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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
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.
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
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
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.)
Dec
19
comment Solutions to the Continuum Hypothesis
I stand corrected. I will have to wait a while to reach the 20k+ stature :) , and, in any case, I will feel best if the owner will undelete it or resubmit it. (Quid, can the owner undelete it without 20k+?)
Dec
19
comment Solutions to the Continuum Hypothesis
Gerry, not when the owner deleted it.
Dec
18
comment Solutions to the Continuum Hypothesis
Dear Andres, I agree. (But it was deleted by the author so I dont know what can be done about it beside asking Grigor to undelete.)
Dec
13
comment Kruskal-Katona for homocyclic groups?
Very nice question. I am quite sure some higher alphabets versions of Kruskal-Katona are known but I don't remember. In fact it seems that this is a Kruskal-Katona theorem for products of paths.
Nov
7
comment What are some very important papers published in non-top journals?
Well, this may apply also to the first and second proofs of Szemeredi's theorem.
Nov
7
comment What are some very important papers published in non-top journals?
I think the importance of publishing in top journals is more emphasized today than used to be a few decades ago. Ohh, the good old times! (And all along I don't think it was considered so important in the area of combinatorics.)
Oct
29
comment Why is “P vs. NP” necessarily relevant?
Joshua, When I say "certain" I do not mean "there exist" but rather certain specific algorithmic problem. I edit my answer to make it clearer.
Oct
8
comment Important formulas in Combinatorics
Dear Tom, Thanks for the addition. As there are few answers that fits to more than one topic and answers that contain more than one formula I represented in the illustration each answer by a single formula and interested viewers can, of course, jump to the full answer.
Oct
6
comment first Chern class of complex vector bundles and first Pontrjagin class of quaternionic vector bundles
Regardless of this particular question, in my opinion, "standard characteristic class stuff" like "standard applications of spectral sequences" like "standard facts about Floer homology" etc. are research-level mathematics as far as mathoverflow is concerned.
Oct
2
comment Proposals for polymath projects
One more remark regarding the level of problems. In four out of the five first polymath projects the problems were well known and central open problems in their field (while not among the famous problems of mathematics). Polymath2 was a more open-ended project but it also reflected a question that experts in the relevant field were thinking about for decades.
Oct
1
comment Proposals for polymath projects
Joel, John and all. I did find a very relevant link about the criteria for choosing the very first polymath project: gowers.wordpress.com/2009/02/01/why-this-particular-problem I think that they are very good (of course, with more projects different criteria can be experimented.)