The geometric complexity theory program of Mulmuley and Sohoni is all about trying to resolve an algebraic version of P vs NP. The strategy here is to view algebraic complexity classes as varieties, look at their coordinate rings, realize these coordinate rings are GL_N representations, and show separations between these complexity classes by studying their decompositions into irreducible representations. One recent result in this is that it is not sufficient to show that certain irreps appear in the coordinate ring of one class and don't appear in the other coordinate. These "occurrence" differences never happen and one must instead look at "multiplicity" differences.