Et donc l'intervalle $[\delta,A]$.

Yeah, I have wanted to visit Greece. My father has been there.

@TedShifrin son image est dans [1/2, 2]?

non, $[1/2,2]$ ...

oui pardon

et donc il y a un valeur de $t\in [\delta,A]$ pour que $g(t)=1$.

oui j'ai compris merci beaucoup @TedShifrin

Bonsoir :)

If I try hard, I can learn French just by observing

LOL, @Avantgarde. Can you learn math just by observing?

bonne nuit

Man I wish I kept speaking french after high school. Maybe I'd still remember it then

You should work on reading some serious math in French.

remembers that Cookie is off the friend list

@TedShifrin I guess. If I know the notations and language

isn't @Alessandro supposed to be un-unsleeping?

I could go and transcribe your lectures into french. Surely that would earn one a spot back on the list ;)

I've read a little bit of math in French. It's easier than literature.

Yeah, unless it's very prose-filled.

(At least I think.)

Best to read Bourbaki then lol

I have no lectures until the end of February so my sleeping schedule is suffering

I actually have a published paper in French (and gave a bunch of lectures in Paris in French many years ago).

I tried to read Albert Camus L'Etranger in french

I have a question: how do you decide what is a 'good' (whatever that means) definition in mathematics?

Camus is great

L'Étranger?

Ah, there it is.

I failed and switched to english D':

It's not bad.

Try reading Proust :P

L'étranger is great

we read it in my highschool course. That was fun.

Alessandro probably can answer Avantgarde's question :) He's our most logical one.

I don't know how it's done in mathematics. I'm in physics. We decide what is 'good' by making sure it explains the experimental data, respects certain previously-known symmetries, etc.

There's a question of making sense ("well-defined") and a question of being interesting.

Yes.

Perhaps as a subsidiary of being interesting, one should check that such an object exists :P

I think at some point I had a hypothetically very interesting idea that didn't exist.

Well, say you're back in the 19th century and inventing differential geometry. What motivates the definition of a manifold, and not anything else? It seems to me that there can be very many possible definitions. I can see, though, that many of them will lead to inconsistencies within the theory, later on. But that should take years and years of trial and error?

Sorta like the apocryphal story of the MIT Ph.D. thesis on functions with Lipschitz exponent > 1. A whole hundred pages on constant functions.

loll

Well @Avantgarde (I'm by no means an expert) but a lot of what we read nowadays has been "optimized" and does not represent the real development of the ideas over the years.

@Avantgarde: You've already thought hard about surfaces, for example, and how so much depends on their being "locally diffeomorphic" to a bit of the plane.

Most math definitions come from generalizing things already thought about ... but not all.

@Antonios-AlexandrosRobotis True

@TedShifrin Yeah, I get that

didn't manifolds first arise out of complex analysis (elliptic integrals and multivalued functions etc.)?

Isn't math the same as Avantgarde explained, except we have much less "experimental data" so we force our definitions to be precise under a much more intense level of scrutiny?

yes, @Antonios is right that most modern textbooks have nothing to do with historical development.

@MatheinBoulomenos I would expect that they originated primarily from the study of multivariable calculus or the vanishing sets/level sets of smooth functions.

no, that's Riemann surfaces, @Mathei ... Gauss thought about surfaces well before that.

But Riemann surfaces also played a part, I'm certain.

I suppose I should look up dates ...

I meant abstract manifolds. Gauss taught about them embedded in $\Bbb R^3$, right?

Ah, well that's a whole different story.

I don't think Riemann surfaces started off so abstract, actually.

Riemann in his Ph.D. (the German equivalent) developed the notion of a Riemannian manifold, abstractly.

embedded, I assume?

No.

Oh wow.

I suppose I should go back and look.

But I don't believe so.

He had an arbitrary Riemannian metric (positive definite symmetric 2-tensor).

I thought the intrinsic study of manifolds started a lot later than that.

Oh wow. That's really cool.

Got an hour of sleep... oh well

um, go back to un-unsleep

Yeah...

Also yeah Riemann defined an abstract metric

It's a cool intuition, that to define length you only need a pointwise dot product...

That's probably how he came up with it

No idea how he came up with his curvature tensor though!!!!!

lol

I think it came from understanding surfaces ...

I'd have to go back to some reading.

hi @TedShifrin

hi Karim

@TedShifrin I wrote first 15 pages of my thesis looks pretty good

I am planning to have it 150 pages for my master thesis

I'm sure they'll end up changing a lot as things progress

yeah

LaTeX is so wonderful. Imagine having to do this all by hand and with an old-fashioned typewriter. That's what I did for my PhD thesis.

@user193319 if you have a topology on $\Bbb C G$, why not take the product topology on $M_k(\Bbb C G)$ if you identify it with $(\Bbb C G)^{k^2}$?

oh man that would have been so hard especially if you want to include images and stuff @TedShifrin

@TedShifrin Oh god I can't imagine that.

How many years did that take

It was the dark ages, guys.

You don't even realize.

Sounds awful.

For example when I was explaining the rational equivalence geometrically as interpolating using a cycle on $P^1 \times X$ you can explain things using 1 single picture.

it would have been super hard to do that by hand haha

If it's a small thesis of about 50 pages, then it's fine.

My thesis was about 100 pages typed double-spaced, I don't remember.

rational equivalence is the algebro-geometer's cobordism

nice

and what do you say for linear equivalence? :)

Yeah there is two ways of rational equivalence the right notion using interpolation or the more algebraic notion both are equivalent

a fibration over $\Bbb P^1$?

@MatheinBoulomenos Hmm...that's a possibility I haven't considered. I'll have to see which suits my purpose. Thanks for pointing that out.

yeah

@TedShifrin I forget what that is

I think it is like a linear systems of divisors

I am not familiar with classical algebraic geometry so much

One of my todo list after my master thesis is done is to read couple of classical algebraic geometry books

to develop more intuition

Better start with understanding linear equivalence of divisors before you're up to rational equivalence.

intersection theory is just mostly pilfered off from classical algebraic topology

what is a good book that talks about linear equivalence of divisors @TedShifrin ?

Well, of course G/H does.

it's applying the vulgarization functor to beautiful geometric pictures to turn them into hideous algebra

LOL, Balarka, your lack of sleep is making you cranky.

as the algebraists always do

Awesome I will read it.

@TedShifrin It's a running gag of mine

vulgarization functor LOL

@BalarkaSen it is bad to judge mathematics I am taking a more democratic approach of wanting to learn all methods and not hold a grudge against different types of intuition.

It's just a joke.

Nobody is judging anybody

Karim: Don't take him so seriously. He started off an algebraist. Mike and I perverted him :P

haha

topology is basically hentai

you see things

lol yeah I agree lol

Bezout's theorem is a older than algebraic topology

algebra is like literotica

the visual aspect is completely gone

too bad for them

algebra is very non-visual yeah. Altough algebraic geometry combines both approaches of transforming pictures into algebra and algebra into pictures.

I don't want it to be visual. I prefer actual proofs to pretty pictures

pretty pictures can be actual proofs too

yeah

Eg ~~~

but ugly pictures sure can't be!

this is the proof of h principle for immersions

Done!

sometimes if I can visualize the situation at hand, then I don't need to be very formal in my reasoning.

That's how you end with garbage proofs, Karim.

Ugly picture is basically synonymous with handwaving

Nah handwaving is pretty important.

haha

I draw better pictures than you do, Demonark. What's your point?

@TedShifrin is that vacuous?

The person I met who understood math the best (among my undergraduate peers) was extremely good at going between handwavey explanations and rigorous proofs.

But yeah I was agreeing with you!

And yet I still get roasted

lmao

roasted

toasted

:explosion noises:

Antonios: Did that look like roasting to you?

I'm just gonna become a contrarian now

there's a picture in an analysis script. No person I asked understood the picture (though the proof was fine, the picture was only for illustration, this wasn't topology after all)

I wish. It's cold over here.

I have zillions of pictures in my books, but not once is a picture alleging to be a proof.

@Antonios-AlexandrosRobotis My policy is that if you can't handwave enough + translate that handwave to paper or pictures, you don't understand it well enough

"now" ? @Demonark

I agree wholeheartedly @BalarkaSen

yeah I completely agree with you @BalarkaSen and @Antonios-AlexandrosRobotis

I can handwave algebra, too

Oh my dude Daminark you are getting roasted

I don't disagree

Handwaving is an important way of remembering the big details or stepping stones of an idea.

yeah

Few of us have the mental faculties necessary to remember all the details without some sign-posts.

@TedShifrin by my standards, my previous self was toned down big time wrt being contrarian

@MatheinBoulomenos I still think you're secretly a topologist

maybe a homotopy theorist

Modern homotopy theory is mostly category theory

how is your semester going @Daminark ?

I rue the day that Demonark became a chat denizen.

ahaha

Being a homotopy theorist, doesn't sound too bad. I'd still prefer doing ring theory or something like that

@MatheinBoulomenos That is true. But it's category theory with a purpose!

Ring theory seems too hard lol

@Adeek going pretty well, classes are fun

nice

Also many people are still getting inspired from a chunk of topology

Though algebra is... taking its time

Eg Mike Hopkins's work on tmf's being inspired from Thom's calculations of higher homotopy groups

using topology

Taking its toll, you said?

@Daminark Look at this beautiful pic

This is the church of topology

Draw some pictures of Sachs-Uhlenbeck bubbling, while you're at it, Balarka.

I don't dislike topology. I like to think about free groups as fundamental groups of graphs sometimes. I don't think Lurie draws a lot of pictures in Higher Topos Theory, though

@Mathei Lurie is a great man. His notes on low dimensional topology is absolutely brilliant

He knows his topology man

@BalarkaSen I feel like I have sinned just by looking at that. Gonna go prove Snake lemma to purify myself

@TedShifrin I remember Mike saying things about that at some point

Have you done the 5-lemma yet, Demonark?

@Daminark you're a good and honerable man, lol

Yup, it was fun

don't forget to prove naturality in the snake-lemma

Prove the Salamander lemma

I had math 250A at Berkeley last year with Sug Woo Shin (who is a really great guy btw). On the day we proved the Snake Lemma, he burst in the door and said in his lovely accent "ARE YOU READY TO FIND THE SNAKE?"

you need some 3d-diagrams for that

250A is grad algebra*

@MatheinBoulomenos applications of geometry to algebra

shneck lemma

Too much sexual innuendo in mathematics.

I love it

that Lang exercise @TedShifrin. No coincidence.

Wait 'til all the sexual harassment in math hits the papers.

we had the snake lemma, 9-lemma, 5-lemma as exercises in our undergrad algebra course. The reasoning was that you don't learn much by seeing a proof done for you

There are really a lot of horrid misogynistic males.