### One Geometry (Math Raps)

#### Posted by John Baez

While Look Around You was a goofy parody of an old-fashioned BBC educational program on math, here’s a rap video with real mathematical content:

- Stephen Sawin, One Geometry: Perelman’s proof of the Poincaré conjecture, to the tune of Snoop Dogg and Pharrell William’s “Drop it Like it’s Hot”. Lyrics available here.

Steve Sawin is an old friend of mine. I met him when he was a postdoc at MIT — straight outta Berkeley, a student of Vaughan Jones. Since then he’s written some very interesting papers on quantum groups, topological quantum field theory, and path integrals. His latest, with Dana Fine, gives a rigorous version of the path integral for supersymmetric quantum mechanics.

But he also has a sense of fun! This song about Poincaré’s conjecture is not the first of his math raps under the name “essiness”. Here are the lyrics to his proof of the Bolzano–Weierstrass theorem. You can hear it on his website.

#### My Name Is… Bolzano Weierstrass

by essiness (aka Slim Dorky)Come on you math majors if you want to be free

From Corporate America you listen to me.

You’ve got a sequence that you built from your approximating tweakins

And you really need to find a convergent subsequence,

So you ask my man Bolzano and his homie Weierstrass,

Who’ve found you a solution with a trick that’s really boss.

Well are you down with that?

We’re down with that!

Well are you down with that?

We’re down with that!

Well you haven’t got much hope unless your sequence is bounded,

So let’s say some interval has got your numbers surrounded.

They’re all greater than $a$, they’re all less than some $b$,

And it’s right there with those points, that your thoughts have to be.

So to your right is $b$, and to your left is $a$,

And in between your sequence tries to wind its way,

An infinity of $x_n$’s, this interval has in it,

Still you don’t know where to look, to try to find the limit.

So you stand in the middle, halfway in between,

$a$ plus $b$ over two, if you know what I mean,

To your left a line segment, half the big one’s size,

To your right the other half, in the same way lies.

You see every $x_n$ lives in the other or the one,

But kid you’ll never believe what this division has done,

Because if every $x_n$ lies in one of these, dude,

Then in one or the other an infinitude!

Well are you down with that?

We’re down with that!

Well are you down with that?

We’re down with that!

So you slide to the side where this infinity lies,

To the middle of an interval of half the size,

This new interval (I say it’s half as long),

Contains an infinite subsequence if my logic ain’t wrong.

Now you do it again, divide the line in two,

And if you paid attention, you’ll know just what to do.

You count up all the $x_n$’s, on the left and right,

There’s infinity in one, though the space is getting tight.

So you do this $k$ times, now we’re really getting small,

One half to the $k$, is our interval.

Yet in this little space, within this little bound,

A whole subsequence can still be found.

Well are you down with that?

We’re down with that!

Well are you down with that?

We’re down with that!

Well you can do this forever, until Tishebuv,

Cuz infinite recursion is the thing that we love.

A chain of nested intervals, each inside the last,

Like little Russian dolls, and they’re getting smaller fast.

But what you have to believe, because then we’re nearly done,

Is there’s exactly one point that lives in every one!

See all those left endpoints, they have to have a supremum,

The same way that the right ones have to have an infimum.

Well this sup and this inf, they live in each of these sets,

So the distance that’s between them is as small as it gets.

They are both the same point, so I say what the hell,

I think that its our limit so let’s call it $L$!

We’re down with that!

Well are you down with that?

We’re down with that!

Well I promised a subsequence and I never tell a lie,

To distinguish it from $x$ I’ll call this sequence $y$.

Recall the $k$th interval, and all the points in its span,

Well I only need one, that’s just how bad I am.

$y_k$’s my name for this point, it lives in interval $k$,

Which makes it quite close to $L$, you see I planned it that way!

Now you can pick epsilon as small as it wants to be,

Cuz I’ve got nested quantifiers and they’re working for me.

I will come back with an $M$, so big I’m sure it will do,

Which of your epsilon is one minus the log base two.

Well the thing about $M$, is that I picked it so good,

That after it the $y_k$’s lie inside of $L$’s ‘hood,

$L$’s ‘hood is epsilon sized, so all those intervals lie in it,

QED, you’ve got a sequence that converges to a limit!

We’re down with that!

Well are you down with that?

We’re down with that!

Well I am outta here now, because my rap is at its end,

But I’ll leave you with this exercise: to prove it in $\mathbb{R}^n$!

## Re: One Geometry

As they explain in the introduction, this is an extension of a similar construction by Bär and Pfäffle.

Christian Bär gave a talk on their approach a while ago here in Hamburg. One of the key ingredients is an innocent-looking and very obvious construction: they look at piecewise geodesic paths.

I was thinking about such paths a bit recently, while trying how to prove the statement, which seems to be true, that

smooth pseudofunctorsfrom the fundamental groupoid of $X$ to some $n$-group $G_{(n)}$ $\Pi_1(X) \to \mathbf{B}G_{(n)}$ are equivalent to smooth $n$-functors from $n$-paths in $X$ $P_n(X) \to \mathbf{B} G_{(n)} \,.$One direction is easy: you differentiate the pseudofunctor and obtain the data of a Lie $n$-algebroid morphism $T X \to g_{(n)}$ from the tangent algebroid of $X$ to the Lie $n$-algebra underlying $G_{(n)}$.

But integrating that up a again to a pseudofunctor on $\Pi_1(X)$ has global subtleties: for any homotopy class of paths, you’d want to pick one representative and then integrate your form data over it. But the result will depend on which representative you choose.

So apparently one has to choose

singled outrepresentatives.Probably choosing a metric and then looking at geodesics might be good way to do this.

But of course that requires a bit care. The best one can hope for is is to deal with piecewise geoedesic paths.

I was wondering if the apparent need to introduce a metric here is a mere nuisance or maybe actually an inidcation that something interesting is going on.

Because the situations where we can make sense of the QM path integral are precisely the situations where we have a particle which is subject to precisely two effects:

a) a gauge background field, b) background gravity.

The first is encoded in a pseudofunctor. The precise abstract right way to encode the latter is less clear, generally.

And it would be nice if both parts could actually be seen to merge nicely. The point is that the path integral is

$\int_{P X} \mathrm{hol}(\gamma) d\mu(\gamma)$

with the hol-part giving the background gauge field and the measure $d\mu$ being the kinetic part, depending on the metric, and coming from a metric. People expect that we should think of this as one measure $\int_{P X} d\tilde \mu(\gamma) \,.$

And when I think of the $\mathrm{hol}$ as encoded in a pseudofunctor (pseudo-

anafunctor really, but let’s not worry about that right now) it seems I do indeed need think of it always in conjunction with a metric.Well, this is how far I got with thinking about this right now.

But when I read Bär-Pfäffle and Fine-Sawin, I am feeling there might be something to this.