By Mark S


2019-03-10 19:20:58 8 Comments

Oftentimes in math the manner in which a solution to a problem is announced becomes a significant chapter/part of the lore associated with the problem, almost being remembered more than the manner in which the problem was solved. I think that most mathematicians as a whole, even upon solving major open problems, are an extremely humble lot. But as an outsider I appreciate the understated manner in which some results are dropped.

The very recent example that inspired this question:

  • Andrew Booker's recent solution to $a^3+b^3+c^3=33$ with $(a,b,c)\in\mathbb{Z}^3$ as $$(a,b,c)=(8866128975287528,-8778405442862239,-2736111468807040)$$ was publicized on Tim Browning's homepage. However the homepage has merely a single, austere line, and does not even indicate that this is/was a semi-famous open problem. Nor was there any indication that the cubes actually sum to $33$, apparently leaving it as an exercise for the reader.

Other examples that come to mind include:

  • In 1976 after Appel and Hakken had proved the Four Color Theorem, Appel wrote on the University of Illinois' math department blackboard "Modulo careful checking, it appears that four colors suffice." The statement "Four Colors Suffice" was used as the stamp for the University of Illinois at least around 1976.
  • In 1697 Newton famously offered an "anonymous solution" to the Royal Society to the Brachistochrone problem that took him a mere evening/sleepless night to resolve. I think the story is noteworthy also because Johanne Bernoulli is said "recognized the lion by his paw."
  • As close to a literal "mic-drop" as I can think of, after noting in his 1993 lectures that Fermat's Last Theorem was a mere corollary of the work presented, Andrew Wiles famously ended his lecture by stating "I think I'll stop here."

What are other noteworthy examples of such announcements in math that are, in some sense, memorable for being understated? Say to an outsider in the field?

Watson and Crick's famous ending of their DNA paper, "It has not escaped our notice that the specific pairing we have postulated immediately suggests a possible copying mechanism for the genetic material," has a bit of the same understated feel...

13 comments

@Timothy Chow 2019-03-17 16:23:44

The first "announcement" of the Green–Tao theorem on arbitrarily long arithmetic progressions of primes was the appearance of their preprint on the arXiv. When I saw that preprint, I posted an article to the USENET newsgroup sci.math, somewhat incredulously asking whether this was the first public announcement, and Tao replied:

It is our first public announcement, yes. Given the track record for announcements for well-known conjectures in number theory, it seems a low key approach is appropriate. :-)

@none 2019-03-18 06:20:21

Tao posted an announcement on his blog soon after the arxiv upload: see entry for April 9, 2004.

@none 2019-03-15 13:44:29

Russell and Whitehead's Principia Mathematica has a long and complicated proof that 1+1=2, given after spending 80+ pages defining arithmetic in terms of logical primitives. The proof is accompanied by the famous comment "The above proposition is occasionally useful."

@Asaf Karagila 2019-03-11 13:49:37

Kurt Gödel, only a few days before Hilbert gives his famous "We must know – We will know!" quote, had just proven that we cannot know.

Namely, any reasonably strong foundation of mathematics, if it has a finitary proof verification process, cannot decide all the true statements. Mathematics, in its essence, is incomplete.

Philosophically speaking, perhaps one of the biggest mic drop moments. Metaphorically, this virtual coinciding with Hilbert's lecture just makes the room even more silent afterwards.

@KConrad 2019-03-11 15:35:55

This took place in 1930 in Koenigsberg, which was Hilbert's hometown. Goedel's talk was a day or two before Hilbert's talk, not at the same conference, and Hilbert may not have even been at that talk. For more, see hsm.stackexchange.com/questions/29/… and maa.org/book/export/html/326610.

@Asaf Karagila 2019-03-11 16:02:34

@KConrad: The silent room is a metaphor here. Because the usual reaction to a mic drop is that the room goes silent for a moment. :-)

@David Roberts 2019-03-13 12:46:03

Apparently von Neumann, in the audience, remarked at the end of Gödel's lecture "It's all over". Unfortunately I do not have a good source for this.

@Timothy Chow 2019-03-17 16:32:30

I recall reading somewhere that nobody really grasped the full import of Gödel's lecture in real time, with the exception of von Neumann, whose nearly superhuman ability to understand new mathematical ideas quickly is well known. If true, this would be a good argument for calling this a "mic drop."

@Tobias Kildetoft 2019-03-13 12:56:26

I think the following anecdote fits well in this category. Note however, that other participants may have experienced these things differently, since they will have had a better background knowledge of the topic.

At a conference in Uppsala in September 2012, Geordie Williamson was scheduled to give a talk. I can unfortunately not recall the precise topic, as I can no loner find the program for the conference.

He starts his talk by apologizing that he is in fact going to talk about a completely different topic, since he had very recently finished some work on this with his collaborator Ben Elias.
He then goes on to describe Soergel's conjecture and some of the ideas that he and Ben have been working on, hoping to make progress on the conjecture.

The talk is quite technical, involving a lot of quite deep ideas and descriptions of how certain geometrical ideas, such as Hodge theory, can be given more algebraic analogues and how these may be put together to make progress on the conjecture.

As is typical of any technical talk, it is very hard to keep track of all the details and how they fit together along the way, so he provides a nice summary in the end:

"In conclusion, Soergel's conjecture is true".

@Zach Teitler 2019-03-13 13:14:14

Well isn’t he lucky that nobody asked him any questions to make him run out of time. To students on MathOverflow: please don’t plan talks like this (with the result at the very end).

@Matt F. 2019-03-13 13:30:31

@Tobias Kildetoft 2019-03-13 14:18:34

@ZachTeitler While I agree that it is usually not a good idea to save the main result until the end, going over time should not be an issue for an experienced speaker with an eye on the time and the ability to adapt the talk on the fly, especially when the statement of the main result takes 10 seconds.

@Vincent 2019-03-11 15:34:07

I'll just let below (famous) 1966 article from the Bulletin of the American Mathematical Society speak for itself...

Lander and Parkin, BAMS 1966

COUNTEREXAMPLE TO EULER'S CONJECTURE ON SUMS OF LIKE POWERS
BY L. J. LANDER AND T. R. PARKIN
Communicated by J. D. Swift, June 27, 1966

A direct search on the CDC 6600 yielded

$$ 27^5 + 84^5 + 110^5 + 133^5 = 144^5 $$

as the smallest instance in which four fifth powers sum to a fifth power. This is a counterexample to a conjecture by Euler [1] that at least $ n $ $ n $th powers are required to sum to an $ n $th power, $ n > 2 $.

REFERENCE
1. L. E. Dickson, History of the theory of numbers, Vol. 2, Chelsea, New York, 1952, p. 648.

@Ed Pegg Jr 2019-03-11 13:44:13

My favorite is non-mathematician Marjorie Rice challenging the proof of "No other pentagon tilings exist" with multiple new pentagon tilings. Schattschneider's article was the primary announcement of the results.

@Noah Schweber 2019-03-11 17:38:57

While an excellent result, how was this a "mic drop"?

@Carlo Beenakker 2019-03-10 20:59:43

The best known lower bound for the minimal length of superpermutations was originally posted anonymously to 4chan.

The story is told at Mystery Math Whiz and Novelist Advance Permutation Problem, and a publication with a cleaned-up version of the proof is at A lower bound on the length of the shortest superpattern, with "Anonymous 4chan Poster" as the first author. The original 4chan source is archived here.

@Robin Houston 2019-03-10 22:05:21

Also: a new superpermutation of 7 symbols, shorter than any that was known at the time (8907 symbols long), was posted as a pseudonymous comment on YouTube in February 2019.

@R.. 2019-03-10 22:29:08

"Mainly devoted to anime" is a rather kind way to put it.. ;-)

@Mark S 2019-03-10 23:32:09

@R.., as I understand it, the 4chan poster answered a question in a forum dedicated to a particular anime program. The anime in question was meant to be non-linear, and watched in any order. The question was effectively "what is the most efficient way to watch all $n$ episodes of the anime serially, in any order." So it was answered in a forum really "devoted to anime," rather than the average 4chan forum.

@Pedro A 2019-03-11 02:04:30

@MarkS More specifically, Suzumiya Haruhi no Yuuutsu, and hence the problem was also named "The Haruhi Problem".

@user2357112 2019-03-11 03:22:52

The wiki wasn't the original place the proof was posted; that was a repost. The proof was originally posted to 4chan (archive available here). (Also, the wiki isn't really an anime wiki.)

@forest 2019-03-11 10:53:49

@R.. It actually is primarily devoted to anime (well, to Japanese culture) and the majority of the boards have that theme. It's just some of the other boards are more well known outside of imageboard culture. Fundamentally, 4chan is a Japanese-themed image board for discussion of Japanese culture, with a few other boards for "random" things.

@Robin Houston 2019-03-11 18:27:13

@MarkS It’s a reply to my top-level comment on the Matt Parker video “Superpermutations: the maths problem solved by 4chan”. Here’s a direct link to the comment (which I don’t think will work on mobile): youtube.com/…

@Robin Houston 2019-03-11 18:36:06

PS. That is no longer the shortest known superpermutation on 7 symbols, as revealed in Matt’s new video: youtube.com/watch?v=_tpNuulTeSQ

@Robin Houston 2019-03-12 09:34:04

It’s far too late for me to edit my first comment, but I should say that the superpermutation posted as a YouTube comment is 5907 symbols long, not 8907. Apologies for mistyping, and for not noticing sooner.

@Count Iblis 2019-03-11 03:05:44

Onsager announced in 1948 that he and Kaufman had found a proof for the fact that the spontaneous magnetization of the Ising model on the square lattice with couplings $J_1$ and $J_2$ is given by

$M = \left(1 - \left[\sinh (2\beta J_1) \sinh (2\beta J_2)\right]^{-2}\right)^{\frac{1}{8}}$

But he kept the proof a secret as a challenge to the physics community. The proof was obtained by Yang in 1951

@Matt F. 2019-03-11 11:05:37

Why do you say “he” when there were two authors? Why do you say “kept the proof a secret” rather than “considered the argument too messy and unrigorous to publish”?

@lcv 2019-03-11 00:13:13

Not math but in physics the statistical interpretation of the wave-function was announced by Max Born in a footnote.

From his paper Zur Quantenmechanik der Stoßvorgänge,

(1) Anmerkung bei der Korrektur: Genauere Überlegung zeigt, daß die Wahrscheinlichkeit dem Quadrat der Größe $\Phi_{n_\tau m}$ proportional ist.

This can be translated as

(1) Addition in proof: More careful consideration shows that the probability is proportional to the square of the quantity $\Phi_{n_\tau m}.$

Because of its implications this is probably the most important footnote in the history of physics. Max Born was awarded the Nobel prize "for his fundamental research in quantum mechanics, especially for his statistical interpretation of the wavefunction".

@Andreas Blass 2019-03-11 01:44:15

I think "Anmerkung bei der Korrektur" is better translated as "Remark added in proof". In particular, it would be a remark by the author, not by the editor. Also, "zeigt" is present tense, "shows" not "will show".

@Matt F. 2019-03-11 02:41:20

The footnote is not the announcement of a probabilistic interpretation, but a correction that the probability is proportional to $\Phi^2$ rather than $\Phi$. Also the paper is not so much understated as preliminary, as indicated right below the title.

@lcv 2019-03-11 03:14:21

@AndreasBlass you're right. You're welcome to provide a better translation than the one I found online. If I remember correctly Born added that footnote once the paper was already in the review process

@lcv 2019-03-11 17:54:28

@MattF. I take it that your comment means that he didn't mean to show understatement with the footnote. I totally agree. This is however, how the statistical interpretation was brought to public attention. De facto so to speak.

@Jeff Strom 2019-03-11 00:22:13

From the Wikipedia article on Frank Nelson Cole:

On October 31, 1903, Cole famously made a presentation to a meeting of the American Mathematical Society where he identified the factors of the Mersenne number $2^{67}$ − 1, or M67.[5] Édouard Lucas had demonstrated in 1876 that M67 must have factors (i.e., is not prime), but he was unable to determine what those factors were. During Cole's so-called "lecture", he approached the chalkboard and in complete silence proceeded to calculate the value of M67, with the result being 147,573,952,589,676,412,927. Cole then moved to the other side of the board and wrote 193,707,721 × 761,838,257,287, and worked through the tedious calculations by hand. Upon completing the multiplication and demonstrating that the result equaled M67, Cole returned to his seat, not having uttered a word during the hour-long presentation. His audience greeted the presentation with a standing ovation.

@Mark S 2019-03-11 00:41:06

I'm interested in the historiography of this urban legend. Is the only source for the above E. T. Bell? If so, must it be considered suspect, because E. T. Bell was a much better mythmaker than a biographer? I'd like to believe it to be true - a broken clock is still right twice a day...

@Gerry Myerson 2019-03-11 00:51:25

This alleged mic-drop was specifically excluded in the original posting of the question, but that has been edited out. The comments on it remain. Of course, if it's true, it's a perfect answer to the question, but did it really happen this way?

@Zach Teitler 2019-03-11 01:18:47

Maybe things were different in 1903, but I would not give a standing ovation for an hour of silent arithmetic. Also I’m sorry but those calculations don’t seem like they would take an hour. None of it seems believable. Still a fun story though.

@Mark S 2019-03-11 02:57:25

@ZachTeitler Maybe $M_{67}$ was a really big deal in 1903? Maybe actually finding the factors was generally greeted with some expression of acclamation? Mersenne antedates Fermat by a dozen or so years, $M_{67}$ was effectively open for just as long in 1903 as FLT was. I'm pretty sure that people stood up and clapped at the end of Wiles' lecture in 1993. Of course Wiles' lecture was not an "hour of silent arithmetic," so maybe that part is a stretch.

@Zach Teitler 2019-03-11 05:32:15

$M_{67}$ would be a big deal any time and finding those factors would have certainly been worthy of acclaim. I just meant that there would be far better ways to present the factorization than grinding through the arithmetic. As an audience member I would be far, far more interested in how Cole found those factors, than in whether he remembered to carry the $3$ or whatever. An hour of that would have been tough to sit through. Although... maybe at one of those 20-minute AMS special sessions, perhaps.... :-)

@Mark S 2019-03-11 15:21:56

@ZachTeitler right, so the "applause/standing ovation" is not so suspect, but the "silent lecture of multiplication" seems bogus and inconsistent with even mathematicians of 1903...

@TomGrubb 2019-03-11 16:22:59

@MarkS I can't comment on the absolute validity of the claims but did some looking out of curiosity; here is a 1963 article which seems to be the source of the wikipedia claims. The article is "The search for perfect numbers" by Gridgeman: books.google.com/books?id=0Dta7OkNhyoC&lpg=PA87

@Peter LeFanu Lumsdaine 2019-03-13 14:22:48

@MarkS: A “silent lecture of multiplication” was surely hardly typical, but it doesn’t seem at all impossible, as an out-of-the-ordinary piece of showmanship. Of course, not everyone could get away with it, but imagine: you go to a talk by a mathematician you already know and respect fairly well, and they start by silently calculating powers of 2. Weird, you think, but you’ll probably give it a chance. By a little way in, you can see they’re calculating some big Mersenne prime (maybe they even “announced” this in writing at the start of the calculation); you’re aware… [cont’d]

@Peter LeFanu Lumsdaine 2019-03-13 14:27:50

[cont’d] …you’re aware that they’d been working on something about Mersenne primes, so you can believe that this is going somewhere. After a while, they reach the value of M(67); then they set up a big product, and start calculating it — and at this point, the penny drops, you can see where this is all going. Many of the audience have probably tuned out by now, but from the ones who have stayed on the ball, a sense of serious anticipation and excitement starts emerging in the room, bringing the tune-outs back to awareness, everyone holding their breath and waiting for the punchline.

@Peter LeFanu Lumsdaine 2019-03-13 14:30:42

[cont’d] And in the end, the punchline comes — and the standing ovation. Overall — hardly normal, and it would take guts and a bit of pre-existing reputation to pull it off, but it doesn’t by any means seem implausible. Of course, that doesn’t mean it did happen this way — it’d still be good to find a more sober and earlier source than Bell — but I think the arguments of incredulity/impossibility here are lacking in imagination. The whole point of the story is that it was an unusual piece of showmanship.

@Timothy Chow 2019-03-18 16:00:52

Given the content of Cole's paper (see projecteuclid.org/download/pdf_1/euclid.bams/1183417760 ), I would think it very unlikely that he would say nothing about his techniques. Perhaps what happened was that he started off his talk by silently writing down the factorization for dramatic effect, and then proceeded to give a "normal" talk. Later, people may have only remembered the dramatic opening.

@Kimball 2019-03-11 01:08:12

Perelman solving the Poincare "conjecture," posting it only on the arXiv, leaving math, and refusing the Clay prize could be interpreted as a kind of "mic drop."

@Samantha Y 2019-03-11 02:40:25

Let us not mince words: " 'I'm not interested in money or fame,' he is quoted to have said at the time. 'I don't want to be on display like an animal in a zoo. I'm not a hero of mathematics. I'm not even that successful; that is why I don't want to have everybody looking at me.' "

@YCor 2019-03-11 11:42:18

The announcement was posting on arxiv (with no buzzword). Refusing prizes occurred several years later hence is not part of the announcement.

@zibadawa timmy 2019-03-11 14:31:22

@SamanthaY Ironically, I think he's succumbed to the Streisand Effect by doing that. A significant fraction of my awareness of Perelman and the results he's credited with is a result of his refusals to be recognized.

@Samantha Y 2019-03-11 18:36:57

@zibadawatimmy Yes, but Perelman refused to be owned by the public, which is quite a different thing than wanting privacy/anonymity. To that end, I feel he succeeded.

@YCor 2019-03-11 19:38:02

@SamanthaY this was a comment to the answer, not to your comment.

@Samantha Y 2019-03-11 23:47:04

@ycor I apologize :( this entire thread strikes me as unproductive, but that doesn't excuse my taking offense at you, especially as we seem to be in agreement. That is (if I may say so), perpetuating mythologizing the Perelman story, or characterising it as a 'mic drop' will only drive a wedge between mathematicians and the general public, by portraying us as either prize-driven or "odd-balls", and even if somewhere between these two, as drama obsessed gossips. After all, (and this is a very general comment) don't we want everyone to be mathematician? Let them see we're human.

@Timothy Chow 2019-03-17 16:16:39

I think Perelman is a good example, but for a slightly different reason: When his first preprint came out, it wasn't even clear that he was claiming that he had proved the Poincare conjecture. See for example mathforum.org/kb/message.jspa?messageID=1692699

@David Roberts 2019-03-10 22:48:57

Applications of algebra to a problem in topology (YouTube) at Atiyah80 was a talk by Mike Hopkins. In it he announced the solution to the Kervaire invariant one problem in all but one dimension (arXiv, Annals).

@Paul Siegel 2019-03-11 14:52:20

I was in the audience as a second year graduate student, and for me it was just another lecture where I could only understand the first 10-15 minutes. A stranger sitting next to me was very excited afterward, but I remember wondering if he was a nut. So yeah, this qualifies as a mic drop in my book.

@Vincent 2019-03-13 10:51:55

Can I ask a question about the video? At the beginning there is some discussion of various $\pi_n(S^0)$, in particular Pontryagin first errornously claiming that $\pi_2(S^0) = 0$ and later realizing that in fact $\pi_2(S^0) = \mathbb{Z}/2\mathbb{Z}$. So here is my question: isn't $S^0$ just two points? What would a representative of the non-zero class of $\pi_2(S^0)$ look like? You can't get both points of $S^0$ in the range of the map, can you? What is going on here?

@David Roberts 2019-03-13 10:54:41

@Vincent from memory he's most likely discussing stable homotopy groups, so that really it's $\pi_{n+2}(S^n)$ for large enough $n$ (in this case, $n\geq 4$), but being sloppy with notation. It would be better denoted $\pi_2^s(S^0)$. See eg the pink diagonal in this table starting at $\pi_6(S^4)$.

@David Roberts 2019-03-13 11:03:56

@Vincent actually, it's saying the same thing, but I think Hopkins would phrase it as looking at the homotopy groups of the sphere spectrum $\mathbb{S}$, which is cooked up out of $S^0$. Then $\pi_2(\mathbb{S})$ is indeed as he describes.

@Vincent 2019-03-13 11:37:59

@DavidRoberts Thanks!

@Gerhard Paseman 2019-03-10 21:06:13

I consider this manner as a mark of a professional mathematician: let others convey the excitement of a discovery. A good recent example was the submission of a paper on bounded gaps between primes. Much of the public excitement was generated by people other than the author, Yitang Zhang.

Gerhard "Can Be Excited In Private" Paseman, 2019.03.10.

@Mark S 2019-03-10 23:17:06

I especially like his understated comment that "I believe one could make it sharper" when asked if he thought $k<70,000,000$ could be reduced.

@Will Sawin 2019-03-11 01:34:09

Well, a distinction can be drawn between the most professional approach, which I guess is to submit the work to the Annals or another top journal, accept invitations to speak about it, etc. followed by Yitang Zhang and the more dramatic (and fun) approach where you post it only to your personal website, refuse to tell people what your talk announcing the result is about in advance, leave math immediately afterwards, etc. It seems that the "mic drop" refers to examples that go above and beyond what you'd do for a usual strong result.

@Einfacher Schreiberling 2019-03-12 19:16:34

And then there is of course Grothendieck who let Borel & Serre publish his proof of the GRR theorem...

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