Dark Matter and Worst Packings

The annual Geometry and Topology Conference at Lehigh University attracted over 60 mathematicians from around the world. Hugh Bray of Duke University talked about a possible explanation for the mysterious dark matter, known only by its gravitational effect, more than five times as abundant as regular matter. Just as Einstein's theory of General Relativity explains gravity as curvature of the universe, Bray similarly incorporates dark matter into the geometry of the universe, altering not just the "Riemannian metric" but also the "connection." Here from Bray's paper are a few pictures of galaxies on the left with a simulation from Bray's model on the right:

Hubert Bray of Duke University explains a new geometric model for dark matter.

Worst Packings. If you're packing your suitcase, you like to have items that fit together closely (even though that might make your suitcase rather heavy). On the other hand, if you're designing a new material, you might want to have components that don't pack tightly, for a lighter material. What speaker Yoav Kallus of Princeton calls "Ulam's Last Conjecture" says that a round ball is the shape that packs worst in space. Kallus has proved that a round ball is worse than any nearby shape with central symmetry. In the plane, however, there are lots of shapes worse than the circle, some important ones shown in the following figure from his recent preprint. The rounded octagon can be packed in many different, equally efficient ways.

Yoav Kallus talks about the Ulam Conjecture that the sphere packs worst of all.