This is more of a negative result than a positive result and so it may not be what you are looking for, but I consider it a breakthrough: <a href="https://arxiv.org/abs/1604.06431">Bürgisser, Ikenmeyer, and Panova</a> showed that a certain very strong form of geometry complexity theory cannot possibly be true.  A key idea in geometric complexity theory is to try to separate the orbit closures of the determinant and padded permanent polynomials.  A particularly optimistic way one might hope to do this is to show that some irreducible representation of $GL_{n^2}(\mathbb{C})$ occurs in one coordinate ring but does not appear at all in the other.  The aforementioned paper proves that this particular hope is overly optimistic.