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There is somebody who once wrote a brief post “Rethinking geometric complexity theory” (ask Google, I am on my phone), suggesting that monoidal category theory should help. I gather Mulmuley’s program involves quantum groups? That will be the hook into TQFT and hence monoidal category theory.
Can you isolate how much quantum groups control the program? By Tannaka duality, Quantum groups are really just a dual placeholder for their monoidal categories of representations. To the extent that GCT is about quantum groups, we’ll have the answer to your quetion.
Haven’t had time to actually explore GCT yet, but I just noticed that the Simons Institute for the Theory of Computing is dedicating this fall to Algorithms and Complexity in Algebraic Geometry and recording everything: http://simons.berkeley.edu/programs/algebraicgeometry2014. For any non - computer scientists who need a crash course in computational complexity theory before exploring GCT, Tim Gowers is your man: http://sms.cam.ac.uk/collection/545358
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