Abstract: Given a mathematical object, one may associate a combinatorial object that captures some of its properties.  Natural examples would be matroids as combinatorial abstractions of linear subspaces and Newton polytopes as combinatorial abstractions of hypersurfaces.  Then one has a class of combinatorial objects that behave somewhat like the original mathematical objects.  Two questions arise: which properties of the original objects do the combinatorial objects encode?; and how to characterize the combinatorial objects that are abstractions of an object in the original category. 

In this talk, we discuss tropical varieties which are combinatorial abstractions of algebraic varieties and contain as examples the theories of Newton polytopes and matroids.  We develop some examples and share the progress that has been made on those two questions.