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Mar
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comment Expert, Intuitive, Organizing Analogies
(But I am not sure if this answers the question as intended.)
Mar
25
answered Expert, Intuitive, Organizing Analogies
Mar
17
answered Important formulas in Combinatorics
Mar
15
revised Experimental Mathematics
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Mar
15
comment What advantage humans have over computers in mathematics?
This is an interesting answer, but it is not clear that computers will be weaker "on creativity."
Mar
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answered Important formulas in Combinatorics
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revised f(f(x))=exp(x)-1 and other functions “just in the middle” between linear and exponential
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Feb
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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
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Feb
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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
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Feb
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answered How many simplicial complexes on n vertices up to homotopy equivalence?