Christos Athanasiadis: Face enumeration of order complexes and real-rootedness

ABSTRACT: Order complexes of Cohen--Macaulay posets form a class of flag simplicial complexes with especially nice properties. Somewhat unexpectedly, their face enumeration is far from being well understood. For example, the question of the unimodality of their h-polynomials remains unsettled even for well studied classes of posets, such as geometric lattices. This talk aims to show that such h-polynomials tend to be real-rooted surprisingly often by discussing examples which involve simplicial and cubical posets and their rank selected subposets, face lattices of polytopes, partition lattices, noncrossing partition lattices and geometric lattices, and methods that can be applied. This is partly joint work with Katerina Kalampogia-Evangelinou.