Mathematics - 2025-02-09

Ben Steffan

00:00

Usually it's $\geq 70\%$ point-set and then fundamental groups

Alessandro Codenotti

Makes sense, I wouldn't know, I only took Topologie I (also with Lück) when I was there as far as (algebraic) topology courses go

But I do distinctly remember that during algebraic geometry I the professor felt comfortable assuming that everyone knows what a (co)representable functor is, but felt it necessary to remind people what "Hausdorff" means!

Ben Steffan

lol

who taught that

Huybrechts, by any chance? :)

Alessandro Codenotti

Oberdieck

Ben Steffan

ah, ok

well most of the AG people here are a bit

Alessandro Codenotti

I don't know whether he's still in Bonn. He had a temporary position at the time

Ben Steffan

00:03

...like that

not to my knowledge

Alessandro Codenotti

A quick google search suggests he's working in Sweden now

Ben Steffan

Perhaps he enjoys the cold

ILikeMathematics

00:21

@BenSteffan Ah, great, thank you! I generally really like lecture notes, they're basically like books but more concise, yet at the same time they cover basically as much! (I.e. my linear algebra notes covered everything and more than my linear algebra book and yet they were shorter), you just need to accomodate them with the problem sets given in the lecture

@BenSteffan Well, that's even better then, if there is more in them than what is usually taught

Ben Steffan

It's a bit of a double-edged sword. They are shorter generally by virtue of omission, and skew towards the lecturer's interest.

As is clearly the case here, on both accounts.

You'd do well to at least supplement this with a textbook.

@ILikeMathematics Eh, as I said: They trade that material in for other things they omit. And the material in those section is taught, but in the follow-up lecture, generally speaking.

ILikeMathematics

Well, atleast for analysis and linear algebra it was the case that they basically covered everything in the book.

@BenSteffan Topologie 1?

Ben Steffan

@ILikeMathematics Topology 1 and following

ILikeMathematics

Ok, well then I will need a set of lecture notes on that too

Ben Steffan

Cofibrations were only picked back up by Lueck this term in Algebraic Topology 1

ILikeMathematics

00:25

I think I remember you typed some notes up on that?

Ben Steffan

I have no notes for Topology 1 & 2, but maybe Qi has? You should browse the sciebo folder containing the notes I linked above.

ILikeMathematics

@BenSteffan Ah, then was it AT 1?

I remember you have some notes

And they looked pretty good, with colors and everything!

Ben Steffan

I have a set of notes for Algebraic Topology 1, but the material for Algebraic Topology 1 & 2 isn't really standardized. You'll probably learn that material to some extent when you take these classes, but not necessarily in AT1.

@ILikeMathematics Thank you :>

ILikeMathematics

@BenSteffan Ah. So each term, different things are taught in AT depending on the instructor?

Ben Steffan

Pretty much, yes.

This term's AT1 was radically different to the one my notes are from

But to be fair, this term's AT1 was much more in line with what's supposed to be taught in that course

ILikeMathematics

00:29

Well, that's weird haha. You'd expect two people who have taken AT 1 roughly both know the basics of AT

Ben Steffan

But AT2 is basically entirely up to the lecturer's discretion

@ILikeMathematics "basics of AT" is a very stretchable term

the bedrock foundations you learn via Topology 1 & 2

ILikeMathematics

@BenSteffan .. and those are more standardized, right?

Ben Steffan

yes, they're pretty standardized

ILikeMathematics

Alright

Ben Steffan

but AT1 & 2 are more "advanced basics,"

and there's a lot of different topics you could teach

the field is very broad

and some people tend to cram a lot of the intended material of AT1 into T1 & 2 already so that they have more leisure to do more advanced things

this is why most of the material of my AT1 notes will end up probably being taught in AT2 next term

since the lecturer decided to not cram the AT1 material into T2

ILikeMathematics

00:34

@BenSteffan So basically, it could happen that in Einf. in die Topologie und Geometrie, you have things that a different instructor would cover in Topologie 1 and vice versa. But roughly, the material presented over the whole set of courses all cover the foundations and some more advanced things

Ben Steffan

Yes. There's a bit of push and pull with when you learn things, but the "basic" basics you will learn throughout the whole cycle, and then what you learn on top of that is variable

ILikeMathematics

Well, then a problem could be taking Einf. from a different instructor than Topology 1, I guess

Ben Steffan

sure, but instructors generally try to ease that transition

Einf. is strictly a bachelor-only course, but Topology 1 is shared with the masters

so you always have master students coming in from elsewhere who haven't taken Einf.

and the lecturers are of course aware of this

ILikeMathematics

Ah, so it goes over a bit of Einf. too

Ben Steffan

not usually, but the material you start out with in T1 doesn't depend that strongly on the material in Einf.

you should know basic point-set topology, that's expected, but for the algebraic topology parts of Einf. the lecturer will usually say something to the effect of "this will play some role at some point in the course; if you're unfamiliar you can look at this and this and this"

ILikeMathematics

00:39

When we talk about notes typed up by someone like the one you linked, did they copy everything from the board and also what the professor said? Or would they also include things they thought about themselves (which would be prone to errors ig if the typer is someone who is taking it for the first time)

Ben Steffan

but generally the idea is to reassure people that intricate knowledge of covering space theory etc. is not required; a broad understanding will do

@ILikeMathematics Usually it's pretty close to the former

Sometimes people flesh out the notes a bit with additional material, but if they do it's mostly material from books etc.

ILikeMathematics

Alright, that's good. At most other unis, professors type up their own notes, for some reason at Bonn it's different

Ben Steffan

A substantial amount of own thoughts is rare; at most people comment on some easy consequences or other characterizations or stuff like that

@ILikeMathematics No, this just depends on the professor

At other unis as well

Many courses here have official notes

This term's AT1 has, for instance: him-lueck.uni-bonn.de/data/script_AlgTOP_I_www.pdf

Even though neither Einf. nor T1 nor T2 from this cycle taught by the same prof. have

not sure why he decided to make notes for this one

but making notes is a lot of work, and understandably not every lecturer wants to do it

although the number of people who do provide some form of notes has increased noticeably since the pandemic

ILikeMathematics

@BenSteffan Yeah that's understandable

Well thank you for the link Ben

Ben Steffan

you're welcome :)

Thorgott

01:40

@BenSteffan I like that he calls it "Aufdicken"

Ben Steffan

:^)

I hope I never have to read the word "Gruppoid" ever again

Thorgott

I think it was Lück's textbook where I first saw "Einhängung"

PM 2Ring

03:59

Hey, why do YOU get to be the president of Tautology Clu-- wait, I can guess.

@BenSteffan As a German speaker, which looks funnier to you? Germanic words in English, or Dutch?

in The h Bar, Feb 5 at 9:58, by PM 2Ring

I love the sound of Dutch. But in written form, it looks hilarious. It seems to be pretty common with closely related languages that they look or sound a bit silly.

And then there's the fascinating phenomenon of asymmetric mutual intelligibility:

Why some speakers can't understand speakers who understand them - Asymmetric Intelligibility

►

There are numerous great anecdotes in the comments of that video.

X4J

04:25

In the definition of composition series at finite group theory it is also required that for each i, H_{i+1} is a maximal normal subgroup of H_i. Is that defined so because of the aim to

relate uniquely a group to its composition series multiset as in Jordan Holder theorem?

or there's another aspect

leslie townes

04:36

x4j: i am not an algebraist but yes, i would imagine something like that is the motivation. if you were to allow arbitrary nested sequences of normal subgroups to be "composition series" it wouldn't be as useful a concept. the same group could have composition series of different lengths, for example, and the quotients in one could have very little to do with the quotients in another.

it wouldn't surprise me if people still find themselves wanting to consider nested chains of subsgroups that aren't composition series. but the terminology "composition series" suggests to me a motivation of wanting to go somewhere like the jordan holder theorem.

and you wouldn't have anything like that for arbitrary chains of normal subgroups.

X4J

@leslietownes yes that's what I wanted to understand, because the maximality property was defined to me in class equivalently as H_i/H_{i+1} being simple for each i in the chain. However the term of maximal normal subgroup was not defined, although personally it makes more sense to me

(when I relate it to the mentioned theorem)

PM 2Ring

07:01

Happy Birthday, H. S. M. Coxeter!

Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated at the University of Cambridge, with student visits to Princeton University. He worked for 60 years at the University of Toronto in Canada, from 1936 until his retirement in 1996, becoming a full professor there in 1948. His many honours included membership in the Royal Society of Canada, the Royal Society, and the Order of Canada. He was an author of 12 book...

Soham Saha

08:35

I was just making up arbitrary rules for a cellular automata, and this popped out:

imbAF

10:31

I have one question about group action and homomoprhism. In wikipedia I read:
"In mathematics, a group action, of a group G on a set S is a group homomorphism from from G to some group (under function composition) of functions from S to itself. It is said that G acts on S"

Group homomorphism is between two groups. While in the above statement we have a group and a set and we talk about group homomorphism. Is the group homomorphism between the group G and a group H, which is the representation of G, and it retains the group structure of G

Ben Steffan

14:26

@PM2Ring Not sure I understand a question, given that most of the vocabulary in either language is germanic

dutch is rather disreputable as a whole, however :^)

PM 2Ring

14:51

@BenSteffan I said "Germanic words in English" because we also have a lot of Latinate words via Old French (as well as a bunch of words from various other places, eg India).

@BenSteffan Ok. That fits the theory, since Dutch is closer to German than English is.

pie

17:52

Is there an upper limit on how many times a number (other than 1) would show up in pascal's triangle? Most numbers like $3,4,5$ only appear once or like $6$ appear twice (excluding the duplicates) but few appear three times like $210$,

I wonder if there a funcrion $f(n)$ such that a number $n$ show less than $f(n)$ in pacals triangle

3003 is the first to appear four times

Thorgott

@Ben If we have a chain complex $D$, we can compute homology with coefficients in $D$ by considering the homology of the chain complex $C_{sing}(X)\otimes D$

do you know what spectrum in general represents this homology theory?

pie

between (1, 10^4) there isn't any number that is repeated 5 times, is there a way to calculate how many times a number would repeat

Xander Henderson

What do you mean by "there isn't any number that is repeated 5 times"?

As far as I know, each number occurs exactly once...

pie

@XanderHenderson $6=\binom{4}{2}$, $6= \binom{6}{1}$, $6$ is repeated twice

210 is the first to be repeated three times.

Xander Henderson

@pie I must be missing some context... what do combinations have to do with ti?

pie

18:00

@XanderHenderson I am asking how many times a number appear in pascal triangle (excluding duplicates in the same row)

This graph shows how many times a number appears in pascal triangle.

Ben Steffan

@Thorgott I've never heard of this construction

Ben Steffan

18:26

hmm, is this really a homology theory?

If you take $D$ to be $C_{\mathrm{sing}}(S^1)$ and you plug in $S^1$ for $X$, then you should recover $H_*(S^1 \times S^1)$. But if this were a homology theory, then you should have an MV sequence in the first copy of $S^1$, so decompose $S^1 = U \cap V$ as usual with $U, V \simeq *$ and $U \cap V \simeq * \sqcup *$

This gives rise to $0 \to H_2(S^1 \times S^1) \to H_1((U \cap V) \times S^1) \cong H_1(S^1 \sqcup S^1) \to H_1(U) \oplus H_1(V) \cong H_1(S^1) \oplus H_1(S^1) \to \ldots$

nevermind, I can't do linear algebra

EE18

18:43

Hopefully this is enough context, but is there a mistake here in the sentence ending with "(chain rule)"? Should it read "of any subset $\Delta$ of $\Gamma$" (no overline)?

Xander Henderson

@EE18 No?

I don't know the theory at all, but the structure in most places is that if $A$ is a subset of some "closed" space, then the "closure" of $A$ is contained in that closed space.

E.g. let $A \subseteq \mathbb{C}$ (where $\mathbb{C}$ is the algebraic closure of $\mathbb{R}$). The algebraic closure of $A$ will be a subset of $\mathbb{C}$.

So my gut tells me that $\Delta$ can be a subset of $\overline{\Gamma}$, and things work out correctly.

Ben Steffan

19:01

@Thorgott If you can work reducedly and focus on the case that $D = \widetilde{C}_{\mathrm{sing}}(Y)$ for some space $Y$, you should obtain that the associated (reduced) homology theory is given by $\widetilde{H}_({{-}} \wedge Y) \cong [\mathbb{S}[], \Sigma^\infty({{-}} \wedge Y) \otimes H\mathbb{Z}]$, but $\Sigma^\infty(X \wedge Y) \otimes H\mathbb{Z} \simeq \Sigma^\infty X \otimes \Sigma^\infty Y \otimes H\mathbb{Z}$, so the representing spectrum is $\Sigma^\infty Y \otimes H\mathbb{Z}$

...couldn't find the mistake in time so this monster is now carved in stone I guess

To reiterate, the representing spectrum should be $\Sigma^\infty Y \otimes H\mathbb{Z}$, using that $\Sigma^\infty$ is (strong) symmetric monoidal

EE18

I think I follow, thanks Xander!

Ben Steffan

so for the unreduced case you get $\Sigma^\infty_+ Y \otimes H\mathbb{Z}$

Xander Henderson

@EE18 Again, I don't actually know the field in that paper at all, so I could be completely wrong. But my gut says that what is written there is fine.

Ben Steffan

What it should be when $D$ is not quasi-isomorphic to $C_{\mathrm{sing}}(Y)$ for some space $Y$ I have no idea

Thorgott

19:42

@BenSteffan isn't every chain complex?

Ben Steffan

you tell me; I barely know the meaning of the word :^)

Thorgott

every chain complex is quasi-isomorphic to a free one

(take a cofibrant replacement in the projective model structure on chain complexes, but projective = free for abelian groups)

every free chain complex should be the cellular chain complex of a CW-complex built via the attaching maps forced by the differentials

and then in turn quasi-isomorphic to the singular chain complex of that CW-complex

Ben Steffan

I can get behind that

Thorgott

well almost, it's false for stupid reasons

Xander Henderson

@Thorgott Most things are.

Except those things which are false for smart reasons.

Thorgott

19:55

obviously can't have homology in negative degrees and $H_0$ needs to be free and non-trivial unless the entire thing is zero

Xander Henderson

"Obviously." :P

Ben Steffan

you can remove the quotation marks because that one is truly obvious

:)

Xander Henderson

"Obviously." :P

Thorgott

so when $D$ is the group $A$ concentrated in deg $0$, do these observations yield the right thing

obviously not

when $A$ is not free, this does not apply

so we're not recovering EML-spectra

Xander Henderson

@Thorgott Okay, but when $A$ isn't free, how much does it cost?

Ben Steffan

19:59

@Thorgott you can probably get around this by shifting, as long as the homology is bounded-below

if it's not bounded below you might get away with taking some co/limit, perhaps

Thorgott

20:12

yeah, I'm more worried about not generalizing EML-spectra

I kinda wanted that

Ben Steffan

why though?

I guess the more natural thing to do would be to ask "what happens when I replace $HG$ with $HG \otimes HG'$" directly on the level of spectra

I don't think I know the answer. $H$ is (op?)lax monoidal, but not strong monoidal

Faoler

20:29

hello I asked a question about solving PDE by lagrange method here https://math.stackexchange.com/questions/5033472/simplifying-characteristic-equations-in-lagranges-method
I really need an answer after some hours only

Ben Steffan

20:43

fascinating things going on on the site today math.stackexchange.com/questions/5033477/…

leslie townes

haha that account coming to life after almost 10 years to ask that

hbghlyj

22:37

A two-dimensional manifold is orientable iff the surface contains no subset that is homeomorphic to the Möbius strip. Thus, for surfaces, the Möbius strip may be considered the source of all non-orientability. I wonder if there is a similar result for higher-dimensional manifolds.

Is there a "n-dimensional Möbius strip" that is the source of all non-orientability in n-dimensional manifolds?

leslie townes

@hbghlyj here's someone asking this for 3-manifolds and getting an answer math.stackexchange.com/questions/3244953/…

hbghlyj

Thanks

leslie townes

i dunno about the general case (somewhat related to the question, i have no idea how big a role orientability plays in the general study of manifolds)

Ben Steffan

Not that big, it's just the checks notes second most important condition after closedness :)

leslie townes

22:56

what's #3? :)

Ben Steffan

dampness

no one wants to work with a damp manifold

Thorgott

@BenSteffan I don't know, was trying to figure out an Example for my thesis (in typical category fashion, the Examples section is the very last one)

Ben Steffan

:')

Thorgott

@hbghlyj I think dimension is a red herring

a manifold is non-orientable if and only if it admits an orientation-reversing loop

the Möbius strip is just a thickened up orientation-reversing loop in dim 2

more generally, you can restrict attention to embedded loops and take the normal bundle

for an orientation-reversing loop, this will be the unique non-orientable disk bundle over $S^1$ of the respective dimension -1

in dimension 2, you get the Möbius strip, in dimension 3, the Klein bottle

Ben Steffan

23:21

"Now that I know what an $\infty$-operad is, what about $\infty$-operads with values in other $\infty$-categories?"

"Hmm, good question" several hundred pages of cryptic remarks follow

I hate this field.

iirc it's also a rather recent result that an $\infty$-operad is the same thing as an $\infty$-operad with values in $\mathrm{An}$

Thorgott

@BenSteffan ah yes, of course

Ben Steffan

hey, at least it's not dendroidal sets...

Thorgott

they might be the answer

Ben Steffan

@Thorgott it's clearly a good sign when one of the most basic sanity checks for the theory proves that difficult

@Thorgott at least we now also know that they're equivalent to Lurie's approach, another fact which took longer to prove that anybody had hoped

Thorgott

category theory is a field where most statements are kind of obviously true and the proof is basically forced upon you by virtue of basic syntax
higher category theory is a field where most statements are kind of obviously true but it took a decade to prove that

complicial sets being an equivalent approach to higher categories as higher Segal spaces is from $2$ years ago

Ben Steffan

23:29

my respect for people who actually work on this grows with every new cursed corner I have to discover

Transcript for

$Mathematics$ Mathematics