point-set-topology

[nimble portal]

Argh

This problem with the closed ball is stumping me

Just because of the origin

I feel like I found charts that show what the problem wants but their images just aren’t open because of the stupid origin LMFAO

[nimble portal]

I caved and looked at MSE

I didn’t read a solution but I read a comment to one and it matched my intuition/my solution approach

Just a tiny difference

So their version says f(U) = R^(n-1) x (-infty, 0]

I got f(U) = R^(n-1) x (-1, 0]

🤔

One thing I don’t get is that they say since f is the composition of continuous open maps, it’s also a continuous open map, but R^(n-1) x (-infty, 0] isn’t open right?

[urban zinc]

Wdym what are you trying?

[urban zinc]

open in what?

[nimble portal]

One second

So my idea was

The image is (ax^1, ..., ax^(n-1), b) which we can write as ((ax^1, ..., ax^(n-1)), b)

Recognizing that as the Cartesian product of R^(n-1) x {whatever values b can take on}

[urban zinc]

I'm confusing what you're doing on the first line

What's the meaning of U \cap M?

[nimble portal]

Uh

I wanted to pick sets in the cover for the domain of my function, but I need to guarantee those sets are subsets of the manifold (and not just any subset of R^n) so I took sets out of the intersection of the cover & the manifold

[urban zinc]

You should write out what the charts are because the open sets you pick are not arbitrary

But also that's not the correct way of writing that

[nimble portal]

So as far as I can tell, b is in [-1, 0], but the dude on MSE got (-infty, 0]

[urban zinc]

(you're saying Ui is an open subset of M but also an element of M?)

[nimble portal]

The charts are gonna be phi restricted to each U_i

[urban zinc]

That's not gonna work

You can't pick an arbitrary open cover

For example what if the open cover is just one set and it's all of M

[nimble portal]

It can't be all of M because then it'd have to contain the entire boundary and then it wouldn't be open no?

[urban zinc]

Oh that's what you mean

What if the cover is just one set, R^n

[nimble portal]

o

[urban zinc]

You need to write out what the open sets are where your construction actually works

[nimble portal]

Okay ignore that part then

I'll get to that later

I just want to know how MSE friend got R^(n-1) x (-infty, 0] for the image and why he says it's open in H^n

Whereas I'm getting R^(n-1) x [-1, 0]

Image of the entire closed ball*

[urban zinc]

wait so what is the domain of your function here?

[nimble portal]

I'm planning to restrict it to an open neighborhood of my manifold

(Ignore the details)

[urban zinc]

It's kind of important what the domain is if you're asking what the image is lol

[nimble portal]

Oh I'm being a silly goose

Domain is the closed unit ball

Here's the MSE post I'm referencing

[urban zinc]

It shouldn't be -∞ that's ridiculous since sigma^-1 maps onto the sphere

[nimble portal]

I just noticed they got the problem from ITM instead of ISM, maybe pi and sigma^(-1) are defined differently there

[urban zinc]

Well R^(n-1) × (-infty, 0] would pretty clearly be open in H^n since it is H^n (except I think usually H^n is the upper half plane instead of the lower half plane but whatever)

[nimble portal]

Nope they're the same here too

[nimble portal]

Ya they're homeomorphic so it's all gravy

[urban zinc]

Ignore the math stackexchange post

[nimble portal]

Aw man

[urban zinc]

Try drawing a picture

[nimble portal]

I have a mental one

[urban zinc]

Maybe try the case where n=1 to make this a bit easier

[nimble portal]

I don't get it

[urban zinc]

Draw out what the charts would be if n=1

I think you'll need 3? if you're following Lee's hint

[pallid karma]

Two questions, why is the channel description have anime in it, and secondly what is popular literature on topology? I picked up Hocking and Young’s Topology from my prof and I’m wondering if that’s known at all

[urban zinc]

1. Why not 2) Munkres's textbook on topology is pretty standard for a first course in topology

[tiny ridge]

what do you mean by popular literature

[pallid karma]

good to know 👍 just graduated and want to learn on my own now

[urban zinc]

I find Munkres kind of dry though, especially the beginning

[pallid karma]

like the go-to (text)book to read. I know for computer finite state machine theory it’s ullman, for calc a lot of schools use the one with the violin, and linear algebra has its own well known one

[urban zinc]

really ullman? I always hear sipser

Anyways there's not really necessarily a go-to, different textbooks move at different paces, have different audiences, and cover slightly different stuff

What's important tbh is that you can learn from it

[tiny ridge]

what eric says

[pallid karma]

that is very true

will be tough to learn now, won’t have a teacher to guide me along

[pallid karma]

well my prof has been using the book since ‘85 might’ve been ‘the book’ from that time (just found a syllabus hanging around inside it)

[tiny ridge]

Want to learn topology? Ignore everything else and just read https://pi.math.cornell.edu/~hatcher/Top/Topdownloads.html. Yes, you too can become an expert in topology from just 50 pages! Bookworms hate this trick.

[pallid karma]

Reading through some of this, I’ve gone over it in chaos and fractals class

Metric spaces, cantor set, very little of Hausdorff metric

I’m guessing dynamical systems and topology go hand and hand?

[tiny ridge]

not particularly

[pallid karma]

Well thanks all for a pretty fast response! Time for more math post-grad!!

[sweet wing]

tbh first chapter you're better off substituting with a generic set theory intro book i usually recommend skipping it or jus skimming and making sure you're aware of the content

[urban zinc]

Really?

Topological conjugacy was one of the first things I read about in dynamical systems

[fervent root]

i was recommended armstrong for topology, what are your thoughts on it, i know it’s not the place for it

and also, does anyone know of any lecture videos online i could use to supplement learning point set topology?

[tiny ridge]

oh armstrong is fantastic

[tiny ridge]

they’re related but only as much as any two things in math

[fervent root]

and is good for a first course?

[tiny ridge]

yep

[fervent root]

ty

[coarse night]

sounds like a marketing campaign 🤣

I swear I've read it before

[tiny ridge]

thats exactly what it is

[marble steeple]

Does anybody have any examples of self - homeomorphisms that are isotopic to the identity? (Any space).

[tiny ridge]

what about the identity homeomorphism

[nimble portal]

Does an open ball about a point in a metric space include that point?

[tiny ridge]

yes

[nimble portal]

I believe so right? Since the distance would be 0 < r

Yeah

[tiny ridge]

yup

[nimble portal]

And if you want it to explicitly exclude your centerpoint you call it a punctured nbhd right

[marble steeple]

I meant other than the identity lol

[tiny ridge]

right on all accounts

[nimble portal]

I think

I know how to fix my proof then

[tiny ridge]

take the identity homeomorphism but swoosh it a little bit

[nimble portal]

omglob

[marble steeple]

fair enough!

[tiny ridge]

heres a serious example. take f(z) = 2z, self homeomorphism of the riemann sphere

its very much not identity, has two fixed points at 0 and infinity tho

but isotopic to identity

f_t(z) = (1+t)z

[marble steeple]

Thank you!

[nimble portal]

Hmmm

If I have a point on the unit circle

What’s the greatest distance r such that a circle of radius r centered at p is tangent to the unit circle

Oh

Just 2 no? LOL

Eheheheheh I think I see

It makes sense to say “The open n-ball of radius r centered at p” right?

As an open neighborhood of p in R^n

I don’t see why not

[urban zinc]

Yes

[nimble portal]

I think I got it 🤔

So what if we took a point on the boundary

Then considered an open n ball of radius 2 about the point

Nope never mind lul

[nimble portal]

Argh I can’t get over like

[nimble portal]

im going insane

just wrote “let p in S” and read it out loud and started cackling

[gritty widget]

[nimble portal]

tteppiejisehfouhweifn

[fickle hamlet]

if x in closA, then every open neighbourhood of x intersects A, why doesnt this imply that there exists a sequence of a in A that converges to x?

[white oxide]

Like, choose a point from the intersection and add it to your sequence using choice?

[fickle hamlet]

ya but i dont think its supposed to be true

cuz then x in closA implies that there is a sequence of a in A that converges to x, but this is supposed to not be true in general topological space

[white oxide]

Well, you have to verify that for each of your previous open set intersections all your latter choices were also in it

[fickle hamlet]

and that would be impossible in some topological spaces?

cuz the books says if there is first countable axiom on the space then it is true

can you give like an example where it is not true

without the first countable axiom

[abstract saffron]

Well, you have the sequences, sure, but what does "converges" mean?

[fickle hamlet]

the sequence converges to x if for every neighbourhood of x, there is N s.t all n>=N a_n is in the neighbourhood

[abstract saffron]

If it's a metric space, then everything is nice. You still can make the notion of convergence using open sets (for any n, there exists an open neighborhood of x that contains all terms after n^th)

The problem is, the limit might not be x

You don't even know if there's a sequence that converges to x

[white oxide]

What you wanna do is possible in so-called sequential spaces by definition, so as a counterexample take any non-sequential space

[fickle hamlet]

havent heard of sequential space can you give me an example of a nonsequential space

[abstract saffron]

that's my point. gotta take some non-Hausdorff space on that

[white oxide]

Cocountable topology on R

The examples are all kind of stupid, but being sequential is also very low on its "niceness" requirements

[white oxide]

I mean, the notion of convergent sequences exists in all topological spaces?

[abstract saffron]

I know, just wanna clarify what it means in this context

We have many, and they might not be equivalent

Depending on the space, of course

[white oxide]

Gotcha

[fickle hamlet]

unrelated but I read that as coconut topology

[white oxide]

Bro is hungry

[abstract saffron]

Can someone explain the notion of handle? I am confused

[bitter smelt]

What about it

[unreal stratus]

It's a thing often attached to objects, typically to allow you to move something like a drawer or a door

[abstract saffron]

Idk, I have read multiple definitions, but still don't quite grasp what it is

Is it a tube D_n x I? Or a disc D_n? What even is its dimension?

And then 0-framed is apprently non-twisting? Why should twisting make a difference, since the attachment is smooth anyway

And to what is it attached? All illustrations have it that the tube (or whatever it is) attach to the boundary at two regions, but apparently it is not defined that way?

[bitter smelt]

A handle requires a few things RE: dimension. You need the dimension of the manifold and the dimension of the handle

I'll get some indices wrong if I just make stuff up, but I like wolframs blurb on this:
https://mathworld.wolfram.com/Handle.html

If M is a manifold together with a (k-1)-sphere S^(k-1) embedded in its boundary with a trivial tubular neighborhood, we attach a k-handle to M by gluing the tubular neighborhood of the (k-1)-sphere S^(k-1) to the tubular neighborhood of the standard (k-1)-sphere S^(k-1) in the dim(M)-dimensional disk. In this way, attaching a k-handle is essentially just the process of attaching a fattened-up k-disk to M along the (k-1)-sphere S^(k-1). The embedded disk in this new manifold is called the k-handle in the union of M and the handle.

[abstract saffron]

Can we do some examples?

Say, we have M to be the 3-ball, what can we do with it?

[bitter smelt]

The boundary of M is S^2. You can attach handles up to the dimension of the manifold, so you can attach a 0 handle, 1 handle, 2 handle, or 3 handle. Not all of them will be interesting. What do you think you get by attaching a 2-handle?

Say you take a simple surgery. No twisting or funny-ness

[abstract saffron]

So we take a S^1 embedded in boundary of M, and identify it with that of a disc D_2?

Doesn't look like a handle to me, geometrically

[bitter smelt]

Well in this case it looks exactly like a handle, which is why I chose 2 lol

But no, that's not what you're doing. A (n,m) handle is D^m \times D^{n-m}. You attach in on the boundary of M to an embedded copy of S^{m-1}\times D^{n-m}

Here it is a (3,2) handle.

[abstract saffron]

So the handle is S^2 x D^1, which is S^2 x I?

A hollow tube, essentially

[bitter smelt]

The handle is D^2 \times D^1

[abstract saffron]

Ah, so... a solid tube?

[bitter smelt]

So a solid tube. When you attach it to the boundary of M is when you specify a copy of S^1

Yes

So you'll get a solid torus

[abstract saffron]

Ahhh

[bitter smelt]

(Up to homeomorphism)

So should look quite like a handle

[abstract saffron]

So you identify one of the ends, which is D^2, with D^2 embedded in the boundary of M

[bitter smelt]

This should be your intuition. Higher dimensions won't be so obviously a handle but yeH

Yep

[abstract saffron]

It's topology, everything is up to homeomorphism 😄

You can't just always smooth the corner, can you?

[earnest ibex]

Can a loop with self intersections be homotopic to a loop without self intersections?

[hidden crag]

Yes

Look at loops in simply connected spaces for example

[earnest ibex]

Would these two be homotopic?

[hidden crag]

yes

[urban zinc]

are you okay 😭

[earnest ibex]

I'm having a hard time seeing a continuous deformation between them, do you know any reference that shows a deformation of this type?

[unreal stratus]

Each can be deformed to a constant loop

[tiny ridge]

If your attaching maps are smooth, you can always smoothen the corners

Imagine a 1-handle attaching to, say, D^2 and convince yourself

[fervent root]

is the second sentence justifying that A bar is closed? A bar contains all its limit point by definition, so it’s already closed

[trail charm]

not necessarily

A bar contains all limit points of A

thus you need to prove that A bar also contains all limit points of A bar

it seems like a very obvious fact and its proof isnt too hard

the whole proof that A bar is closed is

"We first observe that A bar... consequently X \ A bar is open"

X \ A bar is the complement of A bar; X \ A bar is open and thus A bar is closed

[fervent root]

so I can’t apply the fact that a set is closed if it contains all its limit points because A bar contains As limit points, not necessarily all of A bars limit points

[trail charm]

correct

well, i should say that it doesn't follow directly from definition

[golden gust]

is it true that every path connected space contains a simply connected open set?

[unreal stratus]

I'm curious as to whether you can uhhh

Take a circle, attach a circle at each rational point on that circle

And iterate that construction

It's then path connected and at least intuitively has no simply connected open subsets but not sure how I'd go about proving that lol

(Does that sound correct lol)

[lime sable]

what about something ugly like R/Q

i actually don't know much about it so maybe not

[abstract saffron]

Knot theory is soooo deep

It's like, a bunch of fancy stuff put together that you know it's true because you can see it yourself, physically

But you don't know how or why ppl found stuff

[bitter smelt]

Uhh... sometimes.

At some point the pictures stop being real and start being schematic

It is a big appeal though, part of why I like knot theory

[unreal stratus]

Lol I like how different the vibes of our examples are xd

But yeah I imagine some hairy group action would work hm

[lime sable]

actually i think R/Q might be simply connected

[tiny ridge]

R/Q is indiscrete

Take R^2 - Q^2

[lime sable]

good point

[tiny ridge]

you can skiddaddle every rational point while joining any two points by travelling along irrational sloped straightlines

and in any nbhd you can do a little loopy about a missing rational

[tiny ridge]

https://upload.wikimedia.org/wikipedia/commons/e/e6/Apollonian_gasket.svg

sounds right

the apollonian gasket is path connected but has tiny loopies everywhere

[unreal stratus]

Hehe

[unreal stratus]

Okay that was what i was originally doing but then wasn't sure how to construct it more properly lol nice

Cute!

[tiny ridge]

in general there are these batshit julia sets also

which are path connected

but looks like a sponge

[unreal stratus]

Lol

If you trace out the paths from Brownian motion are you gonna a counterexample with probability 1

Probably not quite

[tiny ridge]

oh shit maybe, in dimension 2

[unreal stratus]

Lol that's what I meant ye dim 2

But it may not be loopy enough

I remember this being funny because rather than refer to the Weierstraß function as a reminder that not every continuous function is differentiable anywhere uh

[tiny ridge]

yeah BM is a way better example

[unreal stratus]

my prof just was like yeah brownian motion has probability 1 of giving you a continuous nowhere differentiable function

lol

though we'd not done Brownian motion but yes believble

Tbf I've still not touched it lol

[tiny ridge]

now im curious if your conjecture is true

does the BM return to a given point in arbitrarily small time with probability 1

feels right

yeah this is true

BM in dim 2 returns to any given point infinitely often

now use scale invariance

[urban zinc]

the empty set

[tiny ridge]

i think thats vacuously simply connected

[urban zinc]

hm I know path connected spaces don't even have to be locally path connected, I wonder if you could produce a counterexample that way?

[abstract saffron]

What would be an open neighborhood that is not simply connected?

[unreal stratus]

I mean the examples we gave were like that Eric

Just modified slightly to be even worse behaved

[fervent root]

I’m trying to show that (A u B) bar = A bar u B bar, i’ve shown (A u B) bar is a subset but i need help with the other direction

where bar denoted the closure

[gritty widget]

A is contained in A \cup B so its closure is contained in that of A \cup B

same for B

the direction you already did is arguably the more subtle one. see how it would fail for an infinite union

[gentle osprey]

Couldn't find an attached image in the last 10 messages.

May 23, 2023, 7:18 PM

[fervent root]

,rotate

[fervent root]

like that?

[novel acorn]

I mean coarseness tell you how fine the structure of your space is and how small you can define open sets to be
Imagine this as having smaller components to build a model out of
The smaller the components the more precise you model will be

On the other hand your second question is a very general one

Basically specifying the topology tells you an insane amount of info

How sets are separated connected etc.

That's literally what topology is and what Munkres will teach you

[umbral panther]

Have studied any analysis? Say, Rudin? He starts from metric spaces and then switches to studying the properties of convergence in terms of open sets, rather than metric

[novel acorn]

Why are we interested in topological spaces is a complicated one and also pretty general
Essentially topological spaces are a way to give any set some interesting structure let's say it that way

Most math books are like that lol

Realistically you can always try another source and see if that clocks better with you

I'd advise you to stay your current course
You'll see in time when you get to applications that topologies have an insane amount of ways they can be applied

[umbral panther]

First you do analysis and you have the intermediate value theorem and Bolzano weierstrass. You can rephrase these in terms of open sets. Yes, it would probably be good to review them

So, then, what can you do with open sets? That is topology. Munkres is a very general topology book, general enough to cover many different fields, and also pathological examples. Maybe it would be better to read a more restricted book

[novel acorn]

Lee is a very steep book to read and it requires a pretty good understanding of topology to read the first couple of chapters

[undone ridge]

btw, john lee has a preliminary book to his smooth manifolds book, called Introduction to Topological Manifolds. It deals basically with the basic topology you'll need to have to study the second book, a little bit about the fundamental group and covering spaces, and also a little bit on homology.

hmmm it depends on your objectives; are you trying to study topology just to learn more about manifolds?
it also depends on your background; usually Lee's Topological Manifolds is used at the graduate level

[urban zinc]

Yes the first few chapters of Introduction to Topological Manifolds are really good

It doesn't cover as much as Munkres's book though

Just enough to learn about manifolds

Still might be good to have that intuition

[tiny ridge]

A topology on a set is like an abstract version of a scale or distance. Given two points x, y in a topological space X, if two points x, y belong to an open set U, you can think of that as “x and y being U-close”. If there is a smaller open set V of U such that x, y belong to V, then x, y are V-close and thats a better “measure” of their “distance”.

[undone ridge]

just to make an observation here: there are topics which are treated in Lee's topological manifolds book which aren't dealt with in Munkres' book (like homology, I'm pretty sure Munkres stops with the fundamental group); but when it comes to the extent to which the basic topological facts are treated, Munkres' book is far more complete

[urban zinc]

Yes they are, every open set is built out of basis sets; just like how in real analysis you often prove something about open intervals and then apply it to all open sets. Open intervals are a basis for the topology of R.

[abstract saffron]

Munkres intentionally didn't deal with homology. He has a whole EAT for that

[tiny ridge]

The fact that union of two open sets is open, for example, is like the triangle inequality

[abstract saffron]

If you want to study Manifolds, after Basic point-set, there are other great books

[undone ridge]

and homology is used in manifolds to prove some non-elementary facts (like, for example, invariance of dimension: a nonempty n-dimensional topological manifold cannot be homeomorphic to an m-dimensional manifold unless m=n)

[abstract saffron]

Lee has been classical, but not the only good one out there

[tiny ridge]

Want to learn topology? Ignore everything else and just read https://pi.math.cornell.edu/~hatcher/Top/Topdownloads.html. Yes, you too can become an expert in topology from just 50 pages! Bookworms hate this trick.

[abstract saffron]

They went really, really deep

They covered Green's theorem, Stokes' theorem, div theorem, and alike. Also they treated differential forms very heavily

[urban zinc]

I agree Munkres is pretty unmotivated at the start

[tiny ridge]

its not

[white oxide]

Oh wow, 70% of the content of munkres book I can still remember in only 50 pages. What a steal!

[tiny ridge]

Its all you ever need

[abstract saffron]

For differentiable manifolds, I will recommend Guillemin & Pollack's Differential Topology, in support with Milnor's Topology from Differentiable Viewpoint.

[tiny ridge]

Nope

[urban zinc]

tbh there's nothing wrong with not going deep enough on a first pass

[urban zinc]

urysohn lemma </3

[abstract saffron]

Both of which are light enough to be read in a week. And they give a good overview of what's going on

[warm quiver]

This was when point-set became next level, it was so cool

[urban zinc]

I like Urysohn but I still can't figure out the proof of Tietze extension

[abstract saffron]

They are equivalent, no?

[tiny ridge]

use nets

[abstract saffron]

[urban zinc]

Yes, that's the part idk

[tiny ridge]

I have not read Lee. It ought to be more than enough

[urban zinc]

Munkres's book really hypes up Urysohn as being a hard result, but the proof.. doesn't seem that hard to understand?

[abstract saffron]

I didn't read Lee either, only skimmed through. And it was not a good experience.

[undone ridge]

nononono waiwaiwait

[abstract saffron]

One is more wordy

[white oxide]

Was urysohn the result with that 1/3 trick he hyped up or was that the supporting lemma before it?

[warm quiver]

I think he hypes it up as more so different from the rest of the book, it's not an obvious proof to think of, but not that bad either

[slender magnet]

I strongly recommend trying to prove the theorems (atleast in chapter 2) by yourself when working through munkres,

it seems very dry but by the end of the chapter you will probably understand the motivation behind the number of definitions

[urban zinc]

it's the one where you use dyadic rational numbers to construct a continuous function

[urban zinc]

I guess that's fair

[abstract saffron]

Same goes with Sard's theorem. Sternberg spent half a chapter on that thing, and I still didn't understand shit. Read Milnor, had my mind explode within a morning.

[urban zinc]

It might help to have motivating examples in your mind

[warm quiver]

this was tietze I'm pretty sure

[urban zinc]

Like a basis for the topology of R

[white oxide]

Oh wait, that was urysohns lemma?

[tiny ridge]

Urysohn is tricky. You mimic the rationals with open sets

[white oxide]

Okay yeah, bit of a letdown

[urban zinc]

Or just jump in deep and look up everything as you go

[abstract saffron]

It's point-set. Rarely is it ever interesting.

[urban zinc]

hey untrue

[tiny ridge]

I never read the normal proof of Tietze, I use the ultrafilter lemma instead

[slender magnet]

something about having a continuous function between two disjoint closed sets afair

[warm quiver]

wow cool person alert

[undone ridge]

differential topology is kinda of a step up from analysis on manifolds. like, in the same way ppl usually get used to topological concepts by studying real analysis and then after that going to a proper course in general topology, an analogous thing happens with differentiable manifods: there are some concepts which are easier to absorb about manifolds when you learn them from the analytical perspective... and then after you've got used to them, then go into differential topology.

At least that's what I recommend.

[abstract saffron]

Almost everyone here is self-studying something

[tiny ridge]

It’s much easier trust me

You can fit the proof in a paragraph

[slender magnet]

It is hard, I have trying to read algebraic topology for the past few weeks and breaking my head,

but munkres was really fun to read for some reason??

[abstract saffron]

Well, sooner or later there won't be any courses. The deeper you go, the harder it is.

[warm quiver]

okay, easy, just gotta know what an ultrafilter is first

[undone ridge]

of course, Pollack's book is kind of an exception to this, the topics are dealt with in a pretty light manner, and you could read it while studying about manifolds while doing an analysis course

[abstract saffron]

His books suit my taste. I particularly like his treatment, much more than other authors.

[white oxide]

Hatcher by any chance?

[abstract saffron]

No

[slender magnet]

Started off with hatcher, chapter two was just annoying to read, and then I have been skipping books from there

[abstract saffron]

Welcome to the club

Hatcher is just so bad for the beginners

[undone ridge]

well, read Munkres if you wanna know more about basic topology.

If you have a specific interest in manifolds (and not in learning general topological facts) then to Introduction to Topological Manifolds --> Introduction to Smooth Manifolds

[white oxide]

Almost finished the homology chapter before I couldn't take hatchers way of explaining things anymore

[abstract saffron]

Might as well go with May

[tiny ridge]

How do I create a subthread in this room

[slender magnet]

If you can power through like 20/30 more pages I think the motivation will be so much more evident, and the problems might seem fun too

[slender magnet]

I think it a book meant for a classroom and not self studying

[slender magnet]

I found this book by hajime sato and was reading that but it has wayy too many typos

that's hatcher's point set topo course

[abstract saffron]

Munkres did AT so well. He spent half a book to build simplicial homology, meticulously and with a good amount of examples. Then poof, he introduced Eilenberg-Steenrod axioms, and singular homology and cellular homology become immediate.

Hatcher spent half a book before getting to Homology, and then he started with singular. Bruh moment

[undone ridge]

you don't need to learn about all of the topics dealt with in Lee's topological manifolds book (on a first read you could like, skip the last chapter), but yeah, that's my final recommendation I guess

[slender magnet]

I started reading the eilendberg steenrod book directly like this week 😭 , it's the best written book I have found yet, might read parts of VV Parsolov too

[abstract saffron]

I have seen no one saying they liked Eilenberg Steenrod

[thorny agate]

what is the catch...

[tiny ridge]

No catch

It is my honest recommendation

[abstract saffron]

You do point-set topo. From any book you like. Doesn't have to be Hatcher.

[slender magnet]

I am 25 pages in so prbably why 🗿

[thorny agate]

Hmmmm I've been working through Bredon chapter 1

[undone ridge]

lol the 50 page notes? I mean, you can skim through them (but necessary is a very strong word)

[thorny agate]

Maybe I'll finish that and then see what holes the Hatcher notes plug

(just compiling notes for myself so more sources the better)

[abstract saffron]

I don't know Bredon. I have him, but still procrastinating through chap 2

Apparently my prof likes Abott and Tu

🤷

[thorny agate]

I'll maybe look at chapter 2 idk, I was just recommended chapter 1 as "here's all the topology you ever need to know"

which I interpreted as "learning this will mean that anything you're missing will be easy to pick up"

which is good enough for me

[abstract saffron]

For the context, it is preliminary for the stuff that comes after, so...

[thorny agate]

yea

[abstract saffron]

Chapter 2 deals with IFT iirc

and some other hardcore stuff

[slender magnet]

I should have probably started of with munkres, but I think I have wasted too much time moving from book to book now

[tiny ridge]

That’s a new one

Did you mean Bott and Tu

[abstract saffron]

I guess. I didn't hear him clearly, was just guessing

@plucky lynx one last thing, if you want to have a taste of modern topo, you can try SGA

By the God himself

[abstract saffron]

Once you find the right book, it's soooo worth it

I spent weeks grinding Sternberg, hoping one day I'd finally get what a bundle is.

Should have lurked around library more. I could have found Milnor and Pollock, and could have understood way more by now.

same with AT and Hilton & Wylie

Once I found Munkres, I knew I'd never go back

[slender magnet]

hmm, might check out munkres, because reading eilenberg and steenrod has been exhausting

I spent way too much time reading the proofs of theorems I should have proved fairly easily

[abstract saffron]

Apparently my brain doesn't want to speak English today

The Holy Bible

[slender magnet]

by gronthendieck i think?

[abstract saffron]

Yes

By the God himself

[white oxide]

modern topology, topoi?

[tiny ridge]

I don’t think there’s any topology in SGA, let alone modern

[abstract saffron]

I told you it was to give you a taste, not to learn from

[tiny ridge]

manifold = many folds. True story

Except for one thing

[white oxide]

Yeah why I am asking. Seems kinda random and topoi sound super modern anyways

[tiny ridge]

Topos theory, believe it or not, has very little to do with topology.

[abstract saffron]

Apparently there's something called topoid?

[tiny ridge]

That might be controversial

Huh never seen someone conclude a definition

The more you learn

[white oxide]

I know enough of topoi to confidently say I wouldn't know if they are used in topology or anywhere else unless told

[tiny ridge]

Whats this lol

Why bring in M theory

[abstract saffron]

What's all with these letters? M theory, K theory

[white oxide]

There are two flavors of the latter

[tiny ridge]

But which book is this

[white oxide]

Vanilla (topological) and acid (algebraic k-theory)

[tiny ridge]

I cannot take a book that motivates manifolds by talking about M theory seriously

ok cool

[abstract saffron]

knot theory has a bunch of names, in addition to those from topology, but at least they make sense

[unreal stratus]

M-theory

Is so cool

So manifolds

[abstract saffron]

there's also F-system in logics?

[tiny ridge]

goddamn these string theorists

[abstract saffron]

Or something like that

[abstract saffron]

Can we get the whole alphabet? Lol

In numerical anal we have A-stability

[unreal stratus]

L-theory I saw once ig

[white oxide]

Hmm... yeah

[abstract saffron]

L-theory, kekw

[white oxide]

Might need to work on your abbreviations skills there megumi

[tiny ridge]

L is quadratic K theory

[unreal stratus]

I decided to use the word K for Klasse from my native tongue, German

When I was studying generalised Riemann Roch theorems

[abstract saffron]

Functional Anal has C*-algebra

We're getting there

[tiny ridge]

.

[unreal stratus]

Lol

[abstract saffron]

https://en.wikipedia.org/wiki/T-theory

bruh

[white oxide]

Finally, tree theory

[abstract saffron]

I really start to think we can have the whole alphabet

[tiny ridge]

I think we can safely say that this one does not exist

Look at its wikipedia entry size

[abstract saffron]

https://www.b-theory.com/about-b-theory/

I guess that counts? 😄

[white oxide]

Add some higher category inbetween the paragraphs and it could pass as math

[abstract saffron]

Oh hey, I know Stermfels, he's really known in mathematical biology. I think we can still count it as legit

We do have E-theory, lmao

E-Theory is the name of a category whose objects are C*-algebras and whose hom-sets are homotopy classes of slightly generalized C*-homomorphisms, called asymptotic C*-homomorphisms.

[tiny ridge]

You should do what you want

[abstract saffron]

G-structure from Diff Geo?

I'd take that

[tiny ridge]

H-principles

[abstract saffron]

H for Hasse?

[tiny ridge]

My personal favorite

homotopy

[abstract saffron]

We need I and J, then from N onward

[unreal stratus]

Fr

[tiny ridge]

I-giveup

[unreal stratus]

Mood

I am have to do more fundamental group stuffs for exams bleak

[thorny agate]

Homotopy seems pretty, did some very basis stuff in my algebra course last sem

[white oxide]

T-he voices are getting louder

[thorny agate]

once say I'll properly explore it

[slate timber]

Does j-invariant count? 🙂

[unreal stratus]

Kinda funny doing fundamental group stuff after learning about more alg top cause e.g. I can't use the fact S^2 is nonconttactivle lol

[abstract saffron]

That's our J

I know we have W-algebra

[white oxide]

fr fr

[abstract saffron]

fr fr

[warm quiver]

Also KK-theory

[abstract saffron]

Oh, Y-combinator, from lambda calculus

[tiny ridge]

I need to read more but I’m unsure how to begin

Way too many things to learn

[coarse night]

KKK theory

[warm quiver]

I sometimes wonder if an ignorant soviet mathematician ever did that and then was changed when coming to american publications

[tiny ridge]

Shouldn’t you be asleep mr Adem

[abstract saffron]

Aren't you a grad student?

[tiny ridge]

Yes

[coarse night]

exam

[tiny ridge]

When is it

[coarse night]

tomorrow (25)

[hidden crag]

Rough

[tiny ridge]

Mirror symmetry?

[coarse night]

,ti

[gentle osprey]

The current time for Ryuzaki is 02:54 AM (IST) on Wed, 24/05/2023.

[tiny ridge]

Or is it the quals

[coarse night]

no Vija

[tiny ridge]

Oh right

RIP

[coarse night]

[tiny ridge]

What happened to him

[coarse night]

dm

[abstract saffron]

Which Russell? Bertrand?

[coarse night]

Yeah

[nimble portal]

I give up on this God forsaken problem 😭

I hate manifolds with boundary ⚠️⚠️

CANCELED ❌

[gritty widget]

wait until you hear about manifolds with corners

[warm quiver]

Or orbifolds 😦

[tiny ridge]

stratifolds

[warm quiver]

Smooth sets

[hidden crag]

Y’all just making shit up now

[tiny ridge]

no its all there

[gritty widget]

diffeology

[tiny ridge]

i can confirm

[warm quiver]

Yes, diffeological spaces

[warm quiver]

see

[hidden crag]

Deranged

[warm quiver]

REAL math done by REAL mathematicians

[gritty widget]

my symplectic geometry professor long ago started talking about orbifolds randomly in class once

scared young me

[hidden crag]

They’ve played us for fools

[nimble portal]

r u srs

u can’t be

[gritty widget]

absolutely serious

[nimble portal]

wtf is a smooth manifold with a corner

[gritty widget]

[0, 1]^2

[abstract saffron]

It's a thing

[hidden crag]

I heard about orbifolds in some presentation on symplectic dynamics once

It was about the phd thesis of some guy

[tiny ridge]

the thing with symplectic geometry is

the spaces there are not even orbifolds

theyre like

[bitter smelt]

Feather how are you already to manifolds with boundary, weren't you doing real line with two origins like a week ago

[tiny ridge]

kuranishi atlases

which is sort of like

[bitter smelt]

Oh I guess Lee does it earlier than Tu

[tiny ridge]

intersection of nontransverse things

modulo group action

[warm quiver]

so scary

[abstract saffron]

feather being feather

[nimble portal]

I hate origins too

[abstract saffron]

[gritty widget]

manifold: locally R^n
manifold with boundary: locally H^n = R^{n-1} x [0, \infty)
manifold with corners: locally R^{n-k} x [0, \infty)^k

[nimble portal]

It’s this stupid ugly fucking origin ruining my otherwise nice satisfying thought process

ban 0

[nimble portal]

[tiny ridge]

maybe i should learn the theory of kuranishi structures

[gritty widget]

manifolds with corners are much more finnicky than manifolds with boundaries. you could have a homeomorphism straighten out a corner to be an ordinary boundary component. you need to... do some other stuff

[abstract saffron]

I'll never be done with these obscure objects. I'm still trying to learn wtf is a collar, and I don't even know where it belongs in topology anymore

[gritty widget]

it's what you put on a dog

[nimble portal]

a collar is what I use to help me go to sleep after staring at the problems for four hours and making 0 progress

[abstract saffron]

Great, so you have handles and pants, and now you have collars

I guess next is a hat

[nimble portal]

the implication was that I use the collar to asphyxiate myself

[gritty widget]

...

[nimble portal]

but ig that wasn’t clear 😭

[coral pivot]

wierd

[tiny ridge]

i dont think thats how people fall asleep usually

[gritty widget]

weirdchamp

[nimble portal]

im hibernating

[coral pivot]

well sometimes depriving myself of oxygen helps me pass out quicker!!!

and theres barely any brain damage!

[unreal stratus]

Bruh momentum

[abstract saffron]

I tend to pick a very hard, random problem that I still have in mind, then think about it. Either I solve it, or I fall asleep in the process

[tiny ridge]

poincare grindset

[unreal stratus]

I like how uh

I tried to prove smth about fundamental groups and realised I would've proven it's always abelian if it worked lel

Well the point was uhhhh

[nimble portal]

every group is abelian ;)

[unreal stratus]

Actually this feels cursed

But it was a question asking about loops S^1 -> X based at a fixed point x

but homotopies needn't respect basepoint

[abstract saffron]

Wtf

[unreal stratus]

Isn't that kind of cursed cause it's neither [S^1, X] nor like the pointed version

Lel

But anyway

[tiny ridge]

its ok just not aesthetic enough

[unreal stratus]

Point is that those loops correspond to conjugacy classes of pi1(X,x) having fixed a basepoint

provided X is path connected

But yeah

I do sometimes wonder why we particularly care about this when it's a lil peculiar lol

[warm quiver]

I feel like I did this exercise

[unreal stratus]

Actually I guess this should be equivalent to just [S^1, X] for path connected X in a sense since we can just tack on a path at the end to change the point

[tiny ridge]

i have seen something similar in Hatcher ch 1 ish

[unreal stratus]

I think

Lol

Yeah

[abstract saffron]

And yet ppl still recommend Hatcher to beginners

[warm quiver]

Hatcher is a great book to lecture from

[coral pivot]

also a great book for self studying!

[tiny ridge]

its also nice to solve the exercises

[hidden crag]

funny pictures

[warm quiver]

Yeah has good exercises, and its extra topic choices are great

And the lucida bright font

so yummy

[coral pivot]

hatcher throws you in the deep end of visual intuition and expects you to learn it

which was rough but v useful

[unreal stratus]

Lol

[coral pivot]

and yeah it has a great selection of topics i feel

[unreal stratus]

I agree there

[coral pivot]

Like I find myself refering to its extra chapters very often

[unreal stratus]

Though Spanier is pretty hot in that department

But ye the extra stuff is cool

[abstract saffron]

One question before I fall asleep: some guy threw in some bundles, fibres, and monodromies, and I'm a bit lost. Anyone has a quick dirty explaination?

[bitter smelt]

of what!

[hidden crag]

monodromies

[unreal stratus]

Monodromies

[tiny ridge]

you experience monodromy when you loopy in a bundle of fibres

[hidden crag]

is that a generalization of pi_1 acting on covering fibers

[tiny ridge]

ye

[unreal stratus]

Lel

[hidden crag]

ahh okay then i know what megumi means

[tiny ridge]

just think of m c eschers endless staircase painting megumi

and fall asleep

[bitter smelt]

If you're reading hatcher, check index for deck transformations

[abstract saffron]

best typos ever

[coral pivot]

lol hey migi

[bitter smelt]

hey john! Havent seen you around in a while. you must be on the grind 💪

[unreal stratus]

Teck dransformations

[coral pivot]

kek indeed

[unreal stratus]

Lol I am learn counterexamples for pi1 pain

[hidden crag]

the classification of coverings for nice enough spaces should help you with this stuff megumi

check that out

[coral pivot]

btw random but have you talked to chifan, went to his talk at a conference and it was p good

[tiny ridge]

what are you guys interested in mathematically, as a subbranch of topology, if you had to throw in some buzzwords

[coral pivot]

topology of the non-commutative type kek

[abstract saffron]

in Hatcher?

[warm quiver]

index theory

[hidden crag]

doesn't really matter where

[coral pivot]

oh wow fellow operator person flygon

[hidden crag]

dunno yet

[bitter smelt]

surely not the operator theory conference in iowa? But yeah ive talked to him. im waiting to take the second year analysis until he teaches it. chifan is a very talented speaker

[hidden crag]

i don't have a topic

i just do stuff

[abstract saffron]

knot sure yet

[coral pivot]

a operator alg confy in nashville kek

[tiny ridge]

the best kind

[coral pivot]

yeah I think you will benifit from it a lot

he does von nuemann algebras

[tiny ridge]

cool!

[coral pivot]

and von nuemann algebras is super connected to GGT

so I feel like that might be interesting to u

[tiny ridge]

nice, i might have some questions sometime

[warm quiver]

Wait, I thought you just said interested, I don't have more than a superficial understanding cause I'm still learning K-theory lol, but that's one of the motivations for me learning K-theory at least

[hidden crag]

i'm trying to learn more homotopy theory and cat theory things rn but my courses are getting in my way

[tiny ridge]

no thats cool im not looking for experts

i could never understand the K theory proof of APS it seems black magic

[unreal stratus]

Same here Timo Bleak

[tiny ridge]

i learnt a little bit of how the supersymmetry proof goes but the geometric analysis kills me

homotopy theory is nice!

[hidden crag]

I've been trying to read like mays more concise/Adams Blue book but yeah i'm indecisive about what ressource to use

[coral pivot]

I was reading analytic K homology by higson a while back, had some good stuff on index theory

[warm quiver]

Homtopy theory goes crazy, also Ibsen you already no more than me with knowing a proof, I'm just surrounded by people who know index stuff, so I get absorbed into it a bit

[coral pivot]

use adams kek

[warm quiver]

Gonna probably use this when I actually learn K-homology

[hidden crag]

or maybe goerss jadine or barnes and roitzheim

might just let a random number generator choose

[coral pivot]

yes but one of these books have great latexing!

[hidden crag]

yeah barnes and roitzheim has such a clean version!

[tiny ridge]

goerss jardine is pretty good

[solemn oar]

GJ is the standard resource for homotopy theory.

[hidden crag]

yeah i've been somewhat confused what the "road" is supposed to be

[solemn oar]

May and Adams are more topological and stable stuff

[hidden crag]

right

[solemn oar]

If your goal is abstract homotopy theory then it's GJ and then going into Lurie and that stuff.

[tiny ridge]

but why would your goal be abstract homotopy theory right

GJ is good tho

[hidden crag]

hmm i'm not sure if i'm ready to dive into that

[tiny ridge]

you should at least start with simplicial sets

[hidden crag]

might stick to a more topological side for now

[tiny ridge]

coz theyre pretty useful

[solemn oar]

Yeah then May, Adams, Milnor are your friends.

[hidden crag]

i'm trying to organize a reading course on advanced topics in alg top

[coral pivot]

I think adams was pretty nice after finishing hatcher btw

[hidden crag]

but i am like the only undergrad here interested in this stuff it seems

[coral pivot]

https://people.math.rochester.edu/faculty/doug/otherpapers/Adams-SHGH-latex.pdf

[solemn oar]

I see! In that case def check out Mosher-Tangora.

[coral pivot]

dont forget to use the latex version!

[warm quiver]

it's a dover book too!

[bitter smelt]

whats a good resource to read about h-principle in the context of contact topology? Geiges suggests (his own) H. Geiges, h–Principles and Flexibility in Geometry, Mem. Amer. Math. Soc. 164
(2003), no. 779.

[hidden crag]

Yeah i have that downloaded john, i was just messing with you earlier :D

[tiny ridge]

Eliashberg Mishachev is the standard resource

[hidden crag]

the cohomology operations one?

[solemn oar]

Yeah

[tiny ridge]

Geiges is quite nice also

[hidden crag]

I see, looked at that a while ago shortly

i'll check it out more closely

[bitter smelt]

http://math.stanford.edu/~eliash/Public/271-2018/intro.pdf ?

[tiny ridge]

yes

[bitter smelt]

cheers

[tiny ridge]

its a big messy book but it has everything there

well not everything

but everything that is required to get you started

you may appreciate this blogpost as an outlook on what a typical application looks like without reading the general theory

[hidden crag]

Thank you for the suggestions

[tiny ridge]

https://witheredstumps.wordpress.com/2022/05/17/the-h-principle-for-legendrian-immersions/

[abstract saffron]

You're not alone

[hidden crag]

oh sorry i didn't mean "here" as in this server

i meant my uni

[abstract saffron]

Ah. Still, have you tried talking to ppl?

[hidden crag]

Yeah

[abstract saffron]

I was surprised to learn that some of my peers were also into some very obscure math

[hidden crag]

alg top is rarely offered and the people i took it with last year didn't really like it

[coral pivot]

why talk to ugs when u could talk to grads!

[hidden crag]

the grads here don't do alg top either

[warm quiver]

I have more grad friends than I do undergrad friends now

[hidden crag]

it's just not really a thing at my uni

[coral pivot]

i see

[bitter smelt]

i do alg top!

[warm quiver]

Have you talked to any physics students

[hidden crag]

if it wasn't for a friend of mine who is now at Bonn for grad school i wouldn't be doing alg top either

[coral pivot]

why the sully smh

[warm quiver]

Cause no one at my school does infty cat stuff except for this one physics student

[tiny ridge]

lol

[abstract saffron]

Same! Lmao

[bitter smelt]

its easy to hate on physics

[warm quiver]

Lie infty groupoids

[arctic relic]

I’m a physics student and still sullied

[coral pivot]

many physicists do homotopy adjacent stuff

[arctic relic]

There’s some obscure alg top stuff though

[coral pivot]

and the conference I went to ensured me that category is important in physics

[abstract saffron]

I started algo topo and knot theory because a friend of mine, who is coming to Bonn, insisted on doing it

[coral pivot]

fusion category moment oof

[arctic relic]

https://www.mathnet.ru/PresentFiles/37753/Sergeev.pdf

[bitter smelt]

many physicists do AG adjacent stuff yeah

witten mode

[tiny ridge]

most physicists do not give a shit about what a categorical tqft is

[warm quiver]

Being able to understand physicists is an important skill

[tiny ridge]

they just want to compute their correlators

let them cook

[coral pivot]

haha I agree flygon

[abstract saffron]

Iirc quantum knot theory was from the actual quantum mechanics

Whatever the hell that means

[coral pivot]

I feel like the exchange of ideas between these 2 is important, which is why I am teaching myself physics from a physics perspective

[warm quiver]

Enumerative geometry and physics are becoming more intertwined than ever lol

[tiny ridge]

i think that field now makes very slow progress

it was a thing a decade ago

[warm quiver]

idk, but quantum K theory is still pretty decently sized

[tiny ridge]

of course do not quote me on this

[warm quiver]

from what I can tell

[tiny ridge]

oh

[eager herald]

hi john how’s the qm class goin

[coral pivot]

well I do operators so its impossible to seperate it from physics

[tiny ridge]

i was thinking of gromov witten theory

[coral pivot]

p well, we did some cool stuff recently

like noethers theorem

but today we are just doing unbounded operators since most students here dont know it

oof

[eager herald]

ohh cool

i always wanted to know about noether’s thm

[coral pivot]

we didnt really go into a ton of details though oof

[eager herald]

ah yeah the math review in that class was kinda boring

[warm quiver]

idk maybe my world view is skewed, but my algebra prof does this stuff, and it seems like the big quantum cohomology still shows up, but quantum K theory is popular right now

[tiny ridge]

huh

i mean i could be wrong

[arctic relic]

Where is quantum K-theory being used outside of string theory and the slides I posted earlier?

[coral pivot]

to be fair I think the whole stuff is p niche at the end of the day, so to an outsider maybe it seems small/non-existent

[warm quiver]

idk, people who do schubert calculus

[warm quiver]

I can't claim to know what I'm talking about either

[tiny ridge]

most of math is niche these days

[coral pivot]

right but anything in operator alg is a much smaller field than say, something in homotopy theory or w/e

[abstract saffron]

I'm actually wondering between a PhD in Math and in CS

[tiny ridge]

one of the above has more likelihood of getting you a job for sure

[warm quiver]

see https://sites.math.rutgers.edu/~asbuch/ or https://personal.math.vt.edu//lmihalce/

[arctic relic]

I meant in physics

[abstract saffron]

I know CS is the way to go, and I'm better with algos than with abstract maths, but I always see myself as a math folk who knows how to program more than a CS folk

[coral pivot]

megumi you can do CS that is very mathy kinda

[tiny ridge]

quantum cohomology is apparently useful in integrable systems

[coral pivot]

like, I have been getting into Quantum information lately, maybe that stuff would be interesting to you

[warm quiver]

this is a paper lol https://arxiv.org/pdf/2008.04909.pdf, but I don't know anything in it

[coral pivot]

and also employable!

[arctic relic]

That is string theory

[warm quiver]

whoops, you did say outside string theory

Wait no

I have somthing

Some lattice related stuff

hold on

[tiny ridge]

what kind of physics do you do chernberries

[arctic relic]

The employable kind

[tiny ridge]

condensed mat?

or information theory

[arctic relic]

I do quantum oscillations and methods to compute band structure

Condmat

[tiny ridge]

gotcha

[arctic relic]

I think quantum ergodicity is big in math and physics, some SLE stuff is big for conformal invariance reasons, basic algebraic topology is helpful for topological materials but I don’t think there’s much utility in studying like prespectra, c infinity categories, etc.

After learning about basic spectral sequences, thom isomorphism, characteristic classes I sort of tapped out of algtop

[tiny ridge]

SLE is 2 dimensional CFTs. has any progress been made in high dimensions from the physics side

because i think mathematicians have no clue what are the universal scaling limits in high dimensions

percolations for example nothing is know in 3d

too few conformal symmetries

[arctic relic]

I took a topics in analysis class about it and basically as far as physics is concerned, outside of string theory, you can compute critical exponents of the Ising model for arbitrary lattice structure

[tiny ridge]

that’s interesting

[arctic relic]

Yeah there was an interesting talk by Parisi about universal scaling limits but in 3D I believe

He won the Nobel prize for it

[tiny ridge]

oh ok ill look that up

[warm quiver]

See https://arxiv.org/abs/1702.07162 and this https://arxiv.org/abs/1408.4718, I think

[arctic relic]

Nice although the only application to physics of the first is string theory sadly. interesting how top invariant can be computed using dimer models

[tiny ridge]

did you mean the KPZ universality class, chernberries

[warm quiver]

Yeah, makes sense, the talk I went to seemed to be just that there's this thing in chemistry and it's related to quantum K-theory

[arctic relic]

For the second one, if I see “equivariant” as an adjective I’m not understanding it

[warm quiver]

Wait, but like, isn't that adjective like all of physics

[arctic relic]

I don’t know what that is actually

[tiny ridge]

i think “equivariant” means “constrained by symmetries of a group action” in physics

[arctic relic]

No I meant idk what KPZ universality class is

https://www.nobelprize.org/prizes/physics/2021/parisi/lecture/

[tiny ridge]

eg if you have a phase space M and some symmetries G you can pass to the symplectic quotient M//G which is the constrained Hamiltonian system corresponding to the G action

[tiny ridge]

thanks

[patent quarry]

i’m pretty sure even physicists don’t know how kpz universality should work beyond dimension 1 or 2.

nobody knows critical exponents, for example.

[tiny ridge]

thats my impression from talks i attend

[patent quarry]

should qualify this: i don’t actually know the “critical dimension” (critical wrt physicists’ ability to predict).

yeah there’s even disagreement about the exponents in the limit as dimension tends to infty.

[tiny ridge]

huh

[patent quarry]

coincidentally, today i was reading a recent math(!) paper on (2+1)-dimensional kpz equation that claimed they found some phenomenon that wasn’t predicted in the physics literature.

(i have no idea whether physicists care about kpz anymore.)

[tiny ridge]

i dont really “get” the discrete models of which kpz is a scaling limit

i dont know what the point of last passage percolation is for example

nor do i understand this random tetris shit

[patent quarry]

tetris?

[tiny ridge]

like, you drop tetris pieces using a poisson point process on the boundary of the upper half plane, and these things can randomly stick side by side also

the scaling limit is kpz lol

who cares

too many layers of randomness for me

[patent quarry]

well the entire point is that kpz should be universal.

so the discrete model isn’t supposed to matter.

as for lpp, one perspective is that lpp is the zero-temperature version of a (discrete) polymer in a disordered environment.

[tiny ridge]

i get it, but the reason i care about sle is more to do with how natural the discrete models it arises from are than the actual sle equation

[patent quarry]

maybe you know this and don’t care about polymers though.

[tiny ridge]

oh ok so like the boundary of the arctic circle for dimers is kpz?

[patent quarry]

hmm maybe, i don’t know enough about dimers to say.

[tiny ridge]

i dont think i know this but thatd make sense

[patent quarry]

sorry it’s a bit annoying to define the model lol.

[tiny ridge]

my cat has something to say about kpz, it started walking all over my keyboard for no reason

[tiny ridge]

yeah its alright

[patent quarry]

ok let me put it this way, since you know what lpp is.

in lpp, you put some random disorder on the environment, and then for each realization of the disorder you study some maximizing up-right lattice path between two points, say x and y.

[tiny ridge]

yup

[patent quarry]

instead, fix beta>0, and for each realization of the disorder, consider the corresponding gibbs measure (at temperature 1/beta) on the space of up-right lattice paths from x to y.

[tiny ridge]

ok

[patent quarry]

we have a reasonable understanding of the partition function Z and free energy logZ.

for instance, in the scaling limit, we formally expect log Z to be the hopf-cole solution to the kpz equation.

[tiny ridge]

ok that’s interesting

[patent quarry]

oh right, my original point was just that the discrete model recovers lpp at beta=infty.

[tiny ridge]

yeah

[patent quarry]

namely the length of the lpp maximizer is the free energy.

i should maybe mention that kpz equation is not the universal object.

it is a bit special among the models in the kpz universality class, but the true scaling limit is much harder to describe.

[tiny ridge]

i barely remember the actual equation, all i know is the 1:2:3 scale invariance

[patent quarry]

i think in this polymer example, log Z(beta) is supposed to converge in the scaling limit to kpz equation driven by a spacetime white noise that’s been rescaled by some coefficient depending on beta? idk the details, but beta -> infty should then yield the true universal object.

[tiny ridge]

gotcha

[patent quarry]

also idk how much of this stuff is rigorously known, i’m only very recently learning about how the kpz equation fits into this story.

[patent quarry]

yeah i mean, it arguably isn’t important. nobody thinks of kpz universality in terms of the literal kpz equation.

[tiny ridge]

i took a bunch of courses on probabilistic geometry but zoned out when people nudged me to learn ising model or this stuff. i like percolation, random walks, uniform spanning forests, …

[patent quarry]

also when you prove things it’s usually better to work with hopf-cole transform.

well then sle is for you 😛

[tiny ridge]

yeah but its “done” so people are coming with more and more twisted discrete models

[patent quarry]

though i will say: much of the recent work in kpz universality from the discrete side is focused on developing geometric/probabilistic arguments.

[tiny ridge]

random walks with random traps on percolation clusters in hyperbolic graphs

new paper

zzzz

[patent quarry]

we don’t actually know rigorously that lpp is in KPZ universality class in any real sense, except for a couple special choices of the disorder.

[tiny ridge]

ok interesting i will have to keep an open mind next time i attend a talk

[patent quarry]

it depends who is speaking though.

[patent quarry]

no thanks.

i think actually the best way to see what i mean is to learn how people prove things in first passage percolation. but fpp is horribly understood, so idk if this is actually fun to learn about.

[tiny ridge]

i learnt a bit of fpp from auffinger damron hansen a couple of years ago

it was fun

[patent quarry]

oh ok great. then you know the flavor i have in mind.

[tiny ridge]

yeah.

[patent quarry]

all that stuff about geodesic coalescence has become really powerful in, say, lpp contexts.

people try to take some minimal input from exactly solvable models (a one-point distribution, or a tail bound, or …) and extract information about kpz-type behavior (critical exponents, correlation, scaling limits,…).

[tiny ridge]

i see. maybe i should look at which parts of the argument goes through from fpp to lpp in regards to coalescence

[patent quarry]

well it isn’t really about establishing coalescence, it’s about applying it. think about how we can prove that directed fpp semi-infinite geodesics are unique, provided the limit shape has some curvature.

[tiny ridge]

ah ok

[patent quarry]

this is not a great example, let me think of a better one.

i guess broadly speaking though, the essential features that you try to use in lpp are that the limit shape (morally) has parabolic curvature and (morally) brownian fluctuations.

hmm this is a bit hard to give a good feeling for, sorry.

[tiny ridge]

no worries. i should sit down and think about it someday

[patent quarry]

i have no idea how to get into this field, the literature is so bad.

[tiny ridge]

everyone in my school does random geometry, but i think its not for me. i like it but its too hard for me

[patent quarry]

uh here this might be useful https://youtu.be/mpoRJXlDrqw.

[tiny ridge]

thanks

i took a course in percolation from the mentioned Basu lol

[patent quarry]

idk if this talk itself is good, never watched it, but the paper it’s about is good (extremely long so don’t read it).

oh wow. he is one of the pioneers in this direction.

[tiny ridge]

ye

[patent quarry]

i’ve heard that the original paper to start this kind of geometry-based study was https://arxiv.org/abs/1408.3464.

never read it, and weirdly it is unpublished…but yeah.

[tiny ridge]

interesting

[patent quarry]

anyway it is by basu. you should just ask him to explain to you the ideas lol.

[tiny ridge]

ill watch the talk, thanks a lot

[tiny ridge]

maybe i should

i got kind of sick of random geometers in the middle

riddhi is a nice guy, he teaches and explains very well

[patent quarry]

i understand, they’re a bit cultish.

[tiny ridge]

ill head to bed, but heres a question if some topologist is around. in https://arxiv.org/pdf/1907.06025.pdf, figure 3.2.1 feels misleading to me. the actual picture of the dynamics of the characteristic foliation of Z_PL seems to me to be as follows

they apply some smoothening on the edges. before smoothening the top right and top left edges are attracting and repelling respectively so its believable that the midpoints are the two elliptic points

the hyperbolicity at the base is not so obvious to me.

its most likely what they say it is but it seems reductive for ignore the complicated dynamics of the block. am i paranoid?

there should be a clean conjugacy between the two pictures that i dont see

[pulsar lagoon]

I was going through munkres topology and came across “kuratowski” theorem

I couldnt help but wonder what that might even be useful for

[urban zinc]

https://www.youtube.com/playlist?list=PLN_4R2IuNuuTWD00k9BAB1fo0UldBHnME I found this really nice playlist on intro algebraic topology!

Hope this helps someone

[urban zinc]

Stumping undergrads

[pulsar lagoon]

True, this is ploy made by big math

[quick bough]

does anyone know how to show first the implication (assuming that f is closed, i tried a bunch of things, but always had the problem of V_i being open)

[white oxide]

What does it mean for a set to be closed in the pre-image of V_i? As in, when is a set closed in the subspace topology of the pre-image of any V_i?

[coarse night]

Is your open cover finite?

[quick bough]

nope

[quick bough]

it’s just an intersection with a closed set in X

(with the preimage)

[fresh thunder]

Hi @coarse night

[white oxide]

Right, so let's call the closed in the pre-image C, it is the intersection of a closed K with the pre-image. Since f is closed K must be mapped to a closed set

[coarse night]

Hello

[abstract saffron]

I spent a month grinding alg topo, knot theory, and diff topo for a math exam

Only to learn today that I wouldn't even get to do it, bruh

[quick bough]

yeah, i alr got that, too, but i’m not sure how that helps

[white oxide]

entrance exam?

[coarse night]

Wanted to say something?

[fresh thunder]

I wanna help you

and also I am sure you can help me

[coarse night]

Well ok lol

[quick bough]

@white oxide what now?

[abstract saffron]

Yes

I didn't pass the application round. Can't believe it.

[fervent root]

what is the typical notation for the frontier of a set

my text doesn’t cite one

[unreal stratus]

Frontier can have a couple of meanings right

[quick bough]

anyone got a hint for this? 🥲

[viral atlas]

I've seen prime (like A' to denote the frontier of A)

Otherwise you might as well invent your own suggestive notation

Front(A) or similar

[novel acorn]

Does frontier here mean boundary?

[viral atlas]

I think frontier is boundary\cap the set?

Might have to look it up again

Oh, Google says otherwise

It often refers to boundary\cap complement of the set

[fervent root]

it’s defined to be the intersection of the closure of A and the closure of X-A

maybe i’m using an old ass book if they call it the frontier, so western

is it just the boundary though? if i did this problem correctly it worked out that way

actually yeah that makes sense, the interior is all but the boundary, and the union of the interior and the boundary gives the closure

[lean knot]

X and Y are topological spaces, f is a borel function from X to Y. How can I prove that the continuity point of f is dense in X? (borel function means the preimage of any borel set is still borel. Borel sets are the smallest \sigma algebra generated by open sets)

correction: X and Y are metric spaces

[tiny ridge]

this is just the boundary yes. a point here has the property that any nbhd intersects both A and X - A

thats exactly how boundary is defined

[tiny ridge]

this is not the boundary eg take A = rationals in R

frontier is an antiquated notion btw

nowadays its either the boundary or not called anything

[viral atlas]

Oof I see

I dislike the notion as well

[tiny ridge]

yeah

[fervent root]

i don’t think i’m gonna be able learn very much self studying topology until i figure out how to think in terms and apply the use of nbd

[tiny ridge]

Want to learn topology? Ignore everything else and just read https://pi.math.cornell.edu/~hatcher/Top/Topdownloads.html. Yes, you too can become an expert in topology from reading just 50 pages! Bookworms hate this trick.

[undone ridge]

https://www.youtube.com/playlist?list=PL2Rb_pWJf9JqgIR6RR3VFF2FwKCyaUUZn
I also like this one, despite requiring familiarity with the fundamental group, covering spaces and things like that

[fervent root]

this is so helpful thanks!!!!!!

[undone ridge]

hope it helps someone who's trying to learn more about singular homology

[hidden crag]

I like this one as well

[urban zinc]

thank you!!!

[jagged ocean]

is there a criterion which classifies when a topological space has cohomology groups with a vector space structure? My motivation is that de Rham cohomology groups are always vector spaces, being the quotient of a vector space by a vector subspace, so being a smooth manifold is a sufficient but I imagine unnecessary condition.

[fading vale]

What do you mean by "cohomology groups with a vector space structure?"

vector spaces are just modules over a field so taking cohomology with coefficients in any field will of course give you what you want

[graceful oriole]

Can someone verify this? Suppose Y is a subspace topology of X, and A is a subset of Y. Then cl_Y(A) (A's closure in Y) = cl_X(A) ∩ Y. And does it still hold if we change closure to sequential closure?

[jagged ocean]

Thanks for the clarifying comment: any time I have a cohomology group taking values in a ring, the group is endowed with a module structure over that ring, so if the ring is a field, then I can inherit a vector space structure.

[fading vale]

Right

So what exactly do you mean when you say that a space has cohomology groups with a vector space structure

because you can simply take cohomology with coefficients in a field

[jagged ocean]

No this answers my question, it was just different than I was imagining because I'm new to alg top

[fading vale]

Ah i see

Glad it clicked

[tiny ridge]

Hmph, I understand this better now. It is still not easy to see what exactly is going on IMO

[unreal stratus]

Here's smth fun lol

What is the best way to prove fundamental theorem of algebra using topology

I feel like I've seen two somewhat different ideas

Like the one I'm most familiar with is uhh like

[gritty widget]

best way or favorite way?

[unreal stratus]

Favourite i guess aha

[hidden crag]

i only know one way so i might be biased

[tiny ridge]

I don’t want to prove the fundamental theorem of algebra using topology

It’s algebra

Who cares yes?

[unreal stratus]

I really hope that is a joke lol

[gritty widget]

get a map S^2 -> S^2 with degree equal to the degree of the polynomial

[tiny ridge]

ofc

[tiny ridge]

Sorry to interject but I want to share my descent into madness as I try to draw pictures for this:

[unreal stratus]

Sps p is non-constant polynomial without roots. Restricting to some big circle C, you can show it's homotopic to a suitable scaling of z^n (relative to endpoints), so the map π1(C) -> π1(C^x) corresponds to multiplication by n, Z -> Z

[unreal stratus]

But p extends over C (as a map into C^x) so nullhtpic

[unreal stratus]

Oh how lol

Interesting

[tiny ridge]

Take the polynomial itself

S^2 = Chat

[gritty widget]

S^2 minus a point (infinity) is C, nonconstant polynomial sends infinity to infinity

[unreal stratus]

Oh sure yes sorry yes