Huh, looks like quite a bit of more or less chemical q. around here lately

@BalarkaSen did they do it the cauchy way or the different-PDR-solutions-approach way (or are those effectively the same?)

(i don't know very much about Cauchy sequences, so i'll have to read about those.)

It involves very careful estimations. I have not read that paper but I was informed of the method by someone who is in a workshop on h principles, and lectures are given by one of the two authors of that paper.

The theory of $h$-principles is interesting because it can be used to prove a wide variety of weird "integrability" results (don't worry about that terminology). Have you heard of sphere eversion, @heather?

@BalarkaSen I have not.

according to wikipedia:

> sphere eversion is the process of turning a sphere inside out in a three-dimensional space.

I have no better insight to offer than this video. Watch it at your leisure.

(then watch it 10 more times)

lol ^

okay.

I feel like one can make a Mathcamp lecture explaining sphere eversions.

The idea is so simple

But simultaneously rich

so...i haven't quite finished yet. but why can turning number govern whether or not things are "equal" or can be everted? i assume it is a very complex proof.

They explain it a little. If you deform a curve, the frowns and smiles cancel in pairs.

So the net turning number after and before deformation remains the same

Because -1 + 1 = 0

So turning number MUST remain invariant under deformations of the sort they restrict to

well, yes...but i can't quite seem to connect that canceling to the twisting. it sort of makes sense.

Which twisting do you refer to?

the twisting to make it inside out, or make curves equivalent

You mean at the 2:10 mark?

That's an invalid solution, as you are not allowed to use the 3D space.

They were demonstrating that it was not part of the rules of the game

oh, okay (sorry, popping in and out a bit)

No worries

I love the comments section below that video. So many memes

wow that video

i just finished that video...that's great!

so, how does sphere eversion relate to the homotopy principle?

(and then, how does all that relate to the inverse of a gradient?)

Haha

Long story

Basically the sphere eversion can be thought as a solution of a PDE, kind of.

that sounds like an impressive generalization.

It is!

Let's see if I can formulate it in less vague sense

Say $S_t \subset \Bbb R^3$ is the sphere eversion, paused at time $t$. $S_0$ is the normal sphere, $S_1$ is the outside-in'd sphere.

okay.

The conditions that there are no creases, infinitely tight pinches, singularities of any sort is a differential restriction on $S_t$. The point being, every point on $S_t$ has a tangent space (at self intersection points, there can be multiple tangent spaces: think of desmos.com/calculator/qsa7zqqe15)

sorry, what is a tangent space?

Tangent plane.

oh.

Sorry for using "space"; habit

no, it's all good...they're the same?

that's good to know.

Yeah, the point is every point on $S_t$ has tangent planes passing through it.

okay.

So it's a "smooth surface, with self intersections" in R^3

Now since for any $p \in S_t$, it has a tangent plane $T_p$ tangent to $S_t$ at $p$, it also has an outward-pointing normal direction $N_p$ which is simply perpendicular to $T_p$

At every point on $S_t$ you have these outward pointing normal vectors.

Agree?

right, yeah

I'll write down the rest of the stuff because I need to hit the sack soon. $S_t$, $t \in [0, 1]$ is a movie of sphere eversions with $S_0$ the plain vanilla sphere and $S_1$ the inside-out sphere. For each fixed $t$, there is a normal vector $N_p(t)$ to $S_t$ at any point $p \in S_t$. So $N(t)$ should be thought as a sphere-parameterized family of vectors.

The "differential equation" is that $N(t)$ is always normal to $S_t$, with boundary conditions that $N(0)$ is all outward pointing and $N(1)$ is all inward pointing

The boundary conditions are those because the sphere inside-out'd

So outward pointing normals turned inward pointing

The differential relation corresponding to that differential equation has a solution - that is what the fact that turning number = 1 for both the standard sphere and the inside-out sphere indicates.

that makes sense

thank you. i'll have to keep reading about this =)

So the differential relation has a solution, and it's in fact an open relation so that Gromov's theorem applies and it satisfies the $h$-principle.

Therefore the differential equation has a solution

That gives you a sphere eversion with 0 work

@heather Always happy to talk. This is an amazing piece of math with very weird consequences.

So I encourage curiosity.

one question real quick before you go off to hit the sack:

oh wait, nvm, i figured it out

sorry.

Okay :)

If you have questions feel free to ping. I'll look after I wake up hell knows when and answer to the best of my knowledge.

thank you =)

'Night

have a good night

(for others reading this conversation/general reference: math overflow q&a on this topic)