Julian Pfeifle: Positive Plücker Tree Certificates for Non-Realizability

Abstract: In 2020, Hailun Zheng constructed a balanced, 2-neighborly combinatorial 3-sphere Z on 16 vertices whose graph is the complete 4-partite graph K_{4,4,4,4}. However, it was unknown whether Z is realizable as the boundary of a convex polytope. If so, Z would provide the first example of a polytope with such a graph aside from the cross-polytope.

However, known techniques for proving or disproving the realizability of Z could not cope with this example. In the present talk, we level up the old idea of Plücker relations, and assemble them using integer programming into a new and more powerful structure, called "positive Plücker trees", that proves the non-realizability of Z and many other previously inaccessible families of simplicial spheres.

Zoom meeting ID: 654 5562 3260

Zoom link: https://kth-se.zoom.us/j/65455623260