Chances are, if you're an algebraist, when I say the word "identity" you think of some nice element in a group that fixes everything. Use the word around me, however, and I'll probably think of a "semigroup identity". In layman's terms, these things are rather like trigonometric identities - two expressions which are always equal, no matter which elements of your semigroup you substitute in.
Five years ago or so, the first papers on identities satisfied by tropical matrix semigroups (a subject I now have a considerable interest in) were published. I'll be giving a brief introduction to tropical maths before going into detail about which cases are known, which aren't, and the one irritating case of the identity that works... but nobody really understands why.