Mathematics - 2020-11-06 (page 1 of 2)

Leaky Nun

00:00

@LukasHeger have you heard of the new series Barbaren?

anakhro

@MikeMiller could it maybe be done kind of "piece-wise linear"? Maybe only for simple things that behave nice enough like knots.

Lukas Heger

@LeakyNun no

Leaky Nun

Barbaren (Fernsehserie)

Barbaren (Arbeitstitel: The Barbarians, internationaler Titel: Barbarians) ist eine sechsteilige deutsche Fernsehserie mit Jeanne Goursaud, Laurence Rupp und David Schütter. Als Showrunner fungierten Jan Martin Scharf und Arne Nolting, die zusammen mit Andreas Heckmann das Drehbuch schrieben. Regie führen Barbara Eder (Folge 1 bis 4) und Steve St. Leger. Die Serie wurde am 23. Oktober 2020 auf Netflix veröffentlicht. == Handlung == Die Serie behandelt die Schlacht im Teutoburger Wald in der zweiten Hälfte des Jahres 9 nach Christus. Im Zentrum der Handlung stehen drei Figuren, die sich au...

es ist uber Germania 2000 Jahre vor nun

Rithaniel

But, if I must decide, I want to learn about topology more deeply, differential topology, category theory, number theory (still haven't had a proper course of that), some interesting geometry course or another, complex analysis, some non-standard algebra (non-commutative or non-identity-having rings), and maybe even a set and model theory course

Leaky Nun

why not all @Rithaniel

Rithaniel

00:02

That was my initial answer, yes

Maybe not OR or statistics. Those areas have shown up as being fairly dry

user2103480

@anakhro @AlessandroCodenotti can you think of any absurd space from logic like club sets or a spaces of formulas?

Mike Miller

@anakhro I think if you do PL embeddings of simplicial complexes it's true. I don't want to try to write down an argument.

anakhro

@MikeMiller yeah I was thinking simplicial is probably an easy class. Is every simplicial complex a CW complex, too?

@user2103480 en.wikipedia.org/wiki/Aronszajn_tree

I like these trees.

The nice thing about them is that they give reason for actually being skeptical of Konig's lemma

Or at least, maybe not being as comfortable.

Alessandro Codenotti

@anakhro They are very nice

user2103480

"The stronger proper forcing axiom implies the stronger statement that for any two Aronszajn trees there is a club set of levels such that the restrictions of the trees to this set of levels are isomorphic, which says that in some sense any two Aronszajn trees are essentially isomorphic "

Now that's something that would excite high schoolers

anakhro

00:09

Oh was this supposed to be for high school students?

user2103480

Don't know whether an isomorphism of trees is a homeomorphism under a sensible topology though

Alessandro Codenotti

@anakhro Well ZFC proces that there is an $\aleph_1$-Aronszajn tree, so the next step in cardinality after Koenig's lemma is just false

Above that it's all independent and it gets very complicated

user2103480

Hahaha no I was just kidding about presenting some weird ass homeomorphic spaces as an example of "spaces that are homeomorphic but not obviously so"

And not horned-sphere-weird but logic-weird, which makes it completely detached from anything one would want to present to high-schoolers

Alessandro Codenotti

Getting the tree property at many consecutive cardinals is an active area of research and those things tend to grow very fast in consistency strength

anakhro

@AlessandroCodenotti what have you been working on these days?

@user2103480 yeah I feel like logic is not the way to introduce math to high school students. :P

Alessandro Codenotti

00:14

Some stuff at the intersection of topological dynamics, descriptive set theory and Ramsey theory

More precisely I'm interested in extreme amenability and universal minimal flows of Polish groups

anakhro

I am not familiar with any of these terms to the point where I know how they work together. Could you give me insight on perhaps one potent connection?

Thorgott

that sure is a mouthful

user2103480

extreme amenability sounds dope

Alessandro Codenotti

So for a topological group $G$ extremely amenable means that whenever acts continuously on a compact space $X$, there is a fixed point for the action (an $x\in X$ with $gx=x$ for all $g\in G$)

user2103480

That's less dope than it sounds

Alessandro Codenotti

00:17

It's also called the "fixed point on compacta property" for obvious reasons

There is an important theorem by Veech saying that every locally compact group admits a free action on a compact space (its so called greatest ambit for example), so extreme amenability is in some sense an "infinite-dimensional" phenomenon

Thorgott

now that sounds dope again

user2103480

Where do these things arise?

Alessandro Codenotti

One of the first examples of such groups was discovered by Pestov and is $\mathrm{Aut}(\Bbb Q,\leq)$

Thorgott

order-preserving bijections of Q?

Alessandro Codenotti

yeah

Pestov's proof relied on the infinite Ramsey theorem (and logic people actually tell us that it is equivalent to it over some weak theory but who cares)

anakhro

00:22

Pestov...first name?

Alessandro Codenotti

Vladimir

anakhro

Oh I know him.

Alessandro Codenotti

Oh nice

anakhro

Would have thought there were more math Pestovs.

In any case, so this reliance on the infinite Ramsey theorem is less than ideal?

Alessandro Codenotti

Later Pestov, Kechris and Todorcevic expanded that and classified all the extremely amenable closed subgroups of $S_\infty$, they turn out to be the automorphism groups of countable structures in some language that are Fraisse limits of a Fraisse class of rigid structures with the Ramsey property, which basically says that this class of structures satisfies a Ramsey-like theorem

@user2103480 So they tend to arise as automorphisms groups of structures

@anakhro Well it can't be avoided, so it's fine

Thorgott

00:26

starts sweating profusely

Alessandro Codenotti

There's also a bunch more extremely amenable groups known, orientation preserving self homeomorphisms of $\Bbb R$ (or of $[0,1]$, they are the same group) as well as the unitary group of $\ell^2$ apparently

anakhro

When you say "Ramsey-like", what in particular is like Ramsey?

Alessandro Codenotti

I don't understand the latter very well

Ok so, the classical infinite Ramsey theorem says that $\omega\to(\omega)^n_m$ which I now need a moment to unpack into a readable sentence

Mike Miller

@anakhro Yes, but not the other way around.

For a simplicial complex the n-simplices are your n-discs.

anakhro

@AlessandroCodenotti heh.

@MikeMiller is there a classic counterexample?

Alessandro Codenotti

00:29

So whenever you pick a colouring $c\colon[\Bbb N]^n\to m$ there is an infinite $A\subseteq\Bbb N$ such that $c$ restricted to $[A]^n$ is monochromatic

Mike Miller

I don't know one but I'm sure it's easy to construct.

Alessandro Codenotti

Like the intuitive way is when $n=2$ and $m=2$, then you are building a graph on $\Bbb N$ (you colour pairs of integers depending on whether there is an edge between them) and the theorem tells you that your graph will have either an infinite complete subgraph or an infinite independent family

$n=1$ is pidgeonhole, if you colour $\Bbb N$ with finitely many colours, then there is an infinite monochromatic set

Mike Miller

As a fancy counterexample topological handle decompositions exist in every dimension at least 5, and give CW decompositions. But there are nontriangulable manifolds.

Alessandro Codenotti

For two structures $A,B$ in some language let $\binom{A}{B}$ denote the set of all substructures of $A$ isomorphic to $B$

Then the arrow notation $C\to(B)^A_k$ means that whenever we pick a colouring $c\colon\binom{C}{A}\to k$, then there is a $B'\in\binom{C}{B}$ such that $c$ restricted to $\binom{B'}{A}$ is constant

anakhro

Okay, I think I see.

Alessandro Codenotti

00:40

And a class of structures $F$ has the Ramsey property iff for all $A,B\in F$ and $k\in\Bbb N$, there is $C\in F$ with $C\to(B)^A_k$

So the infinite Ramsey theorem says that when $F=\{$finite linear orders$\}$, then $F$ has the Ramsey property

Anyway I should go to sleep now, it's 1:40am over here

Mike Miller

@anakhro I saw on math overflow in an answer by tom goodwillie that you can take the mapping cylinder of a really bad self map of the interval.

You can picture that as a crumpled curtain, where the stuff at the bottom gets pasted to other stuff at the bottom.

anakhro

@AlessandroCodenotti I will maybe bug you later, goodnight!

@MikeMiller very peculiar.

Mike Miller

So if you crumple it a lot it shouldn't have the local structure of a simplicial 2-complex. Not much in the way of local structure there.

anakhro

Can you say much else about this space?

Mike Miller

Yes, ok. So imagine pinching the bottom of the curtain together once. (Corresponding to, say, the self map |x-1/2|+1/2). This looks like a trough with the front end open.

Perfectly good simplicial complex. But you could also imagine folding together the bottom of the curtain multiple times (say your map of the interval goes up and down and up again).

Then, near points which are in the image thrice, the space locally looks like (<-) x I, where <- means the cone on three points.

Still all good. So keep folding. If you fold finitely many times you will get something homeomorphic to a simplicial complex. But if you fold infinitely many times... and do it in a neat enough way... you should get a local model that looks like (infinite cone) x I.

This infinite cone nonsense can't be done in a simplicial complex.

Certainly not a finite one, which it would have to be, since our space is compact

user2103480

00:58

@MikeMiller one exercise is proving that chain homology defines a functor from chain complexes to the category of Z-graded abelian groups. Now I don't really know what's expected here, since we haven't officially covered chain homology, and my first reflex was "uhh isn't it a functor from chain complexes to chain complexes?" (correct me if I'm wrong)

And now my first guess would be to just take the sum of the homology groups and check whether a chain map translates to a homomorphism between such sums

anakhro

Infinite cone being like the hourglass shape, or just one side?

Mike Miller

I mean the cone on an infinite set. A wheel with infinitely many spokes.

@user2103480 You can think of a graded Abelian group as a chain complex with 0 differential but I don't see the point

anakhro

Oh I see!

user2103480

@MikeMiller You mean one single group, everything else 0, and all d_n = 0?

... 0 -> 0 - > G -> 0 -> ...

Thorgott

I'd assume the group with grading $k$ at the $k$-th place and all maps $0$

Mike Miller

01:10

Yes

epic_math

01:21

Hello!

Rithaniel

Anyone know any "real world" examples of non-associative quasigroups?

Thorgott

does something like positive reals under exponentiation work

Rithaniel

Well, yeah, but I'm thinking like "scrambles on a Rubiks Cube" is a "real world" example of a group

Something concrete, a little bit outside the clarity of purely symbolic descriptions

anakhro

I think of the Rubiks cube thing as more of an action than a group.

Rithaniel

Yeah, the Rubiks Group acts on the Rubiks Cube. So, perhaps I should say "Anyone know of any real world things which a non-associative quasigroup acts on?"

anakhro

01:34

I always liked the magma of rock paper scissors.

Rithaniel

Hmmm, is that closed? I'm thinking of something that returns "victory/loss/draw" instead of "rock/paper/scissors"

redivider

Hello I am trying to solve a diff eq system, puu.sh/GKx5U/590d0eac47.png of these 2 sums in wolfram mathematica. Well and basically I cant figure out how to do it, any help appreciated. Here is my attempt puu.sh/GKx7D/b16f6ab0fe.png

And forget the u(x,t)

anakhro

@Rithaniel xy = winner

So rp = pr = p, rs = sr = r, sp = ps = s

It's non-associative, and commutative, with no identity.

Rithaniel

Ah, I see. What about ss? Just s because tie?

Thorgott

sadly it's not a quasigroup

Rithaniel

01:45

Yeah, I don't know if I've ever seen a quasigroup action on something

anakhro

Yes. And yeah, sorry it's not a quasigroup. But it's an algebraic rephrasing of a children's game.

Rithaniel

Yeah, I like applying algebraic structures to games. I'd like to make a video game that relies on a ring structure full of zero divisors as part of its core mechanics, actually

rschwieb

02:14

@LukasHeger Thanks for the pointer! But how does Joe mathematician see that the Henselianization isn't all of the completion?

anakhro

@Rithaniel "Quasigroups are precisely groupoids whose multiplication tables are Latin squares." <-- maybe can conceive of one from Sudoku, Kamisado, or something that exploits Latin squares.

Also, the averaging operation (on R) is a quasigroup.

Could stretch that to be a real world example. :P

Rithaniel

Ah, operation is the average of two elements. That I can see as something you could see in real life, yeah

anakhro

At least, the non-associativity of it is a common mistake made by students in my experience.

Rithaniel

It's a good place to mine for thoughts, at the very least

anakhro

Are you just looking for such an example out of curiosity, or did you have something in mind for what you wanted to do with it?

Rithaniel

02:21

Curiosity, right now, but, honestly, all things I learn I store away with the intention of somehow using them at some point

Thorgott

you can probably get an example out of discrete heat flow or something

mathguy

Good evening!

Can someone explain what a coxeter group is?

anakhro

02:41

@mathguy what about the definition on wikipedia doesn't make sense to you?

mathguy

Haha I didn't search the definition

Wanted someone to explain

opticalic

03:16

Hello. I would like to know if you guys think Rudin is pedagogical. I'm studying it and I'm still at chapter 2. When he's going to prove Heine-Borel he strongly makes use of past theorems, such as in "Theorem X and Theorem Y shows that (a) => (b)."

From a pedagogical point of view, isn't it more difficult to understand what exactly is going on? Or on the contrary, this is indeed the intended effect, as it's more interest to prove general theorems and then use them afterwards?

skullpatrol

There's no way around the fact that math builds upon itself.

Thorgott

I don't think Rudin is pedagogical at all, but for other reasons

I don't recall Rudin's proof of Heine-Borel in detail, so maybe he's not making the invocation of previous theorems lucid enough for the reader to understand (this certainly is an issue in the book from time to time), but, at least in principle, proving theorems by making use of previously proven theorems is how most mathematics works

opticalic

Maybe the previous theorems are not very clear for me, so that's why I felt a confusion when he summoned them. Thanks for your answers.

skullpatrol

Are you studying independently?

opticalic

03:32

I'm having classes, but the professor is not using Rudin

skullpatrol

Have you discussed your choice with him.

opticalic

I'm a little shy but I will try to ask him what he thinks.

Thorgott

I would recommend against Rudin if you're learning analysis for the first time

opticalic

Anyway, this course goes till 18 December and we are beginning continuous functions now. I'm not sure how quick things will going to be

@Thorgott Yes it's my first time

mathguy

Yeah only a few beginners can understand Rudin

opticalic

03:38

Is it worth the extra effort?

skullpatrol

Sep 21 at 22:47, by Ted Shifrin

I didn't say need, but Rudin is about the hardest text there is for real analysis.

opticalic

"Somehow math students have this macho idea that one must do Rudin to be a real math student."

LOL

I was indeed thinking that if I study through Rudin I would become better at intro analysis

mathguy

About three months ago I was a beginner in real analysis and was gonna buy Rudin. Thank god I bought some other book

It's hard

opticalic

I think I'll go back to the book that professor is using, which is kind of standard here in Brazil

skullpatrol

Wise plan.

epic_math

03:43

I found out that my laptop was corrupted with a virus

It deleted my important files

Even the files which I paid for

Thorgott

Rudin won't help you with understanding analysis on a conceptual level, it would only present you with more results and more ways of applying your knowledge in various ways in the exercises

it can be rewarding to work through Rudin when you're reviewing analysis/for a second course, but it's not good to learn from for the first time

mathguy

Give yourself a medal if you understand Rudin.

Hahahaha

opticalic

I was going to reconsider another book after finishing chapter 2, maybe Pugh or going back to Elon Lages, but I'm staying much more time than I was before (with Elon) in each theorem's proof.

Maybe because when he says "neighborhood" I can't help but think of an interval in the real line, and I was trying to get rid of this, as I would like to understand better the metric space generalization

feynhat

04:04

I like the Red Book of Chairman Tao.

He starts with Peano's axioms sand constructs real numbers. Very few books on real analysis actually care to define what 'real numbers' actually are.

rschwieb

@LukasHeger oh wait, does cardinal iff alone allow you to say it’s proper? Looks like the henselianization would be countable, if it is a subset of algebraic elements over integers

skullpatrol

04:40

Norman Wildberger goes through a list of the most popular real analysis textbooks complaining about that @feynhat

mathguy

@feynhat Ya I also learned real analysis from that book.

epic_math

Hey can I say a joke

opticalic

@feynhat lol

epic_math

Why are math books often sad?

skullpatrol

@epic_math They're always looking for their x?

epic_math

04:52

Ya but the answer in my mind was because they have a lot of problems

Who was the roundest knight at King Arthur's round table?

skullpatrol

Problems with their x

Sir Cumference

epic_math

the real answer is...

Sir Cumference.

skullpatrol

The mostly absent theory of real numbers|Real numbers + limits Math Foundations 115 | N J Wildberger

►

epic_math

I found the shortest proof of RH. Do you want to read its abstract?

skullpatrol

ok

epic_math

04:58

"The proof is trivial and left as an exercise for the reader"

"We can similarly prove the Hodge conjecture, which is also trivial"

"The other millennium prize problems are corollaries of these"

opticalic

05:15

marxists.org/archive/marx/works/1881/mathematical-manuscripts

That analysis book "Red Book" of Chairman Tao remembered me of these

Marx's Mathematical Manuscripts: Two Manuscripts on Differential Calculus

skullpatrol

06:48

6

Q: Will we have a winter bash 2020 this year?

(or some other end of year fun?) At the end of last year, there was discussion suggesting that "winter bash 2019" (with the hat thing) was the last one (see here to learn more). Now that December is near, I would like to know if the 2019 winter bash (with the hat thing) was indeed the last one? A...

Sal

07:20

Dumb question: is this true? $$\sum_{n=1}^{\infty}f(n) = \lim_{N\to\infty}\sum_{n=1}^{N}f(n)$$

Edward Evans

yes, that's the definition

Sal

Thanks!

Edward Evans

:)

epic_math

Hello @Sal!

Hi @EdwardEvans!

Edward Evans

Hi @epic_math

epic_math

07:27

Mathvengers: Endgame

Newton vs Leibniz

Hey does anyone wants to see the personality of different set theory symbols?

Edward Evans

@epic_math youtube.com/watch?v=9p55Qgt7Ciw

Take a look at this :)

epic_math

Yeahhh

zacts

What is a number system?

And what does it mean to create a new number system?

Edward Evans

just a system of representing numbers

zacts

Hum... that might be interesting to me.

Edward Evans

07:38

to create a new number system is to create a different system for representing numbers

it's not that interesting, the history in that video is the interesting thing

zacts

Thanks for the link.

epic_math

What are some examples of conjectures that were widely thought to be true but it turned out that they were false?

mathguy

Oh man what a coincidence I was going to ask this on MSE

I will give the link to you

Will that question be closed?

zacts

Squaring the circle

Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge. The difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square. In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem which proves that pi (π) is a transcendental, rather than an algebraic irrational number; that is...

mathguy

Please also answer it there :)

zacts

07:44

oh, sorry

mathguy

This is the question:math.stackexchange.com/questions/3896170/…

There is a duplicate, and I have mentioned it. It is almost a year old and doesn't have enough answers, so please don't vote to close the question as duplicate.

zacts

answered!

mathguy

Thanks @zacts

zacts

08:05

@EdwardEvans. I don't know. I kind of find the idea of number systems to be intriguing, or an idea like that. I haven't studied abstract algebra yet tho.

Edward Evans

I'm just saying, it's a number system with very restricted uses (can only represent up to 4 digit numbers, for example)

zacts

ah, I see. :-).

mathguy

Are there any conjectures in abstract algebra that an intermediate level reader can understand?

I am really interested in conjectures.

Martin Sleziak

As pointed out in comments, there is a similar question on MO: Examples of conjectures that were widely believed to be true but later proved false

zacts

08:32

What is the primary difference between MO and MSE?

Martin Sleziak

08:52

@mathguy This is a list of open problems on WP, but you'll also find various conjectures there: List of unsolved problems in mathematics.

@zacts Several questions about this are linked in the mathoverflow tag-info on our meta and in math-stackexchange tag-info on MO meta.

user2103480

09:47

@zacts research level and not necessarily research level

Edward Evans

Y0

user2103480

el trompo is ouuuut

georgia was flipped

Astyx

Was it?

Edward Evans

woah

Astyx

I still see 460 votes in favor for Trump

user2103480

09:50

917 for biden now

Astyx

Yus

user2103480

460 is old news

Astyx

When did they start counting again? Is it no 4AM there?

user2103480

idk

its not unusual to count the night through

Astyx

They didn't yesterday

AFAIK

mathguy

09:54

I want terence tao as the president.

user2103480

@mathguy maybe the most scientifically qualified person isnt always the best politician

Astyx

I doubt Terence Tao wants to be president

mathguy

The reason is that he will integrate the people so they will unite.

He can integrate quite skillfully

user2103480

So you're talking about mushing all people to jelly and throwing all the blobs together?

mathguy

Haha haha

He will not differentiate

Benoit B. Mandelbrot = Benoit Benoit B. Mandelbrot Mandelbrot

Edward Evans

10:01

p-adic Hodge Theory just went from 0-100 in 1/c seconds

wait what

really slow

hahaha

man that was insane

mathguy

We all say "man," but what if a woman is here?

Just joking

Edward Evans

First theorem: $(\Bbb C_K)^H = \widehat{(\overline{K}^H)}$, second theorem

$$\Bbb C_K \otimes_{\Bbb Q_p} H_{ét}^n(X_{\overline{K}}, \Bbb Q_p) \cong \bigoplus_{q}\lbrace \Bbb C_K(-q) \otimes_K H^{n-q}(X, \Omega^q_{X/K})\rbrace$$

Astyx

Bless you

Edward Evans

starting to think this might have been a bad idea

Alessandro Codenotti

10:20

Too late to turn back

Astyx

"'cause I'm in too deep"

user2103480

@EdwardEvans me, 6 weeks into every semester

Edward Evans

this is the second lecture

hahahaha

user2103480

@MikeMiller turns out that taking the sum of all homology groups was actually the solution

Nice. My lecturer just said "I'm not interested in algebra, I'm interested in topology"

Astyx

angry Thorgott noises

user2103480

10:33

It was the most beautiful sentence I've heard today

Mike Miller

@user2103480 Depends what a graded Abelian group is. I'm not going to argue with him because I share that opinion

user2103480

@MikeMiller The context were long exact sequences and how we use their properties

for the algebra comment

Mike Miller

Sure I agree with him

I don't like to do the algebra junk til after it shows up topologically

user2103480

10:56

Now the lecturer is trying to convey to a student that pictures suffice as proofs lmao

Mike Miller

His answer ought to be "You can write the proof from this"

This is what I do in a point set class

I train people to extract proofs from pictures

That way they don't whine about this in algtop :p

user2103480

@MikeMiller yeah that was the answer, learning to translate from picture to proof and back is a necessary skill

Mike Miller

Good

This is exactly the reason people say Hatcher's book is nonrigorous. They haven't developed that skill

Balarka Sen

11:23

What's new

Edward Evans

prof nuked us today

so that's new

robjohn

@EdwardEvans tac nuke?

Edward Evans

Strategic hydrogen fusion missile launched from space

Balarka Sen

nucular

Alessandro Codenotti

Hi @Balarka

Balarka Sen

11:32

hi

Alessandro Codenotti

Some days ago you asked me why should one care about extremely amenable groups, I think I have a decent answer now if you're not busy with other stuff

Balarka Sen

Tell me later today? I'm trying to finish an assignment due later tonight :P

Alessandro Codenotti

Sure, good luck with the assignment, what's the subject?

Balarka Sen

Physics lol

Alessandro Codenotti

rip

user2103480

11:34

@BalarkaSen :what type of

Balarka Sen

EM

user2103480

ah

mathguy

Hello @BalarkaSen! I didn't see you in this room for a day. Where were you?

Balarka Sen

Getting work done

mathguy

Oh

I think you are busy today

Mike Miller

11:52

Once you get busy you remain that way til the day you die

mathguy

Hahaha

Thorgott

@user2103480 what does taking sum have to do with functoriality

Mike Miller

He was talking about understanding what the graded group of homology is

Balarka Sen

graded homology group is kind of pointless

unless you talk about the coalgebra structure

Alessandro Codenotti

Put a PTSD warning or something before writing those things, thanks

Thorgott

12:04

ah, so it was just a definition issue

Balarka Sen

@Alessandro -<

Thorgott

@BalarkaSen does that exist in general?

Balarka Sen

ye

Thorgott

how does it work

Balarka Sen

here we go again

$H_n(X; F) \stackrel{\Delta}{\to} H_n(X \times X; F) \cong \bigoplus_{p + q = n} H_p(X; F) \otimes_F H_q(X; F)$

Thorgott

12:09

hello homology my old friend, ive come to describe a coalgebra structure on you once again

Balarka Sen

$F$ is a field or $\Bbb Z$, so the last isomorphism is Kunneth. Actually there's still an arrow for a general $F$ I suppose, but I am not thinking too hard.

lol "general $F$". "let $F$ be a commutative unital ring"

Nah the Kunneth arrow goes backwards for homology

Thorgott

ah ok, but that's only for singular homology, right

Balarka Sen

lol, oh, by "does that exist in general?" you mean "homology of a chain complex"?

then of course not

Thorgott

yeah, the original context was arbitrary chain complexes

which is why I had to ask

Balarka Sen

gotcha

Mike Miller

12:12

@BalarkaSen No, you need F a field. The last map requires Kunneth iso which fails usually over any non-field

You have Tor terms at least

Balarka Sen

There's no arrow for a general F

because (simplex) x (simplex) -> (simplex)

No but Kunneth short exact sequence is split, so you can project.

There's no canonical splitting though

Mike Miller

Right so no coalgebra structure :P

If it's not a canonical splitting you will fail when trying to prove associativity

Balarka Sen

I am aware

Mike Miller

Got it

Leaky Nun

12:27

@BalarkaSen thats what a robot would say

user2103480

great so now I have to translate physicists unrigorous definitions to measure theory

Semigroups? Transition kernels? We don't do that here

Astyx

Every function is a Schwartz function

That's the fundamental theorem of physics

user2103480

@Thorgott If you have a chain map, that induces a homomorphism from the sum of of the homology groups of one chain complex to the other

Balarka Sen

@LeakyNun Haha

user2103480

@Astyx Yeah and we can mix up distributions with measures anytime we want

Balarka Sen

12:32

yeah ^

Astyx

Insert "they're the same picture" meme

user2103480

@Thorgott you can define homology groups but they need not satisfy the eilenberg-steenrod axioms

e.g. excision

Balarka Sen

Eliezer Yudkowsky, minutes after his mind has been uploaded in a quantum computer: "I am aware"

user2103480

haha yeah he probably knows

almost surely

Thorgott

ye

Balarka Sen

12:38

I should just join MIRI

because why not

its a scam but everything is

its not a worse scam than academia

nothing is

Thorgott

12:50

doomer hours

Mike Miller

It's a boring scam @BalarkaSen

user2103480

what's MIRI

Mike Miller

That's the difference

Balarka Sen

@MikeMiller what's an interesting scam

Mike Miller

The kind of scam that lets me tell people for an hour about orientability of topological surfaces

Balarka Sen

12:52

LOL

user2103480

@BalarkaSen TDA

Or convincing business people into letting a neural network control a plane

fun prank

Balarka Sen

TDA is a good scam ngl

most of stat is a scam imo

Mike Miller

I finally figured out the missing piece in my understanding of orientation of surfaces

It's what you need to define w_1 and to prove that orientable => no embedded Mobius

You really have no choice but to define local orientations

Balarka Sen

yeah i got trapped by that once

i tried to extract out all RP^2 summands by finite generation of $\pi_1$

Mike Miller

Yeah I don't think I can present that proof sadly

Balarka Sen

12:59

and then argue $H_1$ would have torsion or something

Transcript for

$Mathematics$ Mathematics