Exploring tropical ideals

The scheme-theoretic approach to tropical geometry has motivated the study of "tropical ideals", which are sets of tropical polynomials that form an ideal whose graded pieces are tropical linear spaces. There are realizable tropical ideals, meaning that they are formed by tropicalizing classical ideals as linear spaces, and there are non-realizable tropical ideals. Three interesting questions are:

1) What invariants of a classical ideal are encoded in its associated tropical ideal?
2) How does the tropicalization of an ideal change as the ideal changes (moving within the Hilbert scheme)?
3) How can one construct non-realizable tropical ideals?

In this talk I will discuss examples, progress and open questions on each of these questions.