I'm a Numberphile!

The popular YouTube channel Numberphile has just released a video featuring yours truly — filming this with Brady Haran was a real pleasure!

It was #3 on Hilbert's list of the most important problems in mathematics - until his student solved it. More links & stuff in full description below ↓↓↓ Featuring Daniel Litt, currently at the Institute for Advanced Study - https://twitter.com/littmath David Hilbert's problems: https://en.wikipedia.org/wiki/Hilbert%27s_problems Max Dehn's famed proof: https://archive.org/details/mathematischean33behngoog/page/n488 Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): http://bit.ly/MSRINumberphile We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science.

A couple of quick mathematical comments: in the video, I compute the Dehn invariant of the cube with side length \(1\) as \(12\otimes \pi/2\) — astute YouTube commenters have already noticed that this is the same as zero! I never really discussed the group structure on \(\mathbb{R}\otimes \mathbb{R}/2\pi\), so I resisted saying this in the video.

For a similar reason, the proof that the Dehn invariant is actually an invariant (under cutting and pasting) sketched in the video is not quite complete. The gap is that I never explain what happens with new edges added (in the middle of what used to be a face) when one makes a cut! The point is that new edges come in pairs with equal length and with dihedral angles which sum to \(\pi\). So the contribution to the Dehn invariant is \[\ell\otimes \theta_1+\ell\otimes \theta_2 =\ell\otimes \pi=0,\] where here I use that \(\ell\otimes \theta=0\) if \(\theta\) is a rational multiple of \(2\pi\).

Anyway, hope you all enjoyed the video!