Seminar in Number Theory and Algebraic Geometry

Abstract: In 1971, McMullen gave a conjectural characterization of the possible sequences of numbers that can arise as the number of simplices of a given dimension in a triangulation of a sphere. The first big result regarding McMullen's conjecture was due to Stanley, and it led to what are now called Lefschetz properties. In this talk we will see combinatorial settings where failure of Lefschetz properties should be expected, and the connection with face counting in spheres.