Flow Proof Helps Mathematicians Find Stability in Chaos

quantamagazine.org

42 points·nsoonhui··7 comments

"As with so much in mathematics, the proof started with coffee"

When I knew that the article wouldn't get straight to it, and make me dig through for the point.

Imagine there was a point to existence...

I wish we had a TL;DR easily. Those lengthy scientific/history articles are doing a good job at killing any excitement about the core ideas.

Quanta's articles are long and I don't invest the time to read every one I see, but the nature of the subject material doesn't lend itself to a TL;DR that captures what is actually interesting.

A true TL;DR might be hard, but if the author finds the morning routine of the mathematician who discovered the proof worth mentioning, it's clear they aren't even trying.

》 if, for instance, you don’t care how fast the particle is moving but only which way it’s facing at any moment in time — you could describe its motion using a three-dimensional state space.

This is assuming we are on a curve across 2D space like the equator and face east or west. This sentence was puzzling me at first glance. In usual 3D space, direction/orientation is represented as a 3 dimensonal vector with length one, so on top of the position dimensions, that makes six dimensions, even if we dont care about magnitude of direction a.k.a. speed. Using angle coordinates changes the support domain, but still three dimensions are needed for direction.

That's the point. A particle moving in 3D space can instead be considered as a curve on a 2D manifold (a surface) embedded in 3D space. So then you only need two dimensions for position and one for orientation.

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