most recent 30 from http://mathoverflow.net 2013-05-24T20:21:22Z http://mathoverflow.net/feeds/question/95837 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time Manya 2012-05-03T10:07:24Z 2012-07-09T07:27:58Z

<p>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!"</p> <p>Thanks in advance.</p>

http://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time/95856#95856 Liviu Nicolaescu 2012-05-03T12:44:12Z 2012-05-03T12:44:12Z

<p>Aigner and Ziegler's "Proofs from the BOOK" contains many good examples.</p>

http://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time/95858#95858 Mark Sapir 2012-05-03T12:47:14Z 2012-05-03T12:47:14Z

<p>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.). </p>

http://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time/95863#95863 Johan Wästlund 2012-05-03T13:18:36Z 2012-05-03T13:18:36Z

<p>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", <a href="http://homepages.math.uic.edu/~kauffman/Bracket.pdf" rel="nofollow">http://homepages.math.uic.edu/~kauffman/Bracket.pdf</a> explains this nicely. </p> <p>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."</p>

http://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time/95878#95878 Andreas Blass 2012-05-03T15:30:04Z 2012-05-03T15:30:04Z

<p>I described an example, Hindman's theorem, at <a href="http://mathoverflow.net/questions/94546" rel="nofollow">http://mathoverflow.net/questions/94546</a> . 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.</p>

http://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time/95879#95879 Luis Solarte 2012-05-03T15:39:52Z 2012-05-03T15:39:52Z

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

http://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time/95881#95881 John Wiltshire-Gordon 2012-05-03T15:49:38Z 2012-05-03T15:49:38Z

<p>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.</p>

http://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time/95887#95887 Artie Prendergast-Smith 2012-05-03T16:07:58Z 2012-05-03T17:07:35Z

<p>[<strong>Edit:</strong> This answer seems to fit the title of the question, though not the actual question in the body.]</p> <p>Resoluion of singularties in algebraic geometry seems like a good example. Hironaka's original proof was over 200 pages and hard to understand: </p> <blockquote> <p>"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."</p> </blockquote> <p>That quote is from Dan Abramovich's Math Review of the book <em>Lectures on resolution of singularities</em> by Kollár; the review goes on to say</p> <blockquote> <p>"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)."</p> </blockquote> <p>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!</p>

http://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time/95888#95888 Benjamin Steinberg 2012-05-03T16:13:48Z 2012-05-03T16:13:48Z

<p>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. </p>

http://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time/95890#95890 Arch Stanton 2012-05-03T16:27:17Z 2012-05-03T16:27:17Z

<p>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.</p>

http://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time/95895#95895 jjcale 2012-05-03T16:40:53Z 2012-05-03T16:40:53Z

<p>Example of a bounded linear operator on a Banach space without non-trivial closed invariant subspace.</p> <p>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 <a href="http://en.wikipedia.org/wiki/Per_Enflo" rel="nofollow">http://en.wikipedia.org/wiki/Per_Enflo</a>). Simpler examples were constucted for example by Beauzamy and Charles Read.</p>

http://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time/95905#95905 Tim Ferguson 2012-05-03T18:46:55Z 2012-05-03T18:46:55Z

<p>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.</p>

http://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time/95918#95918 George C 2012-05-03T21:02:33Z 2012-05-03T21:02:33Z

<p>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 <a href="http://www.math.uiuc.edu/~r-ash/CV/CV3.pdf" rel="nofollow">http://www.math.uiuc.edu/~r-ash/CV/CV3.pdf</a>. 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."</p>

http://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time/96370#96370 Timothy Chow 2012-05-08T20:25:51Z 2012-05-08T20:25:51Z

<p>The <a href="http://mathworld.wolfram.com/AlternatingSignMatrixConjecture.html" rel="nofollow">alternating sign matrix conjecture</a> was first proved by Zeilberger. Zeilberger's proof was <i>extremely</i> computational. A much shorter conceptual proof was later given by Kuperberg.</p>

http://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time/96374#96374 Timothy Chow 2012-05-08T21:04:08Z 2012-05-08T21:04:08Z

<p>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.</p>

http://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time/96377#96377 Zhang Xiao 2012-05-08T21:31:25Z 2012-05-08T21:31:25Z

<p>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.</p> <p>Well, nowadays people can simply learn these topics by referring to the definite account of Milnor or Bott.</p>

http://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time/96951#96951 robot 2012-05-14T22:10:37Z 2012-05-14T22:10:37Z

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

http://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time/96969#96969 Ehsan M. Kermani 2012-05-15T06:27:12Z 2012-05-15T06:27:12Z

<p><a href="http://en.wikipedia.org/wiki/Faltings%27_theorem#Proofs" rel="nofollow">Faltings' theorem</a> (aka <em>Mordell conjecture</em>) can be taken as such an example. Different methods have been used so far with various difficulties. </p>

http://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time/101735#101735 none 2012-07-09T03:53:06Z 2012-07-09T03:53:06Z

<p>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:</p> <ul> <li><a href="http://arxiv.org/abs/quant-ph/0412187" rel="nofollow">http://arxiv.org/abs/quant-ph/0412187</a></li> </ul>

http://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time/101736#101736 Richard Eager 2012-07-09T04:04:08Z 2012-07-09T04:04:08Z

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

http://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time/101744#101744 Per Alexandersson 2012-07-09T07:27:58Z 2012-07-09T07:27:58Z

<p>Widom's formula for calculating determinants of banded Toeplitz matrices. The original paper is hard to understand and uses quite intricate techniques.</p> <p>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: <a href="http://en.wikipedia.org/wiki/Hall%E2%80%93Littlewood_polynomials" rel="nofollow">http://en.wikipedia.org/wiki/Hall%E2%80%93Littlewood_polynomials</a></p>