The Kempf-Ness theorem (see e.g. arXiv:0912.1132) - that the algebraic quotient of geometric invariant theory is also a symplectic quotient - suggests (to me) that certain physical constructions used to compute equivariant cohomologies might be useful to Mulmuley et al's geometric complexity theory. Edward Witten is the wellspring of these ideas and his latest paper (arXiv:1009.6032) continues to develop them. My main concern is that they might not carry across to the objects of interest to complexity theory (e.g. the "class varieties" of arXiv:cs/0612134). But the power of the quantum techniques, and the diversity of approaches possible within GCT, leads me to keep looking...

Cross-posted to Math Overflow.

Edit: I have added some details at MO. What I need next is a much better understanding of the specific unproven propositions of algebraic geometry which GCT has already flagged as important. For example, I don't yet get the relationship between, on the one hand, nonstandard quantum groups and the "plethysm problem", and on the other hand, the Valiant-inspired approach which looks at the orbit closure of permanents and determinants.

  • 1
    $\begingroup$ I voted to close. This feels too much like an open-ended fishing expedition. "Is X useful for Y" questions aren't necessarily bad, but they need to be a lot more narrow and specific. $\endgroup$ – Per Vognsen Oct 3 '10 at 3:04
  • 2
    $\begingroup$ I strongly feel it should be open. GCT is the one plausible approach towards the P vs NP question, and if someone can relate techniques from this new paper, it would be wonderful... $\endgroup$ – arnab Oct 3 '10 at 3:06
  • 1
    $\begingroup$ @arnab: I'm interested in this area too but as it stands the question's too non-specific. If Mitchell has read and understood the paper, he could try to extract some specific suggestions for attacking problems in geometric complexity theory and formulate the question around that. $\endgroup$ – Per Vognsen Oct 3 '10 at 3:08
  • 2
    $\begingroup$ @Per: I think the questioner was specific enough. Someone knowledgeable in GCT can educate us on plausible connections with the Kempf-Ness theorem. I think we're being way too quick to close. We could be stopping interesting answers before there's a chance to post them! $\endgroup$ – arnab Oct 3 '10 at 3:13
  • 2
    $\begingroup$ Also voted to close. Too open-ended, too vague. Questions about GCT are certainly within scope, and there may be a crisp question relating Witten's recent work to GCT, but this isn't it. $\endgroup$ – Jeffε Oct 3 '10 at 6:38