Hey @BalarkaSen

Hi @Ali

@BalarkaSen What have you been looking at today

A bit of Topology, a bit of geometry.

math problem is destroying me

i am not worthy of it

@BalarkaSen What are you doing in topo

Vector bundles and characteristic classes.

oo cohomology

.

@BalarkaSen Have you ever heard of Jordon algebras?

Nope

What are those?

Take a field

define $x\circ y=\frac12(xy+yx)$

like a baseball field

@ForeverMozart what problem

actually screw field make it an associative algebra

@syzygy do you know what a continuum is?

set theory?

topology

thanks not interested^^

lol

@BalarkaSen Make sure the product commutes and satisfies $(x^2\circ y)\circ x= x^2 \circ (y\circ x)$

general topology or algebraic topology?

general

@Lozansky a contour integral would be like the average value of f(z), but modulated or "twisted" by the direction of the contour at every point

@AliCaglayan Er. Fields are commutative, so $x \circ y = xy$?

general topology is not worthy of my attention

sorry not interested

i dont do alrebraic

Right, @Ali.

So why do I care?

Ok here is the fun part

Take this construction $F(L, J)=\operatorname{Der}(L) \oplus (L \otimes J) \oplus \operatorname{Der}(J)$

Where Der is the derivation algebra

L is a lie algebra

and J is a jordan

presumably one should motivate the construction first

Go ahead.

make my day

Then like read this

math.nsc.ru/LBRT/a1/files/mccrimmon.pdf

and search magic

the first hit should give you a square

cba writing that out

but there is a symmetry that comes out

yeah, the FT. baez also writes about it in the octonions, haven't gotten to it yet

non associative ugh

I probably on some list at Uni for typing magic tits all the time

wtf

Why's it magic?

LOL :D

@BalarkaSen There is no reason for it to be symmetrical

mm. I see.

This shows it a bit better

Freud-Tits for short

@ForeverMozart I heard they use this stuff in control theory

Sigmoid-Freud-Tits theory I believe

i came across a book on control theory today

but now I am just being cheeky and avoiding diff geo homework

rather accidentally though

i was looking for some things on commutative algebra

@Ali What kind of diffgeo are you doing

i first thought google was trolling me, but apparently not

@BalarkaSen first course stuff so I can get a degree

I don't know what a first course should entail

Basically Teds notes

Me too.

But other than that I was speaking to someone who looks at 7 dimensional manifolds

eu.wiley.com/WileyCDA/WileyTitle/productCd-EHEP000815.html

i recommend you buy his book

only 300$

I am working on surfaces right now.

The idea is to motivate a cross product from imaginary octonions etc

You get G2 symmetry and everything its very cool

Heard of G2 stuffs. Don't know anything about it.

There was another guy who specialises in isothermic surfaces

Essentially how can I minimise properties of surfaces when embedding them in spaces

Interesting ones being generalisations of soap films and embedding them in interesting manifolds

you can use techniques from harmonic analysis to prove cases and so on

all very nicely linked

Minimal surface theory sort of stuff?

Yeah

On tuesday I went to a graduate seminar

There was a talk on stabalisers of subspace series of infinite dimensional vector spaces

I can find the paper actually

By that I suppose you mean actions on flag varieties.

people.bath.ac.uk/gt223/paper37.pdf

@BalarkaSen Thats what I thought but the guy said he didn't know about that terminology?

Maybe he just didn't listen to me because I am an undergraduate who knows

you wrote paper?

nono

I met the guy that did

so a series ain't a flag. ok

I am done talking about random bits of math for today though

So he finds stabalisers of these things

then takes their HP radical

then tells me that if the dimension is uncountable then shit breaks down

morning

and provides an example to go with it

Good morning @MikeMiller

The counterexample he gives is actually very simple

Anyway I have to go take care everyone

see ya

so now the noobs are gone i can talk about real math

i do

When it's Ravi Vakil who's asking, he's probably looking for a better answer than that

why do i keep forgetting that i need to write down the unit normal to calculate the 2nd fundamental form

Why am I starting to enjoy complex analysis?

What's wrong with me?

why are we both asking rhetorical questions to the wind? :P

Better yet, why isn't the wind answering? :(

why, why oh why

ok, I give up

you must solve problem

'Simply' seems to have some sort of common meaning across contexts. When something has the $P$ property, is it generally understood how 'simply $P$' relates to $P$?

Interesting. Ruled surfaces cannot be positively curved.

Sure, but remembering the meaning of the sign that should hardly surprise you.

is there a "closed form" for the inverse function of $x+\sin(x)$ using the W function and complex exponentials or something?

@MikeMiller I have an intuitive idea. If the shape operator was orientation preserving near $p$, then moving infinitesimally along the integral curve of any tangent direction (from $p$ to $p_v$, $v$ being the tangent vector) one would curve the same way. But a line passes through $p$ by defn of being ruled, which would mean curvature of all of those integral curves, in particular the lines of curvatures (integral curve of the principle directions) are lines. Does that sound right?

Eh, it doesn't. I don't see a better, more accurate way to put what I am saying.

Hell if I remember what a shape map is. I'm just suggesting you think pictorially.

I think I do understand the picture, I am trying to make it a bit more rigorous.

I don't quite understand how much local information the principle curvatures at a point contain.

@MikeMiller: If I have a parametrized surface, and if Gaussian curvature at a point is $0$, does that mean there's an isometry from an open subset of the surface containing that point to R^2?

These are the kind of things that I don't understand.

More generally, what if I ask this for any two surfaces w/ the same Gaussian curvature at two points?

$\int_0^1\int_0^1 \frac{1}{1-xy}dxdy=\int_0^1\int_0^1 \sum_{n=0}^\infty(xy)^n dxdy=\zeta(2)$, does anyone know an elementary way of evaluating that integral? I'm looking for a solution to the Basel problem I can show to people with a high school level of calculus and Euler's original proof sounds too much like magic. I tried substituting $xy=u$ but the arclength of the hyperbola doesn't have a nice closed form. If you evaluate the inside integral, the outside one gets hard

@Balarka Of course not! That's only true if it's locally zero.

As in, $0$ everywhere on the open set?

yes.

Gaussian curvature zero means that the metric agrees with the standard metric on a chart at that point to second order.

Hm.

I'm curious about a counterexample for a point, though.

literally anything

dude, the curvature of R^2 is zero, so if you're locally isometric to R^2...

If I'm locally isometric to R^2, I'm locally of curvature 0, yeah. But I am having a hard time seeing (or writing down) a surface which is curvature 0 at a point but not on a neighborhood around that point, which then will constitute a counterexample.

maybe I'm being dumb

top of a torus

z=x^4+y^4

I think I understand the torus example pictorially (by top, I am supposing you mean the top of a donut lying on the table, not the top when standing vertically, as in the morse theory picture)

In which case, one of the principle curvatures on the "topmost longitude" does vanish, if my intuition is correct. So $\kappa$ vanishes on any point on that.

Thanks for the equation too, I'll check it.

if should be visibly clear why it's zero there too

ah, yes, it is

Am I missing anything interesting?

now I understand what you meant by k = 0 being equivallent to metric agreeing upto second order to the euclidean metric

Huh? @MikeM ... the top of the torus is a usual circle. Are you slicing in some other way?

@TedShifrin I am asking dumb questions :)

@Ted: He means the top of the donut while lying on the table.

Right. That's a usual circle.

So you see the principal curvature in that direction is 0 because the Gauss map is constant as you move along the curve.

Right.

Now maybe what Mike's talking about is what the normal slice in that direction looks like.

But that should be a curve in a plane.

Well, you've stayed up past your bedtime again. What happened to rebooting sleep clock?

I am not sure what you mean there. I was asking for an example of a surface which has curvature zero at a point but not uniformly zero on any neighborhood.

and that does it, doesn't it?

Ohhhh .... Sure ... I thought that equation had something to do with the top of the torus.

ah, nah.

that equation was just another example

It's actually cool to think about what the normal slice is along that direction. You do see something that's got 4th power in it.

@TedShifrin Yeah, I really should not stay up anymore.

G'night. Diff geo will still be there tomorrow.

Thanks, @MikeMiller. That was an enlightening conversation. @Ted: heh, yes. I'll get back to you tomorrow.

Hello people

I've seen a lot of criticism towards online blackjack, even with leave dealers. Has anyone in Math.SE done some honest analysis on their hands?

I clearly searched for opinions through Google, but it's hard to trust any source due to indirect promotion or sites being sellouts.

good question. i am sure someone here knows about this, but sadly i don't

Quick differential equation problem:

Given $\frac{dy}{dx}=x^2y^2$

Oh wait

I just realized my mistake

forget it

Wait but theres still a mistake

Okay given what was given

We get $\frac{1}{y^2}\ dy = x^2\ dx$

And, integrating both sides, $-\frac{1}{y} = \frac{x^3}{3}$

Shift the negative: $\frac{1}{y}=-\frac{x^3}{3}$

I think I see my mistake again

Gosh I'm a clutz

Heya @dsillman ... stop being a klutz.

Don't forget a constant of integration.

VERY important.

Jesus! My headphones were up very very loud. that @dsillman blip nearly gave me a heart attack

Don't blame us.

You're way too young to get a heart attack.

Yeah right! With the cholesterol in american heart valves, I'm expected to die at 24! Live it out!

Well, exercise plenty (and that doesn't mean with blaring headphones) and eat healthfully. ... Speaking of which, I'm going for a hike to go mail in my ballot. See ya later :)

Cya around