Hi @fin

Akiva Weinberger

5:18 PM

4 mins ago, by Balarka Sen

@AkivaWeinberger Isnt it true that if you have an immersed disk bounding a knot in S^3 with an embedded collar then its actually an unknot.

Wait really?

Wow that sounds super useful

Ted Shifrin

Sorry I couldn't do quantum mechanics out of context.

Balarka Sen

Yeah, that's the disk theorem [See en.wikipedia.org/wiki/Dehn%27s_lemma, although they state it for PL disks, it's true for smooth ones as well]

fin

@TedShifrin lmao dont worry

Akiva Weinberger

Yeah OK I'll look at a proof later but I kinda assume that sort of proof strategy is gonna work here

Balarka Sen

Yeah

Vrouvrou

5:30 PM

@LukasHeger sorry to ask you again but if $u_n$ converge weakly to u in W^{1,p}(\Omega)$ how to get that $\nabla u_n(x)\to \nabla u(x)$ a.e

SMC not Sven Magnus Carlsen

Could anyone try to help me understand why in the proof of theorem 3.2 page 13 (here numdam.org/item/10.5802/aif.2098.pdf) is it so obvious that $\pi_{1}(C)\simeq H$? For what I have understood it should be $p*(\pi_{1}(C))\simeq H$

thank you in advance

Or if it is not that how can i obtain a surjective homomorphism from $H$ having one from $C$? For how it is left is should be trivial but I could not find any argument

Vrouvrou

if $u_n \to u \in L^p(\Omega)$ do we have that $\nabla u_n\to \nabla u$ dans L^p(\Omega) ?

SMC not Sven Magnus Carlsen

ok im dumb, p* is always injective

Akiva Weinberger

0

Q: If an embedding of $S^2\times\{I\}$ has nice edges, are the edges ambiently isotopic?

Let $f_t(x)$, $~t\in I$, $~x\in S^1$ be an embedding of $f:S^2\times I\to\Bbb R^3$. (Thus for example the images of $f_{t_0}$ and $f_{t_1}$ are disjoint if $t_0\ne t_1$.) It's known that the images of $f_0$ and $f_1$ need not be ambiently isotopic. An easy construction of this is to "ribbon-ify" ...

@BalarkaSen Outsourcing to main

Peter John

math.stackexchange.com/q/4288020/922120 Is there anyone can answer this question? I wrote the statement for general topological space but it may not true in general. As @love_sodam advised, I first tried to prove the statement of simplicial complex. But even though I restrict the space, I can't prove. And idea for this problem?

Balarka Sen

5:43 PM

Upvoted

Akiva Weinberger

Hm should I change $x$ to $\theta$

nah it's fine

Vrouvrou

6:06 PM

someone have an idea :math.stackexchange.com/questions/4290222/…