Gil Kalai
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 Feb 6 awarded Favorite Question Feb 4 comment Examples of major theorems with very hard proofs that have NOT dramatically improved over time (cont) Not easy by any means, but conceptually more digestible. Kedlaya's contribution was to adapt that technique to the pp-adic setting once rigid cohomology had strong enough foundations; the real breakthrough in simplification was in the earlier work of Laumon et al." 3) Nick Katz provided ("Four lectures on Weil II") an alternative proof of one of the main parts of Weil II using "relatively" elementary facts on étale cohomology and the so-called Larsen alternative. Feb 4 comment Examples of major theorems with very hard proofs that have NOT dramatically improved over time Here are the old comments: 1) "Is this true? I'm not an expert in the area, but the paper arxiv.org/abs/math/0210149 by Kedlaya seems to give an alternate proof (and the first section lists others)." , 2) This is not true. The technique of ℓℓ-adic Fourier transform introduced by Laumon et al. has led to a much simpler approach to the main results of Weil II (vast and much more useful generalization of Deligne's first paper on the Weil conjectures). Feb 4 awarded Revival Feb 3 comment Examples of major theorems with very hard proofs that have NOT dramatically improved over time Indeed there were some comments (to a deleted answer) regarding major simplification by Laumon and others for Deligne's proof. Feb 3 comment How many simplicial complexes on n vertices up to homotopy equivalence? Dear Vidit, This is my expectation. But maybe I am pushing it to assume it will be doubly exponential. Since you dont specify the dimension it is reasonable to think it will typically be n/2. Which also mean that the complex will typically be simply connected. The possibilities for sequences of Betti numbers are well known. I think there are roughly $2^{n^2}$ possible sequences of Betti numbers. The point is that these sequences can be very different for different characteristics. But I admit that the guess that the number is doubly exponential is pretty wild. Feb 3 awarded Nice Answer Feb 3 answered How many simplicial complexes on n vertices up to homotopy equivalence? Feb 3 answered Classifying two-faces of four-polytopes Feb 3 awarded Nice Answer Feb 2 awarded Nice Answer Feb 2 revised Examples of major theorems with very hard proofs that have NOT dramatically improved over time added 1044 characters in body Jan 31 comment realization spaces of 3-dimensional polytopes (But it sounds too good to be true...) Jan 31 comment Proposals for polymath projects Dear Vasselin, Hmm, that's very interesting. Maybe it is interesting to demonstrate the question even based on GRH. I supposed it is easier to show that positive fraction of primes divides $2^n+5$ than that for a positive fraction of them 2 is a primitive root of unity. We can also relax the question by perhaps looking at $2^n + x$ where we let $x$ ranges over an interval (depending on n) to make it easier. But in view of your comments maybe this is not a good polymath question. Jan 31 comment Examples of common false beliefs in mathematics Thank you, Maxime! Jan 31 revised Proposals for polymath projects update: polymath11 Jan 31 comment Proposals for polymath projects Thanks, V. for the good (but short lived) comments. Since this is not in my area I am not sure at all it is a good question. I realize it is very difficult but my uneducated hunch is that it is not hopeless and potentially fruitful. Jan 29 answered “Database” of simplicial polytopes/spheres Jan 29 comment What polytope is this? Bounded sums with choice of coefficients Dear Brendan, it looks like a relative of the "hypersimplex" (in this case) the convex hull of centers of edges of a simplex. (More generally, centers of $k$-faces.) Jan 29 revised Tverberg partitions with less than (r-1)(d+1)+1 points added 1 character in body