The University of Michigan Combinatorics Seminar
Forman introduced discrete Morse theory as a tool for studying CW-complexes by collapsing them onto smaller, simpler-to-understand complexes of critical cells. We will address two questions: (1) under what conditions may several gradient paths be simultaneously reversed to cancel several pairs of critical cells to further collapse a complex, and (2) how to prove that particular gradient paths are reversible in discrete Morse functions on poset order complexes, by a correspondence with reduced expressions for permutations. As an application, we give a discrete Morse function for graded monoid posets and deduce from the Morse inequalities upper bounds on the Poincaré series for affine semigroup rings (via the bar resolution) in terms of the Gröbner basis degree for its toric ideal of syzygies. This application is joint work with Volkmar Welker.