psi = {xa,ya}⊗{xb,yb}⊗{xc,yc}⊗{xd,yd}

and we specify that the eight coordinates {xa,ya,xb,yb,xc,yc,xd,yd} are all real. Then the above outer product yields a 16-dimensional vector

psi = {xa*xb*xc*xd, xa*xb*xc*yd, xa*xb*xd*yc, xa*xb*yc*yd, xa*xc*xd*yb, xa*xc*yb*yd, xa*xd*yb*yc, xa*yb*yc*yd, xb*xc*xd*ya, xb*xc*ya*yd, xb*xd*ya*yc, xb*ya*yc*yd, xc*xd*ya*yb, xc*ya*yb*yd, xd*ya*yb*yc, ya*yb*yc*yd}

such that our eight coordinates define a five-dimensional submanifold in this 16-dimensional space (three of the eight coordinates are redundant, hence five-dimensional).

We imagine that this five-dimensional surface is very bumpy, but here is the surprise — its Weyl tensor vanishes, and hence it is conformally flat.

And this is true AFAICT for any number of spins, not just four.

Who can point us engineers in Seattle toward a book that will tell us how to generalize this surprising result (surprising to us) to complex coefficients?

And be gentle — it will be our first time.

]]>Thanks and happy blogging!

]]>If you want some spouting about when P!=NP will be solved, check out Bill Gasarch’s P/NP poll. To me, the most striking aspect of this poll is that those who’ve thought about the problem the most are usually willing to speculate the least.

I believe that P!=NP, I believe this is *provable* in standard axiom systems, and I believe the proof will require entirely new mathematics. Keep in mind, this is not the first time a mathematical question was asked decades or centuries before anyone had the tools to answer it.

We know that a non-relativizing, non-naturalizing lower bound technique will be needed, and existing interactive proof results tell us that such techniques are possible, but we don’t have any that are strong enough to answer even much “easier” questions, like NEXP vs. P/poly, or whether SAT has polynomial-size, depth-3 circuits consisting entirely of mod 6 gates.

Beyond that I won’t speculate. 🙂

]]>My favorite quote ever from a theory paper is in a paper by Papadimitriou and Yannakakis (it was either the one showing that many optimization problems are Max SNP-complete, or the one showing that the VC dimension is log-SNP-complete):

*“As usual in complexity theory, we have decreased the number of questions without, alas, increasing the number of answers.”*

Holography is a framework for *classical* algorithms, so it’s outside the scope of my challenge. I see it as a design strategy that was implicit in some early polynomial-time algorithms (for matchings, determinants, etc.), and that made explicit has led to some neat new algorithms for things like counting problems on planar graphs.

(i.e., classical simulation of quantum systems, and new algorithms, quantum or classical)

]]>Nothing more subtle than biting vaginas.

]]>