I was going to write a review of Arkani-Hamed's lecture Havermann pointed out on video, but gmail deleted it. After 26 minutes of watching I quit -- let's just say I'd be amazed if a single human being comprehended it, and this is not a "lecture" but rather a "comedy sketch about how not to present a lecture." A few random observations: at one point he says he has "conformal invariance" and goes on about some related stuff like "dual conformal invariance" and "the Yang yin" (whatever those are) as though they matter a lot... if conformal invariance really is needed then that excludes the actual laws of physics from consideration via these methods. At another point he says he has (which is central to his work) "a new way of thinking about permutations" -- a remark which seems on its face absurd. But he does not reveal what that new way might be (at least not in the first 26 minutes), and waves his hand at 2 diagrams plus a permutation on the screen, where there seems to be little or no logical connection between them although supposedly the connection is central. Anyhow, if the main goal/claim of his work is (which he never said in the first 26 min, so this is just a total guess -- he felt no need to state his goal or main claims, if any, during the first 26 min): MAIN GOAL (?): Any physics Feynman-diagram complex amplitude (or perhaps |amplitude|^2 or something?) can be written as the volume of a certain polytope in an infinite dimensional space If that really is the claim, yes that sounds important. (It by the way is quite amazing for such a polytope to have finite volume, but it is possible, e.g. hyperbricks have volume corresponding to any infinite product.) Re computational complexity, we have the following remarkable contrast: Imre Barany and Zoltan Furedi: Computing the volume is difficult, Discrete and Computational Geometry 2 (1987) 319-326. shows no deterministic polytime algorithm can approximate the volume of a convex polytope to within an exponential(dimension) factor, contrasting with M.Dyer, A.M.Frieze, R.Kannan: A random polynomial time algorithm for approximating the volume of convex bodies, Journal of the ACM 38,1 (1991) 1-17 which shows you can approximate volume within a factor (1+epsilon) in time polynomial in #dimensions and 1/epsilon, using a randomized algorithm with success probability >=3/4. The fact that Arkani-Hamed's polytope is "in an infinite dimensional space" seems however to exclude any such algorithm, the above results were in finite dimensional spaces. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)