PC Seminar: Counting combinatorial 3-spheres using Shannon entropy

Abstract: How many combinatorial d-spheres are there with m facets? That is, how many simplicial complexes with m maximal faces are there whose geometric realizations are homeomorphic to the unit sphere in Euclidean (d+1)-space? 
While this has been solved for d=1 (cycle graphs) and for d=2 (triangulations of the 2-sphere), it is still an open problem for d≥3. We prove an upper bound on the number of 3-spheres, by estimating the entropy of a sphere picked uniformly at random. For this we use a corollary of Shannon’s noiseless encoding theorem from a recent paper by Palmer & Pálvölgyi. 

This is a seminar in our seminar series on Probability and Combinatorics (PC).