"Current breakthroughs" I cannot address. So I will just list three recent relevant papers. The first is his summary of the CGT program:
(1) Mulmuley, Ketan D. "The GCT program toward the P vs. NP problem." Communications of the ACM, 55.6 (2012): 98-107. PDF download.
Next, Ketan's just published paper, the 5th in a series:
(2) Mulmuley, Ketan. "Geometric complexity theory V: Efficient algorithms for Noether normalization." Journal of the American Mathematical Society, 30.1 (2017): 225-309. Earlier arXiv version.
And here is a quite recent posting to the arXiv, commenting "on how [the] algebraic natural proofs barrier [they detail] bears on geometric complexity theory":
(3) Grochow, J. A., Kumar, M., Saks, M., & Saraf, S. (2017). Towards an algebraic natural proofs barrier via polynomial identity testing. arXiv:1701.01717 Abstract.