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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: Bürgisser, Ikenmeyer, and Panova 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.