Examples of theorems with proofs that have dramatically improved over time

Question

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.

A sister question: Examples of major theorems with very hard proofs that have NOT dramatically improved over time

Answer 1

[Edit: This answer seems to fit the title of the question, though not the actual question in the body.]

Resolution of singularities 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, Cossart, Piltant... Please tell me any I missed!

Answer 2

The prime number theorem, Newman's short proof is only three pages long.

Answer 3

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.

Answer 4

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.

Answer 5

I think that Ax's proof of the Chevalley-Warning Theorem qualifies.

The Chevalley-Warning Theorem is an affirmative solution of a conjecture made by L.E. Dickson in 1909 and taken up more seriously by Artin [I don't seem to have onhand much information about when Artin first got involved with this; if you do, please let me know] in the 1930's. The conjecture is that every finite field is a C1 field: namely, a homogeneous polynomial in more variables than its degree always has a nontrivial zero.

Chevalley's Theorem is stronger than that: it says that if you have polynomials $P_1,\ldots,P_r$ in $n$ variables with coefficients in a finite field, then if the sum of the degrees is less than $n$, it is not possible for there to be exactly one simultaneous zero. Warning sharpened this to showing that the set of simultaneous zeros is divisible by $p$, but in fact every proof I've seen of Chevalley's Theorem -- so in particular, Chevalley's proof! -- easily adapts to prove Warning's generalization.

(Warning's real contribution was a second theorem giving a stronger lower bound on the number of common zeros, assuming that there is at least one. But that is not the result I am talking about.)

Let me be honest: there is nothing clunky or technical about Chevalley's proof. It is completely elementary, has a clear moral, and takes a bit less than two pages. In an undergraduate course, it would fill one lecture nicely.

So how much room for improvement can there be? Well, Ax's proof literally takes ten lines. See for yourself. The big idea is that $\sum_{x \in \mathbb{F}_q} x^i$ is $0$ when $0 \leq i < q-1$. You could safely assign the proof of this as an exercise in any undergraduate course in which you cover the cyclicity of the unit group $\mathbb{F}_q^{\times}$.

Can anyone think of another serious conjecture made but unresolved by mathematical luminaries which turned out to have a ten line proof?!? I can't.

Answer 6

The isosceles triangle theorem (pons asinorum), that the angles opposite the equal sides of an isosceles triangle are equal, was originally proved by Euclid by constructing several auxiliary lines. Pappus' proof uses no auxiliary lines, but only side-angle-side by "flipping" the triangle over to its mirror image.

Answer 7

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.

Answer 8

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.).

Answer 9

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

Answer 10

I described an example, Hindman's theorem, at https://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.

Answer 11

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.

Answer 12

Gauss's first proof of the Quadratic Reciprocity Law relied on an intricate induction argument and was not particularly illuminating. Later, Gauss's third proof (based on the Gauss lemma) and especially his sixth proof (based on quadratic Gauss sums) gave more insight. Perhaps, the proof with the biggest "wow effect" was given by Zolotareff using his lemma expressing the Legendre symbol as the sign of a permutation.

Answer 13

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."

Answer 14

Emanuel Lasker's original proof of the Lasker-Noether Theorem was 98 pages long, but the modern proof fits into a few paragraphs and is standard undergraduate fare.

Answer 15

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.

Answer 16

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.

Answer 17

The Krylov–Bogolyubov theorem states that a continuous map on a compact metric space admits an invariant measure. The original article is 50 pages long, but nowadays this is a one-liner. This is because all the measure theory involved has been neatly repackaged in functional analytic terms.

Answer 18

Example of a bounded linear operator on a Banach space without non-trivial closed invariant subspace.

The first example was given bei Enflo in 1975. Enflo submitted the full article in 1981 and the article's complexity and length delayed its publication to 1987 (see https://en.wikipedia.org/wiki/Per_Enflo). Simpler examples were constucted for example by Beauzamy and Charles Read.

Answer 19

Fermat's theorem on two squares is another example from number theory where the original proof was clunky or technical, but more illuminating and "wow" proofs, such as Minkowski's proof using the geometry of numbers and the so-called Zagier's one sentence proof, were later discovered.

Answer 20

I suggest Gödel's second incompleteness theorem. The first theorem states that every consistent, sufficiently strong, effectively presented formal system contains an undecidable formula. The second theorem states that such a formal system does not prove any theorem that implies its own consistency. Gödel never published a proof of the second theorem, after logicians accepted that it could be proved by encoding a proof of the first theorem within the formal system in question. However, actually to perform this encoding would be technically very difficult. Modern treatments derive the second theorem very easily after establishing the Hilbert–Bernays provability conditions.

Proofs of the first theorem have also been greatly simplified. Gödel's treatment required extensive technicalities to establish that certain existential quantifiers were bounded by a specific positive integer. These proofs can be replaced at the cost of modest other efforts.

Answer 21

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.

Answer 22

Kottman proved that in any infinite-dimensional Banach space one can find a sequence $(x_n)_{n=1}^\infty$ of unit vectors with

$$\|x_n-x_m\|>1$$ whenever $n\neq m$. The original proof is quite messy, but there is a yet another proof, attributed to Starbird, which can be found in Diestel's book Sequences and series in Banach spaces. It uses essentially linear algebra and the Hahn-Banach theorem only.

Answer 23

de Branges' proof of Bieberbach's conjecture was, if I recall correctly, one or two hundred pages. (And wrong, but correctable.) Now there is a two-page proof.

EDIT: those 2 pages are not a complete proof (oops!), rather they are shortening Weinstein's four-page proof.

Answer 24

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:

• https://arxiv.org/abs/quant-ph/0412187

Answer 25

Tverberg Theorem (1965): Let $ x_1,x_2,\dots, x_m$ be points in $ R^d$, $ m \ge (r-1)(d+1)+1$. Then there is a partition $ S_1,S_2,\dots, S_r$ of $ \{1,2,\dots,m\}$ such that $\cap _{j=1}^rconv (x_i: i \in S_j) \ne \emptyset$.

Tverberg's theorem was conjectured by Birch who also proved the planar case. The case $r=2$ is a 1920 theorem of Radon which follows easily from linear algebra consideration.

(The first thing to note is that Tverberg's theorem is sharp. If you have only $ (r-1)(d+1)$ points in $ R^d$ in a "generic" position then for every partition into $ r$ parts even the affine spans of the points in the parts will not have a point in common.)

The first proof of this theorem appeared in 1965. It was rather complicated and was based on the idea to first prove the theorem for points in some special position and then show that when you continuously change the location of the points the theorem remains true. A common dream was to find an extension of the proof of Radon's theorem, a proof which is based on the two types of numbers - positive and negative. Somehow we need three, four, or $ r$ types of numbers. In 1981 Helge Tverberg found yet another proof of his theorem. This proof was inspired by Barany's proof of the colored Caratheodory theorem (mentioned below) and it was still rather complicated. It once took me 6-7 hours in class to present it.

What could be the probability of hearing two new simple proofs of Tverberg'stheorem on the same day? While visiting the Mittag-Leffler Institute in 1992, I met Helge one day around lunch and asked him if he has found a new proof. To my surprise, he told me about a new proof that he found with Sinisa Vrecica. This is a proof that can be presented in class in 2 hours! It appeared (look here) along with a far-reaching conjecture (still unproved). Later in the afternoon I met Karanbir Sarkaria and he told me about a proof he found to Tverberg's theorem which was absolutely startling. This is a proof you can present in a one hour lecture; it also somehow goes along with the dream of having $r$ "types" of numbers replacing the role of positive and negative real numbers. Another very simple proof of Tverberg's theorem was found by Jean-Pierre Roudneff in 1999.

For further details see these blog posts (I,II).

Answer 26

I hope I'm not repeating an answer already given (or worse still, an answer I gave and have forgotten!) but the solution to the derivation problem for $L^1$-group algebras would seem to qualify. The problem had been open since the late 1960s, resisted the efforts of several serious researchers, and was finally solved by Losert in a tour de force in the Annals:

V. Losert, The derivation problem for group algebras. Ann. of Math. (2) 168 (2008), no. 1, 221–246.

Some years later, Bader, Gelander and Monod obtained a novel fixed-point theorem for affine group actions on $L$-spaces, from which the positive solution to the derivation problem follows almost immediately (modulo some standard reductions that were known to specialists since the early 1970s).

U. Bader, T. Gelander, N. Monod, A fixed point theorem for L1 spaces. Invent. Math. 189 (2012), no. 1, 143–148.

The same paper also gives another example to answer the OP's question. As well as giving a remarkably quick proof of the derivation problem for $L^1(G)$, the FPT of Bader–Gelander–Monod gives a very quick proof that every derivation from a ${\rm C}^*$-algebra into its dual is inner — this was originally obtained by Haagerup in his "nuclear implies amenable" 1983 paper, and the proof had been simplified by Haagerup+Laustsen in an article published in 1997 conference proceedings, but the proof via the BGM-FPT leaves the earlier ones in the dust.

Answer 27

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

Answer 28

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

Answer 29

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: https://en.wikipedia.org/wiki/Hall%E2%80%93Littlewood_polynomials

Answer 30

Szemeredi's theorem and its special case Roth's theorem have been given quite conceptual proofs by Hilel Furstenberg using ergodic methods which I think is quite natural while the initial proofs were extremely complicated.

Answer 31

One historical example that should probably be on this list is the Abel-Ruffini Theorem, which states that there is no general solution in radicals to polynomials of degree 5 and higher. Attempting to clarify the bigger picture of why the proof works may have been one of Galois's motivations for his development of what is now Galois Theory- and the proof we have now is quite insightful and illuminating.

Answer 32

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."

Answer 33

The original proof of Muller-Schupp theorem saying that finitely generated groups with context-free word problem are exactly the virtually free groups, is really involved (though nice) and uses accessibility and Stallings results on ends.

But there is a short and elementary way to prove Muller-Schupp theorem using rewriting systems, as was recently done by Volker Diekert.

Answer 34

The Amitsur-Levitski Theorem (the standard non-commutative polynomial of order $2n$ vanishes identically on $M_n(k)$) qualifies. The original proof (1950) is messy, with no clear logical structure and takes 17 pages. The natural proof was given in 1976 by S. Rosset in a 2-pages article.

Answer 35

an example from number theory (where such simplifications are not uncommon), Bertrands postulate

for any integer $n > 3$, there always exists at least one prime number $p$ with $n < p < 2n − 2$

was first simplified by Ramanujan and then later by Erdos who also proved a more general case.

another interesting case study here may be Lindemanns proof of transcendence of PI which is subsumed by later more general results. as Wikipedia states "Weierstrass proved the above more general statement in 1885. The theorem, along with the Gelfond–Schneider theorem, is extended by Baker's theorem, and all of these are further generalized by Schanuel's conjecture."

another "possible/controversial" famous/legendary case study here is Fermats Last Theorem; Fermat scribbled in the margin of his book that he had a remarkable proof, but modern consensus is that he must have been mistaken based on the 2020-hindsight of Wiles complex proof. however, strictly speaking, it has not been proven impossible that there exists a short proof.

it seems that later simplifications of proofs is a natural process of the historical/evolutionary progress of mathematics so that results once thought more arcane/inscrutable/complex become more accessible with the polishing/systematization of ideas/techniques.

Answer 36

The first proof of the Hopf invariant one theorem due to Adams is very technical. It involves decomposing $sq^{2^n}$ as a composite of secondary cohomology operations when $n\geq 4$. Then Atiyah and Adams came up with a proof that uses $K$-theory which both admired for its elegance and simplicity.

Answer 37

Manjul Bhargava's proof of the 15 theorem was dramatically simpler than Conway and Schneeberger's original proof.

Answer 38

Godel's original proof of his Incompleteness Theorem was immensely more complicated Aaronson's one-paragraph derivation of Incompleteness from Undecidability of Halting Problem. Even including a proof of Undecidability of Halting it would be much shorter and clearer than the original proof.

Answer 39

1. Chirka's proof ("On the propagation of holomorphic motions", 2004) of Slodkowski's theorem ("Holomorphic motions and polynomial hulls", 1991) is much simpler. (Slodkowski's paper is not that long, but uses a lot more difficult mathematics. A number of people had previously attempted to give alternative proofs, but these turned out to contain gaps.)

2. The original proof by Baker that repelling periodic points are dense in Julia sets of transcendental entire functions used the Ahlfors five islands theorem (a very deep result). The proof by Duval and Berteloot (Une démonstration directe de la densité des cycles répulsifs dans l’ensemble de Julia), building on work of Schwick, takes less than a page, and uses only very elementary results (notably, Zalcman's rescaling lemma for normal families). Even for rational functions, this is probably the simplest proof (simpler than the original ones of Fatou and Julia) currently in existence.

Answer 40

Some results by Donaldson were simplified via the Seiberg-Witten invariants.

From Wikipedeia: Seiberg–Witten invariants are similar to Donaldson invariants and can be used to prove similar (but sometimes slightly stronger) results about smooth 4-manifolds. They are technically much easier to work with than Donaldson invariants; for example, the moduli spaces of solutions of the Seiberg–Witten equations tend to be compact, so one avoids the hard problems involved in compactifying the moduli spaces in Donaldson theory.

See also this MO answer by Dylan Thurston.

(Added Jan 7, '16) An additional piece of information I learned today from the Zabrodsky's lecture delivered by Peter Ozsváth is about the existence of exotic smooth structure on $\mathbb R^4$. The original proof was based on Freedman theorem and on Donaldson theorem. Results based on new knot invariants such as the Knot Floer homology (and simpler combinatorial descriptions of these invariants) can replace the "Donaldson side" of the proof by a much simpler argument.

Answer 41

In machine learning and statistics, the technique of Rademacher complexities has a way of radically simplifying otherwise complicated proofs -- in particular those involving margin-based generalization bounds.

Some examples include Support Vector Machine margin bounds, which used to be proved via intricate combinatorial fat-shattering techniques of https://homes.di.unimi.it/~cesabian/Pubblicazioni/jacm-97b.pdf http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.42.6950&rep=rep1&type=pdf http://www.sciencedirect.com/science/article/pii/S0304397500001341 and are now proved in 3 lines (see the Thm. 4.3 in http://www.cs.nyu.edu/~mohri/mlbook/ ).

Another example is margin-based boosting bounds, which were originally proved via an involved covering and sampling argument, http://projecteuclid.org/euclid.aos/1024691352 and now has a very simple Rademacher-based proof (Thm. 5.7 in https://mitpress.mit.edu/books/boosting ).

Answer 42

Faltings' theorem (aka Mordell conjecture) can be taken as such an example. Different methods have been used so far with various difficulties.

Answer 43

DeMoivre's theorem. The pre-calc version of the proof relies on a lot of triangle geometry to establish trigonometric sum formulas and then uses induction. If you use Euler's Identity, it's a one line proof. (Of course, then there's a large amount of analysis implicit in the background.)

Answer 44

See Ostrowski's proof of Luroth theorem in Schinzel's book "Polynomials with special regard to reducibility"

Answer 45

The four color theorem’s first proof by Appel and Haken, which actually contained mistakes, was proved again much more succinctly and successfully by Robertson, Sanders, Seymour, and Thomas (see for instance here). As far as I know, there is no better proof as yet, and unlikely there will ever be one, but who knows.

Answer 46

Liouville's Theorem that there exists a transcendental number had its proof greatly improved by Cantor who showed that a mere counting argument suffices.

Liouville's argument needs facts about how rationals interact with polynomials, and makes use of the metric structure on the reals.

Cantor's argument merely uses the fact that each integer polynomial can be described in finitely many symbols and has only finitely many roots. And it uses this to deduce a stronger result: that almost no reals are algebraic.