Mathematics - 2017-05-25 (page 1 of 3)

Dair

00:08

@MikeMiller Yeah, cigarettes are not the way to go. As Erdos has shown, it's all about methamphetamines. /s

Dodsy

00:23

@Zee I also am a smoker.

@MikeMiller I don't think cigarettes improve my mathematical ability.

I need a friend.

Daminark

02:48

Today in "Nuking a potato from orbit: proof edition" is proving that the degree of composition of functions is the product of degrees via forms

Semiclassical

I prefer "nuking with a potato from orbit"

or I would, if I knew how to do it

Balarka Sen

nuking the orbit with a potato would be very good

Daminark

Orbiting a potato with a nuke?

Balarka Sen

That's just dangerous

Daminark

Anyway you get the idea, probably a very high powered tool to do the job but why not?

Balarka Sen

02:52

Um, I don't get why that's a nuke. It's perfectly reasonable to define degree of a map as the number the integral of the top dimensional form is multiplied by when pulled back.

Indeed, in algebraic topology, you define degree of a map $f : M \to N$ to be $f^*(1)$ where $f^*:H^n(N) \to H^n(M)$ is the map induced on top cohomology (de Rham in smooth world), and both groups are identified with $\Bbb Z$.

Daminark

I mean given that we defined it via intersection number

Balarka Sen

Sure, if it's defined as the cardinality of a generic preimage that's obvious

But then it's nonobvious why that's equivalent to the version I mentioned.

Daminark

I mean yeah, we did prove that

This is what I'm saying, using the degree theorem to prove this statement when it's possible to prove it outright via intersection numbers is probably not what Neves was going for

Eric Silva

i actually usually think of it in terms of forms

Daminark

Lol

I mean in fairness forms just seem like a cool way to think about stuff

Balarka Sen

03:01

I know you'll hate me for saying this but I can't help but thinking of them as a tool for computation

a very cool tool which saves a lot of work, sure, but i never think about them if i have a geometric alternative

I mean this is probably a personal failing

Daminark

shrugs

I haven't gotten deep enough into that side of the world to say anything

Mostly I've liked that they give a nice way of looking at various things which I've appreciated

Eric Silva

i mean @Balarka maybe you just haven't done the problems that really are natural in forms :P

Daminark

Degree, Cauchy integral theorem

Eric Silva

like lots of things carry the same data as forms, but sometimes things really are natural in that language

Balarka Sen

@Eric You're probably right.

Secret

03:09

[Random] It is not known whether $\tau$ has special numerical properties unique to it but not $\pi$

More generally, it is not known whether given any real number $x$, whether $nx$ with integers $n$ can have new properties that are not found in the base itself

Daminark

Is that so?

@Secret

Secret

03:35

Actually I am not sure if what I said is correct, but I think it holds for multiples of $\pi$ I guess, but not so much for integers (e.g. 0 and 1 are very different, 3 and 2 are different because one is odd and one is even (and even numbers are part of an ideal of the integer ring

so 2, 0 has properties that 3 does not have, and 1 has properties unique to it not shared by other integers

Daminark

Hmm

BAYMAX

Hi,Is there anything that does symbolic calculations?Like $a = \alpha x + \beta y$ and $b = \alpha x - \beta y$ then when i enter $a.b$ it must return me the correct result as $a.b = \alpha^{2}x^{2} - \beta^{2}y^{2}$ ?

Balarka Sen