:(

Then why would you be teaching people one year younger

In high school

Also 21 in grad school, how many grades did you skip?

You can theoretically be 20 in high school, in the Netherlands

@0celo7 what is le energy

Also, grad school is a bit different here

@BalarkaSen It's a functional that gives you harmonic maps

It's used in Ricci flow, which is where this question came from

went over my head

@BalarkaSen You know how in Riemannian geometry, there's something called a length functional?

@Krijn what do you mean

I grew up in Germany but I dunno anything about college there

@0celo7 Eats paths in path spaces, spits out arclength?

@BalarkaSen Basically

Something we want to do in Riemannian geometry is figure out what the shortest paths are, this turns out to contain a lot of information about the topology of the manifold

But the length functional has a square root in it and leads to nastiness

Sorry, I have to run for a bit, be back in a while.

Ok

@0celo7 You choose you Bachelor subject (maths for me) and take classes in just that subject for three years. Then it's followed up by 2 years of Master's. After that you can do a PhD

@Krijn I knew that

So Bachelor is a bit like undergraduate and Master/PhD is a bit like graduate

But not quite the same I think

We do 4 years here, where the first two are mostly BS

I got out of a lot of the BS

I did 4 years of maths, so far

So that should indicate my level, I guess

And I started uni at 17, so this makes me 21 now

OK, back.

@0celo7 Right

@BalarkaSen The length functional is $L[c]=\int\sqrt{g(c',c')}dt$, for reference

I do remember.

bounds are based on the parametrization

$g$ is the metric

Mhm, that much Riemannian geometry I have overheard from people.

You can skip that bit.

It turns out the critical points of this functional are shared with the energy functional $\frac{1}{2}\int g(c',c')\,dt$

Hmm

energy functional because of the resemblance to $\frac{1}{2}mv^2$ from basic physics

That's interesting.

@0celo7 Ah.

@BalarkaSen People also call it "energy functional" in general relativity, where the "mathematical" energy and the "physical" energy are different

Can be confusing

I can imagine.

So it turns out you can generalize that from maps $c:(0,1)\to M$ to maps $f:M\to N$ where $M$ and $N$ are Riemannian manifolds

Just use the very same definition, maybe?

The issue is how to interpret $g(c',c')$

No, one has to be careful. $c'$ is not a vector anymore.

Yeah. Let $g$ be the metric on $M$ and $\gamma$ be the metric on $N$

The integrand becomes $$e(f)(x)=\frac{1}{2}\gamma^{\alpha\beta}(x)g_{ij}(f(x))\frac{\partial f^i(x)}{\partial x^\alpha}\frac{\partial f^j(x)}{\partial x^\beta}$$

If you want this in a coordinate free form, you have to mess with the bundle stuff from earlier

Hmm, OK.

You use the metrics $g$ and $\gamma$ to put a metric on $T^*M\otimes f^*TN$, and then $$e(f)=\frac{1}{2}\langle df,df\rangle_{T^*M\otimes f^*TN}.$$

Ah

Nice.

Then the energy functional is $$\mathscr{E}[f]=\int_M e(f)\, \mathrm{vol}_M$$

Right.

And you can do all sorts of fun PDE stuff to that to get information about harmonic functions

Apparently harmonic maps $S^2\to M$ are important for Ricci flow

What are harmonic functions between Riemannian manifolds again? Energy-minimizing functions?

These different formulas for how to compute a householder reflector are really confusing. Half the time it's written to add to the old vector X, the other half the time you subtract from it. Both mine and another online calculator are broken when I try one particular input from a PDF example. hmm.

@BalarkaSen Yeah, harmonic maps satisfy the Euler-Lagrange equations of that functional

Yet the pdf version gets it fine, it seems.

Right

Gotcha.

it's a generalization of the Laplace equation $\Delta f=0$ on $\Bbb R^n$

The general case is treated in Jost, but the stuff in that Ricci flow book I sent you seems to be pretty sensible

$\langle df, df\rangle$ is precisely $\Delta f$ on $\Bbb R^n$, not?

It's $||\nabla f||^2$

Oh, I am confusing two things.

$\Delta f$ is dot product of $\nabla$ with itself as an operator. Sorry, being dumb.

Yes, it's $\|\nabla f \|^2$. I agree.

I want to learn more about this stuff, but I know far too little PDE

Next summer

@BalarkaSen I think you can link the existence of harmonic maps to the curvature, somehow

I know nothing of this, but interesting.

I think where it really shines is in the study of Riemann surfaces, but I know nothing of this

I just know Jost has a chapter on "Harmonic Maps from Riemann Surfaces" in his geometric analysis book

Yeah, I have heard that it's useful in complex geometry.

@BalarkaSen Are we discussing the Poincare-Hopf theorem?

Must an upper-right triangular matrix be square?

I wrote that on an old note but it seems false. Doesn't have to be right?

From wikipedia:

> A square matrix is called lower triangular if all the entries above the main diagonal are zero. Similarly, a square matrix is called upper triangular if all the entries below the main diagonal are zero.

It is a bit ambiguous. It doesn't say it has to be square to be upper triangular..

what is a diagonal for a non-square matrix

Well it doesn't really have one.

there's your answer

Alright, thanks.

I've got a problem...

Let's say you're in the midst of computing a householder reflection.

You get a vector: x = [4 0]

Using formula: $v = x - sign(x_{1}) e_{1}$ you get: $v = [0\ 0]$

Then there's a problem when it comes to computing: $u = \frac{v}{||v||}$

Because it's a division by zero...

Well never mind, that shouldn't happen.

I found what the problem was I was asking about earlier. The difference between adding and subtracting from vector x in the reflector formula was that the formulas for subtracting from x actually had $p = -sign(x_{1})$ while the others had $p = sign(x_{1})$. So it's effectively all just adding.

fml

@TedShifrin in our physics building, there used to be a sign along the lines of "Robert's Rules of Order for Theorists." One of the mandates was to avoid fighting words like "trivial" and "inelegant." :)

Dammit

Bott & Tu is so vague at times

And they flat out misquote shit

@0celo7 I know some calculus of variations stuff, but probably not enough for what you want.

but what are you after?

@Semiclassical the issue resolved itself

ah, okay

@Semiclassical but the question was math.stackexchange.com/questions/1874963/…

whitney looks like a topologist

@Semiclassical I need a function $g(s), s\ge 0$ such that $g(s)=1$ for "small" $s$, $g(s)>0$ $\forall s$, and $\int_0^\infty g=1$.

What's hard is satisfying the condition near the origin and the integral normalization constraint...

Ideally, "small" can be adjusted to be smaller.

@Semiclassical I constructed a monstrosity with a plateau function, a ramp function, and a Gaussian that does the trick