I should expand on this... Geometric complexity theory is about posing and solving complexity-class problems in a geometric context. For example: Computing the permanent of a matrix is in #P, computing the determinant is in P. Permanents and determinants can be thought of as algebraic subvarieties, and showing that a certain mapping cannot turn determinants into permanents (conjecture 3.2 in the third paper) would show that P is not #P. The idea is to use certain new constructions from mathematical physics (e.g. first paper) to understand such mappings. The space of orbits under the mapping is a quotient of the target space, and there is a history of using techniques from physics to understand these particular quotient spaces. The big development I see brewing is the relation between the "geometric Langlands" program and the quantization of the 5-brane in M-theory, for which a whole new approach to quantization has to be invented. So I'd like to see what happens if you take those new ideas and push them along the chain from first paper to third paper. (But that first paper is just supposed to be representative of the ongoing work, I'm not saying it specifically contains the technically appropriate concepts.)
Meanwhile, for P versus NP, there is a particular barrier to proof which Mulmuley and collaborators hope to get around because of the specialness of the varieties. Right now I'm trying to understand what it is about the algebraic-geometric facts which gives them the right logical and combinatorial properties to evade this "naturalization barrier". Then I want to look at models of CEV (e.g. for a population of DFT agents) from this perspective, to see if it will help with the difficult problems there (the interplay between self-enhancement, utility function discovery, and utility function renormalization).
Another thing, although they don't discuss it, it looks like this method also might get around the relativization barrier since the associated varieties when you have an attached oracle are going to look very different.
At the start of 2010, I resolved to focus as much as possible on singularity-relevant issues. That resolution has produced three ongoing projects:
As I put it the other day, the paper is about "CEV, adapted to whatever the true ontology is". I have ideas about how CEV should work, and about what the true ontology is, and about the adjustments that the latter might require. These ideas are tentative, and open to correction, and the objective is to find out the facts, not just to insist on an opinion. Indeed, I would be open to hearing that I ought to be working on something else, if I want to attain maximum relevance to the AI era. But for now, I have my plan, and I take it seriously as a blueprint for what I should be doing.
The relevance of string theory might seem questionable. But it matters for physical ontology and for epistemology of physics [ETA: which matters for general epistemology and hence for AGI]. String theory is also a crossroads for many topics central to pure mathematics, such as algebraic geometry, and their techniques are relevant for many other fields, even discrete ones like computer science. In the theory of complexity classes, there is a self-referential barrier to proving that P is distinct from NP. There is a deep proposal to overcome it by transposing the problem to the domain of algebraic geometry, and I've just begun to consider whether a similar method might illuminate problems like self-enhancement, utility function discovery, and utility function renormalization (for concreteness, I plan to work with decision field theory). Also, if I can speak string, maybe I can recruit some of those big brains to the task of FAI.
"Investigation of academic options" should be self-explanatory. A university is one of the few places where you might be able to work full-time on matters like these. Unfortunately, this outcome continues to elude me. So while I set about whipping up a stew of private microloans and casual work so as to keep a roof over my head, it's time for me to try the Internet option as well.
I find that life costs me AUD$1000/month (AUD is currency code for "Australian dollars"). I'd do better with more, but that's my minimum budget, the lower bound below which bad things will happen. So that's also the conversion rate from "money" to "free time".
I figure that there are three basic forms of cash transaction: gifts, payments, and loans. A gift is unconditional. A payment is traded for services rendered. A loan is a temporary increase in a person's capital that has to be returned. These categories are not entirely distinct: for example, a payment refunded (because the service wasn't performed) ends up having functioned as a loan.
I am interested in all three of these things. The brittle arrangement which allows me to speak to you this way does not presently extend to me owning a laptop or having easy access to Skype, but I do have a phone, so the adventurous can call me on +61 0406 979788. (I'm in the eastern Australian timezone.) My email is mporter at gmail.com, and I have a Paypal account under that address.