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Update: Please restrict your answers to "tweets" that give more than just the statement of the result, and give also the essence (or a useful hint) of the argument/novelty.

I am looking for examples that the essence of a notable mathematical development fits a tweet (140 characters in English, no fancy formulas).

Background and motivation: This question was motivated by some discussion with Lior Pachter and Dave Bacon (over Twitter). Going over my own papers and blog posts I found very few such examples. (I give one answer.) There were also few developments that I do not know how to tweet their essence but I feel that a good tweet can be made by a person with deep understanding of the matter.

I think that a good list of answers can serve as a good educational window for some developments and areas in mathematics and it won't be overly big.

At most 140 characters per answer, single link is permitted. Tweeting one's own result is encouraged.

Update: I learned quite a few things, and Noam's tweet that I accepted is mind-boggling.

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    $\begingroup$ I have discovered a truly remarkable proof of this theorem which this Tweet is too small to contain. $\endgroup$
    – Glorfindel
    Apr 26, 2017 at 8:26
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    $\begingroup$ I feel like most of the answers are mis-interpreting the question. This doesn't ask for a result whose statement is in 140 characters; that would be too broad: most paper titles fit in them. It asks for a result whose essence is tweetable: given the tweet alone, a mathematician with good knowledge of the field can fill in the details and complete a proof. So I am going to downvote almost all of them. $\endgroup$ Apr 26, 2017 at 10:16
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    $\begingroup$ @FedericoPoloni: what is it, a kind of joke? Could you please indicate to me where exactly in the question it is written that the tweet should be such that "a mathematician with good knowledge of the field can fill in the details and complete a proof"? Honestly, I do not think that your personal interpretation of the locution "essence of a notable mathematical development" should be taken as a rule here. $\endgroup$ Apr 26, 2017 at 10:21
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    $\begingroup$ Perhaps the next big MO question should be, "What is the essence of a mathematical result?" I myself lean toward Federico's interpretation- pithifying a theorem's statement does not necessarily clarify or illuminate the ideas at play. $\endgroup$
    – Neal
    Apr 26, 2017 at 12:52
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    $\begingroup$ My initial intention was indeed that the "tweet" gives more than just the statement of the result but also the essence of the argument/novelty. To demand that a mathematician in the field can fill the details is too much to ask for. $\endgroup$
    – Gil Kalai
    Apr 26, 2017 at 12:58

89 Answers 89

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det(exp(X)) = exp(tr(X)). Remarkably, the RHS does not involve off-diagonal elements of X.

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The prime numbers contain arbitrarily long arithmetic progressions.

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1_000_000 $ problem solved: every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. Hint: use the Ricci flow.

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Lagrange: every integer is sum of 4 squares. Pf: 4-squares identity + every real prime splits in the ring of Hurwitz quaternions. Link

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Banach–Tarski paradox: a solid sphere can be divided in a finite # of parts which can be joined to form 2 spheres identical to the original.

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    $\begingroup$ Too many characters. :-) $\endgroup$
    – Gro-Tsen
    Apr 26, 2017 at 10:02
  • $\begingroup$ @Gro-Tsen true. Let's cheat a bit on the grammar, and use # for something else than a #hashtag. $\endgroup$
    – Glorfindel
    Apr 26, 2017 at 10:06
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    $\begingroup$ "finite # of parts" -> "5 parts" save some more bites. Feels like code-golfing… $\endgroup$
    – Dirk
    Apr 26, 2017 at 12:07
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    $\begingroup$ Can the idea in the proof be distilled into 140 characters? "Banach-Tarski: AC + free group actions create unmeasurable ghost sets in the sphere; these become two same-volume spheres, from one!" $\endgroup$
    – Neal
    Apr 26, 2017 at 12:57
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    $\begingroup$ A long haired cat or a samoyed can be used very effectively for demonstration... $\endgroup$
    – chx
    May 1, 2017 at 19:46
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Here are three.

Wiles theorem: Every elliptic curve over Q is parameterized by modular functions.

Faltings Theorem/Mordell Conjecture: A curve of genus at least 2 has only finitely many rational points.

Faltings Theorem/Shafarevich Conj: There are only finitely many abelian varieties with good reduction outside a given finite set of primes.

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    $\begingroup$ "Wiles theorem" actually has several other authors; I think all five can be accommodated within the 140-character limit. $\endgroup$ Apr 29, 2017 at 14:14
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Borsuk–Ulam theorem:

If you crumple a ball and run a steam roller over it there are at least two antipodal points touching.

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No surjection f from S to its powerset. If so, let T = {x in S | x notin f(x)}. Must be some y such that f(y)=T. Then y in T iff y notin T!#

(140 characters, nothing fancy)

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This question is actually much older than what it appears to be. The famous fundamental anagram of calculus (Newton, 1676)

6accdae13eff7i3l9n4o4qrr4s8t12ux

is an anagram of the Latin

Data aequatione quotcunque fluentes quantitates involvente, fluxiones invenire; et vice versa

(less than 140 symbols). In modern English it means

Given an equation involving any number of fluent quantities to find the fluxions, and vice versa

or, in Arnold's interpretation

It is useful to solve differential equations

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Gödel's first incompleteness theorem - any consistent formal system capable of basic arithmetic is incomplete. Link

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If Ax = y underdetermined w unique sparse soln x0, minimizing L1(x) recovers x0 under mild conds. Shannon-Nyquist bad (or sick) thm! Link

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Few things multiply nicely (division rings usually noncommutative) but few things multiply nicely (finite ones are) #Wedderburn

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A rectangle tiled by rectangles with an integral side has an integral side because it has mass zero under a certain periodic signed measure.

(Tweeted here)

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This is based on a real tweet of Ryan O'Donnell on the proof of MIP*=RE which disproves Connes' Embedding Conjecture from 1976.

MIP* = RE, by Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, Henry Yuen: https://arxiv.org/abs/2001.04383 . There is a multiple-entagled-quantum-provers proof system for the Halting Problem, and Connes' Embedding Conjecture is false.

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How about a proof that, given a set, there always exists a bigger set?

For every set A, there is a set that doesn't inject into A. Take the set B of ordinals that inject into A. B is an ordinal and is not in B.

I think one can also fit a proof of Sylvester's theorem in 137 characters.

Take n points not on a line. Let L be a line containing >1 points minimizing the distance to a point off L. It contains exactly 2 points.

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  • $\begingroup$ Can you ensure that B is not a proper class within the tweet limit? $\endgroup$ Apr 29, 2017 at 12:53
  • $\begingroup$ @Hagen von Eitzen: No. The tweet assumes the familiarity of the reader with the axiom of replacement. $\endgroup$
    – Burak
    Apr 29, 2017 at 13:44
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Littlewood's example of a 2 line dissertation fits here, I believe.

An integral function never 0 or 1 is a constant:

$\exp(i\Omega(f(z)))$ is a bounded integral function

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    $\begingroup$ Just seen this. Shouldn't it be "entire function"? $\endgroup$
    – Yemon Choi
    Mar 9, 2019 at 19:48
  • $\begingroup$ @YemonChoi It is the way Littlewood writes, but you are right, current terminology is "entire". Shall I change it nevertheless? Can't decide... $\endgroup$ Mar 10, 2019 at 5:29
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Every set of integers with divergent reciprocals contains 3-term arithmetic progressions. Erdos and Turan conjectured; Bloom and Sisask proved; Bateman and Katz's method for cap set used. link. #TWTMTH

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There is an algorithm to compute étale cohomology in finite time (but we don't know how long it takes) — link

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Weyl's law: pack a domain with tiny squares. In the limit, hear the domain's volume.

and

Polya's theorem: pack a square with tiny domains. In the limit, the domain is always higher-pitched than Weyl's law thinks it should be.

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  • $\begingroup$ Pardon my ignorance, but I can't make sense of this. How do I hear something by/after packing? $\endgroup$
    – Dirk
    Apr 27, 2017 at 5:39
  • $\begingroup$ Which Pólya's theorem does this refer to? Searching, I find the Pólya enumeration theorem. $\endgroup$
    – cfh
    Apr 27, 2017 at 7:06
  • $\begingroup$ @cfh "On the Eigenvalues of Vibrating Membranes," 1961, Proc LMS (onlinelibrary.wiley.com/doi/10.1112/plms/s3-11.1.419/abstract). A domain that tiles Euclidean space has every Dirichlet eigenvalue greater than the value predicted by Weyl's law. $\endgroup$
    – Neal
    Apr 27, 2017 at 12:59
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    $\begingroup$ @Dirk Curious ignorance needs no pardon :) c.f. xkcd 1053. This refers to the eigenvalues of the Laplacian on a domain with (Dirichlet) boundary condition. One can interpret them as squared frequencies of standing waves, so associate them with harmonic tones of a domain. Weyl's law says that the asymptotics of this sequence encodes the volume of the domain. As the Laplacian is elliptic, there's a variational principle that ends up allowing one to compare eigenvalues across domains; Weyl's law can be proven by tiling with squares (which have computable eigenvalues) and making that comparison. $\endgroup$
    – Neal
    Apr 27, 2017 at 13:12
  • $\begingroup$ That's pretty cool, thanks for the explanation! (Now try to explain that in a tweet...) $\endgroup$
    – Dirk
    Apr 27, 2017 at 14:10
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Cap set problem solved: polynomial method, punchline going back to (a+b)²=a²+b²+2ab. link

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    $\begingroup$ I actually tweeted it and a few more under the hashtag #TWTMTH $\endgroup$
    – Gil Kalai
    May 1, 2017 at 18:06
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A finite(dimensional) domain has to be a field: all nonzero "multiply-by-an-element" maps are self-embeddings.

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Cole 1903: 2^67-1=147,573,952,589,676,412,927=193,707,721 × 761,838,257,287 #MoreThanThreeYearsOfSundays

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There is a way of pretending that any reduced ring is Noetherian and a field. Grothendieck's generic freeness lemma is then quite easy. link (Section 11.5)

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**ΕΥΡΗΚΑ num=Δ+Δ+Δ (Every positive integer can be written as the sum of three triangular numbers, Gauss, July 10, 1796)

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Aubrey de Grey's strategies for finding the chromatic number of the plane: A) prolonging life to 1000 years and waiting for a solution B) Constructing a unit-distance planar graph that requires 5 colors!!! Based on SAT-solvers and a lot of Moser Spindles. link

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$S$ unit sphere of an $\infty$-dim Banach isn't compact. Pf: for $H$ closed hyperplane, $\bigcap(S\cap H)=\varnothing$ (Hahn-Banach), but no finite subintersection is empty.

Note: $\infty$, $\bigcap$, $\cap$, $\neq$, and $\varnothing$ are unicode characters, so this is actually tweetable!

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    $\begingroup$ Shouldn't $\ne$ be $=$? $\endgroup$ Apr 16, 2019 at 13:13
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    $\begingroup$ Thanks @AndrésE.Caicedo, I correct! $\endgroup$ Apr 16, 2019 at 19:26
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Stark-Heegner theorem: There are nine imaginary quadratic fields with class number one. Link

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I can't resist. Birkhoff's theorem:

For an erg measure-pres trans, orbit-wise avgs of a fn agree with the spatial avgs.

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Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos & generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.

Not in English, 268 characters so two tweets, @Glorfindel 's comment, but I couldn't not post it.

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  • $\begingroup$ Dear Ethan, warm regards--Gil $\endgroup$
    – Gil Kalai
    Apr 26, 2017 at 16:04
  • $\begingroup$ The canonical answer $\endgroup$
    – David Roberts
    Nov 5 at 9:15
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Trichotomy: the rational points on an algebraic curve are parametrizable; form a finitely-generated abelian group; or form a finite set.

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