But then he threw me when he linked to what appears to be a real proof. I read through it, and found no signs it was tongue-in-cheek itself.
So I'll admit I'm at an impasse here; if he's got a real mathematical construct that is usefully better than what was presented, I don't get the tongue-in-cheek tone; if it really is tongue-in-cheek, I don't get what the real proofs are doing there, unless the proof is itself the best-disguised joke I've seen in the math world. (Generally I've got a pretty good eye for mathematician humor, even in fields I know little about.)
Aaronson's proof involves an observation that PostBQP = PP, in other words, all of the problems you can solve with an especially powerful quantum computer can also be solved with an even more especially powerful classical computer, and vice versa. Since the quantum computer is easier to reason about, you get a hard result as a freebie.
BQP is not a tongue-in-cheek thing; it's also not a well-understood thing -- just "anything which quantum computers can do in this way quickly with bounded errors" -- but it's certainly better understood than this amplituhedron is. The real problem he's highlighting is that complexity classes like BQP have scary names.
For the sake of this discussion, you can imagine a similar problem with "programmable imperative syntax" and "monads". Really all that monads are, in a computing context, is a way of formalizing a programmable imperative syntax as a purely functional programming construct. The issue is that when you say "functional" and "monads" then people kind of just shut down, they don't want to hear about it. Does that make sense?
Sort of like the number omega, which contains in each digits proofs to every theorem, but is not computable:
It doesn't meet the definition of a framework: http://en.wikipedia.org/wiki/Software_framework
Not every one.
Wow... yes... you really straightened me out there. Way to go.
After looking at the linked to papers, I'm assuming this is not a joke, and he is not putting down (making fun) of the Amplituhedron, nor making up the Unitarihedron (just abstracting his work with a media-friendly name to itit).
What he is joking about is simply that mathematicians should named their work as a (insert-something-here)-hedron, as that would cause the media to pick it up and report on it.
Is this correct?
He could have given context but prefered being condescending.
(If it was a joke, the paper he linked to -- http://www.scottaaronson.com/papers/pp.pdf -- probably wouldn't be exactly what he described it as.)
Yesterday in QFT class the professor  explained the name by saying that the guy who came up with it was a "very good salesman". The reality is that probabilities can in a great deal of cases be expressed as the volume of something in some phase space, for a trivial example, consider the area under the curve between two x-values on the normal distribution's probability density function -- it's the 2-volume of a section of the Gaussiohedron.
So after branding quantum information science as the study of the "unitarihedron", Aaronson dragged out one of his more miraculous results, which is that PP is closed under union and intersection as a direct result of PP = PostBQP, where PostBQP is a complexity class derived from BQP using an idea called postselection. That proof was actually in his dissertation (if I recall correctly), written in 2006, and it was an extremely impressive proof which did not use a word like "unitarihedron" to describe its methods.
So, while it was a joke, if you read the article in the context of the physics community's reaction to the amplituhedron story, the included proof really just helps to drive the point home.
The methods now called "amplituhedron" have been under continuous development for the past ten years or so. My QFT professor seemed sort of enthusiastic about the development. They do seem to represent a step forward in our understanding of quantum field theory, though the reaction of the media has been, in the eyes of some commentators, characteristically buzzword-driven.
As he has not objected to this in the ten or so years it's been around, I doubt he'd object to my quoting him either. I thought what he said was illustrative.
FWIW I have met Arkani-Hamed and he didn't strike me for his salesmanship.
(Disclaimer- I haven't the background in complexity nor physics to have any idea, obviously. But geometrical calculations playing a key role in a theory that "did away with space" seemed like a wonderfully cool paradox.)
> The Bayesian says, "Uncertainty exists in the map, not in the territory. In the real world, the coin has either come up heads, or come up tails. Any talk of 'probability' must refer to the information that I have about the coin—my state of partial ignorance and partial knowledge—not just the coin itself.
See, QM doesn't actually work like this. There is no hidden variable. This has been proven.
Yet this is not the case when you send a single photon through a half-sieved mirror. Even when you know everything there is to know about this experiment, you just don't know which sensor will go off, and if you repeat the experiment, you will find that previous results tells you nothing about the next one. Perfect independence.
When total knowledge of the setup yields only partial knowledge about future observations, one is very tempted to look for "probabilities" in the setup itself. But it's a cop-out, a non-explanation. The fact that there is no hidden variable does not mean amplitudes (or their squared norm) are probabilities.
So, what to make about our uncertainty about the perceived outcome of quantum experiments? Well, it's simple: the laws of physics as we know them are deterministic. When you send a photon through a half-sieved mirror, the actual result is always the same: the world splits in half. There will be one blob where the photon hit the first sensor, and one blob where the photon hit the second one. Subjectively, the result is the same: we still observe the Born statistics. But step further from the amplitudes as probabilities cop-out: the Born statistics are now more of an anthropic problem.
Damn, we should compare notes, we're researching the same thing :-)
I need to make a mental note that if the mathematician or physicist tries too hard to get on TV /news and sell something, he might not be as great as others.