As a one-time mathematician, this was a really fascinating article. The similarities seem to be entirely coincidental, but what would have been my doctoral dissertation was also about generalizing some concepts from smooth manifolds to a "non-smooth" setting, and the crux of my work also hinged on optimal transport.

Actually I feel optimal transport is a pretty underrated concept in both pure and applied math, and I would have loved to explore it had I continued in academia. But oh well, one must make choices in life...

This is all really cool, but is getting new singularity theorems really a positive sign? Like, my understanding was that it was generally hoped that an improved, quantum theory of gravity would eliminate such singularities -- that such singularities were generally considered to be non-physical artifacts that occur in GR due to its deficiencies at the most extreme scales (where quantum gravity would be relevant), not that they are in fact real and physical. So I'd consider it a better sign if these predicted black holes, which we see, but without singularities!