Abstract:  Questions regarding the complexity of the simplex method in linear programming  turn out to be related in special cases to questions about partially ordered sets.  Exploring this connection led us to study lattices arising as 1-skeleta of simple polytopes, obtaining the homotopy type of the intervals in these posets as well as a geometric way of constructing lattice-theoretic joins.  In recent joint work with Christian Gaetz, we generalized this to the setting of directionally simple polytopes, motivated by questions about Bruhat interval polytopes.  This allowed us to  generalize a result of Athanasiadis, Edelman and Reiner that the reduced expressions for any fixed permutation are highly connected under braid moves and commutation moves, obtaining an analogous statement for the BCFW bridge decompositions of reduced plabic graphs (a family of graphs arising in the study of the totally nonnegative real part of the Grassmannian); we also determined the homotopy type for each interval in posets arising as 1-skeleta of Bruhat interval polytopes.   This talk will not assume background in this area and will mention several open questions.