## Examples of theorems with proofs that have dramatically improved over time

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!"

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Seems related to the earlier MO question mathoverflow.net/questions/43820/…. – Tom De Medts May 3 2012 at 10:13
@Tom: Yes, thanks. I'd be happy to collect some more examples though, especially of proofs that (now) seem to be a particularly good "fit" for a theorem. – Manya May 3 2012 at 11:35
I guess this is such an example: mathoverflow.net/questions/24913/… – Steve D May 3 2012 at 15:59
I don't know the original proof, but I heard that the trick of Rabinovich provided a drastic improvement of the proof of Hilbert's Nullstellensatz. – Peter Arndt May 3 2012 at 21:42
It would be also interesting to hear of theorems where people didn't think that the proof could be much improved, but then were proven wrong. – David Corwin Jul 9 at 6:02

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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Witten's proof of the positive energy theorem using spinors drastically simplified the original proof by Schoen and Yau.

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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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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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I think that Gelfand's proof of Wiener's $1/f$ theorem qualifies.

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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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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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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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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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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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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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 Worth mentioning that Read was subsequently the first to construct such an operator on $\ell^1$ – Yemon Choi May 3 2012 at 19:00 Related: Aronszajn and Smith's theorem that a compact linear operator on a Banach space must have a nontrivial invariant subspace was later given a dramatically simpler proof by Lomonosov. – Timothy Chow May 8 2012 at 20:36 Did the earlier theorem also get hyper invariant subspaces, as Lomonosov does? – Yemon Choi May 9 2012 at 0:16

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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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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[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:

"Even A. Grothendieck [in Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, 7--9, Gauthier-Villars, Paris, 1971; MR0414283 (54 #2386)] admitted openly that he did not completely understand Hironaka's proof."

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

"One can [nowadays] devote a few weeks in a first course on algebraic geometry to give just a complete proof of resolution of singularities in characteristic 0 (Chapter 3 of the present book, which is largely self-contained)."

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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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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Do you know a good online source where the proof is well explained (not just outlined etc.)? I am equally interested in good textbooks discussing this approach. – GH May 3 2012 at 16:56
In typical US universities, undergrads do not learn Mayer-Vietoris, but your point is of course still correct. – Henry Cohn May 3 2012 at 17:19
According to Tom Hales, there Jordan's proof should never have been controversial. An objection arose that he assumed the polygonal case without proof --- but that's a trivial omission! See mizar.org/trybulec65/4.pdf As for the idea that a student can prove it using Mayer-Vietoris, I disagree. Yes, a good undergrad can learn Mayer-Vietoris, but in order to use it here, you also need that the circle (or in generality, the sphere) is an ENR, which is a separate and clearly nontrivial result. Remember, the hard case of the Jordan theorem is the fractal case. – Greg Kuperberg May 9 2012 at 19:19
What is an ENR? Also, can someone respond to my first comment? Thanks in advance. – GH May 12 2012 at 18:24
@GH: Euclidean neighborhood retract. – Boris Bukh May 15 2012 at 10:47

The prime number theorem, Don Zagier's proof have just three pages of length.

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As the title of Zagier's paper makes clear, this proof is due to Donald Newman: "Newman's short proof of the prime number theorem," Amer. Math. Monthly 104 (1997), 705-708. – Timothy Chow May 8 2012 at 20:50

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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I think Furstenburg's Stone-Czech proof of van der Waerden's theorem would also fit the bill. – Benjamin Steinberg May 3 2012 at 16:11
Thanks, and thanks. – Manya May 4 2012 at 8:05

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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Is it the case that using the Jones polynomial sort of hides away all that mathematical background, or does it somehow clarify the main idea of the proof (give one a sense of why it is true?) – Manya May 4 2012 at 8:17
The first chirality proof, by Max Dehn in 1914, was indeed a lot more involved than the Jones polynomial proof. It involved finding the automorphisms of the trefoil knot group. – John Stillwell May 8 2012 at 23:51

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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Aigner and Ziegler's "Proofs from the BOOK" contains many good examples.

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 Why the downvote? – Johan Wästlund May 3 2012 at 16:59 I don't know who down-voted what. I don't think that I did. – Liviu Nicolaescu May 3 2012 at 17:39 @Liviu: Thanks. Though some of the proofs in that book (as the authors themselves admit) are not necessarily the nicest or cleanest versions. Are there any proofs in there that you think are particularly good? – Manya May 4 2012 at 8:20 My favorite from that book is Sperner's proof of Brouwer's fixed point theorem. – Liviu Nicolaescu May 4 2012 at 12:18