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I am looking for examples of theorems that may have originally had a clunky, or rather technical, or in some way non-illuminating proof, but that eventually came to have a proof that people consider to be particularly nice. In other words, I'm looking for examples of theorems for which have some early proof for which you'd say "ok that works but I'm sure this could be improved", and then some later proof for which you'd say "YES! That is exactly how you should do it!" Thanks in advance. |
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Jordan's proof of the Jordan Curve Theorem was complicated enough that people still argue about its correctness. These days, an undergrad can prove it after learning the Mayer–Vietoris sequence. |
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[Edit: This answer seems to fit the title of the question, though not the actual question in the body.] Resoluion of singularties in algebraic geometry seems like a good example. Hironaka's original proof was over 200 pages and hard to understand:
That quote is from Dan Abramovich's Math Review of the book Lectures on resolution of singularities by Kollár; the review goes on to say
I know almost nothing about this topic, but some names I know associated to the various approaches to simplification of Hironaka's proof are Bierstone, Milman, Encinas, Villamayor, Hauser, Cutkosky, Włodarczyk, Kollár. Please tell me any I missed! |
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Boone-Novikov theorem of existence of groups with undecidable word problem which originally has very long and complicated proof now has several (self-contained) proofs of length $\le 10$ pages (see Cohen, Daniel E. Combinatorial group theory: a topological approach. London Mathematical Society Student Texts, 14. Cambridge University Press, Cambridge, 1989. x+310 pp.). |
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The alternating sign matrix conjecture was first proved by Zeilberger. Zeilberger's proof was extremely computational. A much shorter conceptual proof was later given by Kuperberg. |
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A favorite of mine is the chirality of the trefoil knot, which can be proved easily using the Jones polynomial or some of its relatives. Louis Kauffman's paper "New invariants in the theory of knots", http://homepages.math.uic.edu/~kauffman/Bracket.pdf explains this nicely. I don't know how it was proved before the Jones polynomial, but quoting from p. 204 of Kauffman's paper, "In the old days (before 1984) this was something that required a lot of mathematical background." |
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Aigner and Ziegler's "Proofs from the BOOK" contains many good examples. |
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The prime number theorem, Don Zagier's proof have just three pages of length. |
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The Riesz-Thorin interpolation theorem is an example. As I understand it, the original proof published by Marcel Riesz was rather messy. Thorin found a much simpler proof of the theorem using complex analysis about ten years later. |
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I described an example, Hindman's theorem, at http://mathoverflow.net/questions/94546 . The short version is that Hindman's original proof was unpleasantly complicated, whereas a later proof by Galvin and Glazer is now accepted as the standard proof. On the intuitive level, it's a definite improvement. Formally, though, from the viewpoint of reverse mathematics, Hindman's original proof is "better" because it uses far weaker set-existence assumptions. |
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Kurosh's original proof of the subgroup theorem for free products used messy Kurosh systems. This was improved by covering space proofs (or equivalently covering groupoid proofs). One might argue the Bass-Serre theory proof is now the right one. |
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Example of a bounded linear operator on a Banach space without non-trivial closed invariant subspace. The first example was given bei Enfo in 1975. Enflo submitted the full article in 1981 and the article's complexity and length delayed its publication to 1987 (see http://en.wikipedia.org/wiki/Per_Enflo). Simpler examples were constucted for example by Beauzamy and Charles Read. |
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There are several examples from Tauberian theory. Around 1930, Karamata surprised people by giving much simpler proofs of Littlewood's original Tauberian theorems for power series. Wiener's Tauberian theorems were later given much slicker and arguably more conceptual proofs using operator theory. |
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I think that Gelfand's proof of Wiener's $1/f$ theorem qualifies. |
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Faltings' theorem (aka Mordell conjecture) can be taken as such an example. Different methods have been used so far with various difficulties. |
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If you are prepared to allow an example from mathematical physics, then Penrose's proof that a ball moving relativistically appears as a circle to an observer. This had been proved previously by brute strength calculations with Lorentz transformations. Penrose reformulated it in terms of actions of the action of the Lorentz group on the celestial sphere. Since these are just conformal transformations, which take circles to circles, the boosted sphere appears circular. |
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The global (or homology) version of Cauchy’s theorem was given an elementary proof by John Dixon. I believe this is mentioned in Rudin's Real and Complex Analysis. A proof is available online at http://www.math.uiuc.edu/~r-ash/CV/CV3.pdf. This states "The elementary proof to be presented below is due to John Dixon, and appeared in Proc. Amer. Math. Soc. 29 (1971), pp. 625-626, but the theorem as stated is originally due to E.Artin." |
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It occurs to me that Morse theory is a good example. At the time of Morse, algebraic topology (even the notion of CW complex or cell complex) is barely developed, which made his combinatorial arguments extremely difficult to read. Well, nowadays people can simply learn these topics by referring to the definite account of Milnor or Bott. |
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PP (the class of languages decidable by a probabalistic Turing machine in polynomial time) is closed under union and intersection. This was conjectured by Gill in 1972 and stayed an open problem for 18 years, til resolved by Beigel, Reingold, and Spielman (BGS) in 1995, with a complicated proof involving rational functions. The same result fell out as an almost-corollary of Scott Aaronson defining quantum postselection for unrelated reasons: the new proof is less than a page. See: |
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Witten's proof of the positive energy theorem using spinors drastically simplified the original proof by Schoen and Yau. |
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Widom's formula for calculating determinants of banded Toeplitz matrices. The original paper is hard to understand and uses quite intricate techniques. Now, a quite simple proof can be found in Böttchers "Spectral Properties of Banded Toeplitz Matrices". Actually, it also follows quite directly from the formula on Hall-Littlewood polynomials here: http://en.wikipedia.org/wiki/Hall%E2%80%93Littlewood_polynomials |
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