"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).

In the above paper, KM proves that [*Noether's Normalization Lemma* (NNL)](https://en.wikipedia.org/wiki/Noether_normalization_lemma) 
is not as intractable as it might seem
(from Gröbner basis theory—exponential in $n$).
He shows that, "in practice, NNL for explicit varieties can be solved efficiently
and correctly with a high probability."
The next issue is achieving deterministic polynomial-time.
Here he only partially succeeds, showing that 
"for some interesting cases of explicit varieties,"
NNL "can indeed be solved deterministically in quasi-poly$(n)$-time."
This brings some instances of NNL 
"from EXPSPACE to quasi-P, assuming the hardness hypothesis for the permanent
in geometric complexity theory."

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).