<p>The Kempf-Ness theorem (see e.g. <a href="http://arxiv.org/abs/0912.1132" rel="nofollow noreferrer">arXiv:0912.1132</a>) - 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 (<a href="http://arxiv.org/abs/1009.6032" rel="nofollow noreferrer">arXiv:1009.6032</a>) 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 <a href="http://arxiv.org/abs/cs/0612134" rel="nofollow noreferrer">arXiv:cs/0612134</a>). But the power of the quantum techniques, and the diversity of approaches possible within GCT, leads me to keep looking... </p>

<p>Cross-posted to <a href="https://mathoverflow.net/questions/40897/is-wittens-new-method-of-quantization-useful-for-geometric-complexity-theory">Math Overflow</a>. </p>

<p>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. </p>

CC BY-SA 2.5
2010-10-03T03:04:04.170Z
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.
Per Vognsen
CC BY-SA 2.5
2010-10-03T03:05:47.077Z
The question is too broad. Also it's a yes/no question, written in the title, followed by some rambling text. It's also not clear that this is at all related to TCS.
Dave Clarke
CC BY-SA 2.5
2010-10-03T03:06:59.180Z
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...
arnab
CC BY-SA 2.5
2010-10-03T03:08:42.730Z
@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.
Per Vognsen
CC BY-SA 2.5
2010-10-03T03:08:54.850Z
Perhaps then the question should be rephrased.
Dave Clarke
CC BY-SA 2.5
2010-10-03T03:13:02.433Z
@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!
arnab
CC BY-SA 2.5
2010-10-03T03:26:35.843Z
@Mitchell Porter: It is nice that you put the links to/from the crosspost on MathOverflow, but next time please [avoid simultaneous crossposting](http://meta.cstheory.stackexchange.com/questions/225/official-faq-for-theoretical-computer-science/231#231).
Tsuyoshi Ito
CC BY-SA 2.5
2010-10-03T04:20:57.483Z
I'm wondering if we also need a policy on "open questions": while I understand very little of what this question is about, it seems to be speculative in the extreme, like the very early questions that were posted on MO after the deolalikar proof came out. Just because GCT is important, we can't allow any kind of question that MIGHT bear upon it.
Suresh Venkat
CC BY-SA 2.5
2010-10-03T06:38:34.783Z
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.
Jeffε
CC BY-SA 2.5
2010-10-03T10:12:20.503Z
When I say something like this over coffee, it seldom generates useful discussion.  This is a great way to start, but a question about a new area needs to be more specific.
András Salamon
CC BY-SA 2.5
2010-10-03T17:36:18.593Z
I must admit, I would be interested to see how such a discussion might develop and I tend to agree with arnab about being too quick to close. Hence I'm voting to reopen. Would be good if the author could make this a little more directed though.
Joe Fitzsimons
CC BY-SA 2.5
2010-10-03T21:57:23.893Z
Yes, hopefully the author comes back to give the question more background and make it more specific.
arnab
CC BY-SA 2.5
2010-10-04T01:59:39.533Z
Also voted to reopen. I think we are closing questions way too fast. There's no harm keeping the question open a little while longer to give the author a chance to improve the question. Closing the question discourages the author. Since MO has not closed the question, the OP might decide to stick with MO and never post questions here because the people here seem to be so unfriendly.
Robin Kothari
CC BY-SA 2.5
2010-10-04T06:19:21.697Z
The background the OP just linked to on MO might be helpful there, but makes very little sense to me over here. Even though there are now 4 votes to reopen, I really don't see how this question in its current form can generate useful answers.
Suresh Venkat
<complexity-classes><algebra><gct>
Is Witten's new method of quantization useful for geometric complexity theory?
Mitchell Porter
# Is Witten's new method of quantization useful for geometric complexity theory?

(Asked by: username_0 on 2010-10-03)

The Kempf-Ness theorem (see e.g. [arXiv:0912.1132](http://arxiv.org/abs/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](http://arxiv.org/abs/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](http://arxiv.org/abs/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](https://mathoverflow.net/questions/40897/is-wittens-new-method-of-quantization-useful-for-geometric-complexity-theory).

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.

**Tags:** complexity-classes, algebra, gct
> **Comments:**
> **username_1**: 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.  
> **username_2**: The question is too broad. Also it's a yes/no question, written in the title, followed by some rambling text. It's also not clear that this is at all related to TCS.  
> **username_3**: 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...  
> **username_1**: @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.  
> **username_2**: Perhaps then the question should be rephrased.  
> **username_3**: @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!  
> **username_4**: @Mitchell Porter: It is nice that you put the links to/from the crosspost on MathOverflow, but next time please [avoid simultaneous crossposting](http://meta.cstheory.stackexchange.com/questions/225/official-faq-for-theoretical-computer-science/231#231).  
> **username_5**: I'm wondering if we also need a policy on "open questions": while I understand very little of what this question is about, it seems to be speculative in the extreme, like the very early questions that were posted on MO after the deolalikar proof came out. Just because GCT is important, we can't allow any kind of question that MIGHT bear upon it.  
> **username_6**: 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.  
> **username_7**: When I say something like this over coffee, it seldom generates useful discussion.  This is a great way to start, but a question about a new area needs to be more specific.  
> **username_8**: I must admit, I would be interested to see how such a discussion might develop and I tend to agree with arnab about being too quick to close. Hence I'm voting to reopen. Would be good if the author could make this a little more directed though.  
> **username_3**: Yes, hopefully the author comes back to give the question more background and make it more specific.  
> **username_9**: Also voted to reopen. I think we are closing questions way too fast. There's no harm keeping the question open a little while longer to give the author a chance to improve the question. Closing the question discourages the author. Since MO has not closed the question, the OP might decide to stick with MO and never post questions here because the people here seem to be so unfriendly.  
> **username_5**: The background the OP just linked to on MO might be helpful there, but makes very little sense to me over here. Even though there are now 4 votes to reopen, I really don't see how this question in its current form can generate useful answers.  

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## 0 answers

