"Current breakthroughs" I cannot address. So I will just list three
recent relevant papers. The first is Ketan Mulmuley's 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](http://ramakrishnadas.cs.uchicago.edu/gctcacm.pdf).

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](https://arxiv.org/abs/1209.5993).

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](https://arxiv.org/abs/1701.01717).