Professor Patricia Hersh is a potential recruitment candidate.Forman introduced discrete Morse theory as a tool for studyingCW-complexes by collapsing them onto smaller, simpler-to-understandcomplexes of critical cells. Chari provided a combinatorialreformulation based on acyclic matchings for their face posets. Injoint work with Eric Babson, we showed how to construct a discreteMorse function with a fairly small (but typically not optimal)number of critical cells for the order complex of any finite posetfrom any lexicographic order on its saturated chains. I willdiscuss this construction as well as two more recent results abouthow to improve a discrete Morse function by cancelling pairs ofcritical cells. A key ingredient will be a correspondence betweengradient paths in poset "lexicographic discrete Morse functions" andreduced expressions for permutations.As an application, in joint work with Volkmar Welker, we construct adiscrete Morse function for graded monoid posets which yields upperbounds on Poincare' series coefficients for affine semigroup rings(by way of the Morse inequalities). These bounds are determined bythe degree of a Gr"obner basis for the toric ideal of syzygies andrelated data.I will begin with a brief review of discrete Morse theory.