Natural poset (partially ordered set) structure arises in polyhedral fans, Coxeter groups and hyperplane arrangements. In nice cases, there exist stratified algebraic varieties realizing the above poset structures, namely toric varieties, Schubert varieties and Schubert varieties of hyperplane arrangements respectively. I will discuss some examples of deriving combinatorial consequences from the corresponding algebraic varieties. I will also talk about how to generalize the above arguments to the bad cases, when the corresponding algebraic varieties do not exist (for example, non-rational polyhedral fans). Speaker(s): BOTONG WANG (University of Wisconsin-Madison)