Ben Smith

Staircases and tropical polyhedra

A staircase, a union of shifted copies of the positive orthant, is a seemingly simple but fundamental object in mathematics. It occurs in wide range of areas including order theory, commutative algebra of monomial ideals and multicriteria optimization. However, despite its simple premise, the combinatorics of staircases are deceptively subtle. In this talk, we shall consider a new viewpoint of staircases: as "polyhedra" over the max-plus semiring. We will develop a face poset for these tropical polyhedra and show that this structure captures new combinatorial data of staircases. Finally, we will show how these tropical face posets can shed light on existing problems in order theory and commutative algebra.