Mathematical "urban legends"

  • When I was a young and impressionable graduate student at Princeton, we scared each other with the story of a Final Public Oral, where Jack Milnor was dragged in against his will to sit on a committee, and noted that the class of topological spaces discussed by the speaker consisted of finite spaces. I had assumed this was an "urban legend", but then at a cocktail party, I mentioned this to a faculty member, who turned crimson and said that this was one of his students, who never talked to him, and then had to write another thesis (in numerical analysis, which was not very highly regarded at Princeton at the time). But now, I have talked to a couple of topologists who should have been there at the time of the event, and they told me that this was an urban legend at their time as well, so maybe the faculty member was pulling my leg.

    So, the questions are: (a) any direct evidence for or against this particular disaster? (b) what stories kept you awake at night as a graduate student, and is there any evidence for or against their truth?

    EDIT (this is unrelated, but I don't want to answer my own question too many times): At Princeton, there was supposedly an FPO in Physics, on some sort of statistical mechanics, and the constant $k$ appeared many times. The student was asked:

    Examiner: What is $k?$

    Student: Boltzmann's constant.

    Examiner: Yes, but what is the value?

    Student: Gee, I don't know...

    Examiner: OK, order of magnitude?

    Student: Umm, don't know, I just know $k\dots$

    The student was failed, since he was obviously not a physicist.

    Since every finite CW complex is weakly homotopically equivalent to a finite topological space, that does not sound *so* bad... :)

    I heard a version of this too, from sources I thought were reputable (but I forget who, probably more than one.)

    Perhaps not an urban legend per se, but when I was learning algebra, my professor, in an attempt to impress upon us the necessity of checking that certain maps are well-defined, told us the story of a classmate of his who got several years into his Ph.D. thesis before realizing that the maps he was investigating weren't well defined. Horrified, we asked him if this was true. "No" he said, "but that's one lie you'll never forget!"

    I had heard an urban legend about Milnor sitting in on a dissertation defense, but in this story the speaker airily discussed a certain natural transformation, that Milnor expressed doubted it was natural, and that Milnor was right. According to the story, the thesis was completely invalidated because the student hadn't checked naturality, and had to start all over (or something like that).

    Maybe I hung around the wrong crowd, but I've never heard Igor's story. True or not, it'd be a shame if that story somehow got lost before my generation.

    I also had heard an urban legend about Milnor sitting in on a dissertation defense (long ago when I was a graduate student a Brandeis). In the version I had heard the class of new topological space that were studied was observed by Milnor to consist only of the empty set.

    Well, and of course there is the old chestnut of the (supposedly Harvard) student who wrote a thesis about the class of functions satisfying a Lipschitz condition of order \(1 + \epsilon \) :-) But getting back to the original question, why not just ask Milnor?

    @Dick: is that the one Qiaochu recorded below?

    Willie Wong asked:"@Dick: is that the one Qiaochu recorded below?", and it no doubt is. Perhaps I will add more detail. I first heard it when I was grad student myself at Harvard (so if you know when I got my degree you will realize how old a legend this is !). Moreover the story as I heard it was that the thesis advisor was Garett Birkhoff.

    Mathematical urban legends have been collected by Steven Krantz in the book, Mathematical Apochrypha (and I think there's a second volume). A few refer to the thesis defense.

    For some decades after the alleged event, a story persistently circulated that a certain professor at a certain institution had fallen out of a window while lecturing. I would tell you some specific reasons why this might have somewhat plausible, but then too many people might guess who it was.

    I should proofread before hitting the "add comment" button.

    Though this question and its answers are very entertaining, I think it is a little unfair to close other questions as "offtopic" which are even closer to mathematical research as this one ...

    @ Mariano: I gather that what is meant is that the space is just a finite set of points with the discrete or indiscrete topology.

    Another good source of such legends is *Absolute Zero Gravity*, by Betsy Devine and Joel E. Cohen.

    I have to agree with Martin. This is a very entertaining thread but it seems quite outside the mandate of MO.

    Martin, you should definitely raise your point on meta. Objectively, you're completely right. But I'm enjoying this as long as it manages to remain open.

    So I'm a hypocrite :) Blame Ryan and Martin for convincing me.

    The details in the question appear garbled to me. However, I was present when Katz asked Milnor about the story that he had once asked a question at a thesis defense which had sunk the thesis. Milnor looked embarrassed, and said that it had happened. He added that he hadn't been trying to trip up the student --- he had asked the question simply out of curiosity.

    In case anyone comes upon this question later, the decision to close it after 70 (!) answers was made here :

  • This happened just last year, but it certainly deserves to be included in the annals of mathematical legends:

    A graduate student (let's call him Saeed) is in the airport standing in a security line. He is coming back from a conference, where he presented some exciting results of his Ph.D. thesis in Algebraic Geometry. One of the people whom he met at his presentation (let's call him Vikram) is also in the line, and they start talking excitedly about the results, and in particular the clever solution to problem X via blowing up eight points on a plane.

    They don't notice other travelers slowly backing away from them.

    Less than a minute later, the TSA officers descend on the two mathematicians, and take them away. They are thoroughly and intimately searched, and separated for interrogation. For an hour, the interrogation gets nowhere: the mathematicians simply don't know what the interrogators are talking about. What bombs? What plot? What terrorism?

    The student finally realizes the problem, pulls out a pre-print of his paper, and proceeds to explain to the interrogators exactly what "blowing up points on a plane" means in Algebraic Geometry.

    Anna--Pedro Teixeira and I had a couple of papers where we introduced operations of the following sort: one starts with a function f on [0,1] and replaces it by the function x-->f((x+7)/25), x in [0,1]. We originally called such operations blowing-ups but thought it more prudent to call them magnifications in the published papers.

    Was it Vikram Mehta ?

    Efim Zelmanov told a similar story: he was stopped by the KGB on his way to a conference and questioned at length about his books on "free groups" and "radicals".

    Thank god the diligent TSA caught their nefarious *scheme*.

    I hope I'm not being a killjoy, but I've heard versions of this story so many times over the years that I'd be quite interested to find out if this one is alleged to be true and by whom. Do you know?

    Anna's story is about my officemate in grad school (who is a friend of Anna's brother, one of our classmates). He was flying from London to the US right after the waterbombing incident. He studies DP1's. They ended up strip searching him. Eventually he got through.

    I remember being joking about the potential that something like this could happen with some mathematicians probably in 2004 or so. We were at an airport, but were careful not to discuss even joking about it too near security.

    Georgia Benkart once (probably in 1979 or 1980) related a vaguely similar story about some famous mathematician and the Killing form. In my imagination, the event she described took place in the McCarthy era, but honestly I don’t remember the details. Maybe it was Zassenhaus. Then again, maybe it was someone else, though I’m pretty sure it was Zassenhaus she told about in the artichoke story.

    Well, you know what they say about algebraic geometers: they blow up families, and then the come back afterwards to make sure that they're flat.

    +1: "annals of mathematical legends"

  • Since this has become a free-for-all, allow me to share an anecdote that I wouldn't quite believe if I hadn't seen it myself.

    I attended graduate school in Connecticut, where seminars proceeded with New England gentility, very few questions coming from the audience even at the end. But my advisor Fred Linton would take me down to New York each week to attend Eilenberg's category theory seminars at Columbia. These affairs would go on for hours with many interruptions, particularly from Sammy who would object to anything said in less than what he regarded as the optimal way. Now Fred had a tendency to doze off during talks. One particular week a well-known category theorist (but I'll omit his name) was presenting some of his new results, and Sammy was giving him a very hard time. He kept saying "draw the right diagram, draw the right diagram." Sammy didn't know what diagram he wanted and he rejected half a dozen attempts by the speaker, and then at least an equal number from the audience. Finally, when it all seemed a total impasse, Sammy, after a weighty pause said "Someone, wake up Fred." So someone tapped Fred on the shoulder, he blinked his eyes and Sammy said, in more measured tones than before, "Fred, draw the right diagram." Fred looked up at the board, walked up, drew the right diagram, returned to his chair, and promptly went back to sleep. And so the talk continued.

    Thank you all for your indulgence - I've always wanted to see that story preserved for posterity and now I have.

    David, do you remember what the topic was?

  • Here's another great one: a certain well known mathematican, we'll call him Professor P.T. (these are not his initials...), upon his arrival at Harvard University, was scheduled to teach Math 1a (the first semester of freshman calculus.) He asked his fellow faculty members what he was supposed to teach in this course, and they told him: limits, continuity, differentiability, and a little bit of indefinite integration.

    The next day he came back and asked, "What am I supposed to cover in the second lecture?"

    A little more context from (the version I heard) of this one: the mathematician’s previous experience had been in the USSR — I think in Moscow. (This hopefully adds something useful without getting too specific…)

    This is really not specific, because in the USSR one was taught these concepts in school.

    well, and in large parts of the rest of europe, too.

  • Although David Hilbert was one of the first to deal seriously with infinite-dimensional complete inner product spaces, the practice of calling them after him was begun by others, supposedly without his knowledge. The story goes that one day a visitor came to Göttingen and gave a seminar about some theorem on "Hilbert spaces". At the end of the lecture, Hilbert raised his hand and asked, "What is a Hilbert space?"

    This appears in Krantz's _Mathematical Apocrypha Redux_ (the second edition of the book mentioned by Gerry Myerson in the comments). According to this version the speaker was von Neumann, the lecture occurred in 1929, and Hilbert is quoted as saying "Dr. von Neumann, ich möchte gern wissen, was ist dann eigentlich ein Hilbertscher Raum?" (The translation given: "Dr. von Neumann, I would very much like to know, what after all is a Hilbert space?")

    (By the way, I thought this book would be mostly funny stories, but it is full of sobering tidbits about how mathematicians were affected by anti-Semitism, the Nazis, the Great Depression, and McCarthyism. Interesting stuff.)

    There is another version of this story in which Weyl is the one being asked by Hilbert.

    In Emilio Segre's version of this (very popular) story, as recounted in Segre's Autobiography, the speaker was Enrico Fermi and the year was the late 1930's. "Fermi attended [Robert] Oppenheimer's seminars; coming out of one of them once, he said: 'Emilio, I must be getting senile. I went to a learned theoretical seminar and could not understand anything except the last words, which were "And this is Fermi's theory of beta decay."'"

    To this day, in the building in Göttingen in which Hilbert worked, there is an actual "Hilbert space" (the German mathematical word for space is "Raum" which also happens to be the German word for "room").

    I heard also from somewhere that Hilbert was very disappointed at the answer and said "is that all?", or words to that effect, since he thought the definition so trivial.

    This is similar to Lars' comment above. On the bottom floor of the math building at Lund University (Sweden) you'll find a students' cafe named "Hilbertrummet" (which means both "the Hilbert room" and "the Hilbert space").

    Congrats on the first golden badge! :-)

  • A legend that I heard from my father, who heard it from ... ... ...: Levi-Civita was teaching a course in a room on (what Americans call) the second floor of a building. One day, as a prank, his students "borrowed" a donkey from one of the fruit vendors on the street in front of the building. Somehow, they brought this donkey up the stairs into the lecture hall and had it standing there as Levi-Civita entered to begin his lecture. Levi-Civita set his notes down on the lectern, looked up at the class, commented "I see we have one more today," and proceeded with his lecture.

  • Here is a story I heard many years ago, and have no confirmation of:

    Apparently, there was Asst Professor X at a provincial department Y, and he was up for tenure. Professor X's advisor was a famous Japanese mathematician Z at an Ivy League school. Naturally, he was asked for a letter, which he duly sent. The letter said:

    X has a very nice body of work, he proved the following interesting theorems, extended such and such results, used such and such techniques... and so on for two pages. The last sentence was: all in all, X is a very good second-rate mathematician.

    The committee was mortified, but figured that the rest of the letter was so good, they should call Z, since maybe since English was not his native language... So, call they did, and the phone conversation went about the same as the letter: did this, improved that, ..., all in all a very good second-rate mathematician.

    The committee then said: look, we don't understand why you say he is second-rate!!!

    to which Z replied: well, I really can't understand why that would be a problem -- after all, you are a third rate department.

    Who cares if it's true or false? That's what makes urban legends so fun! It's shocking, and yet we feel sure that somewhere some similar incident must have happened...

    Andre Weil's law of university hiring (according to Wikipedia, undocumented): "First rate people hire other first rate people. Second rate people hire third rate people." This always left me wondering, who hires the second rate people? Maybe Igor's story answers my question.

    "And third rate people hire fifth rate people."

    @Gerry The second rate people always hire.

    Maybe the fourth rate mathematicians hire the second rate ones, and that's how they keep their jobs in the hiring department?

    Ah, I understand. Nth-rate people are hired by mth-rate people, where m is the closest integer to n/phi.

    @Philip, that's a really perceptive (unfortunately...) article, and certainly something I had seen happen...

  • The following story is a bit strange to be true, but we all believed it as students, and I think I still do believe that a somewhat weaker version of events must have indeed occurred.
    Michael Maschler (most famous in Israel as author of the standard math textbooks for middle-schools and high-schools) was in the middle of teaching an undergraduate course- I think it was Linear Algebra- when one afternoon he walks into the lecture hall and announces the discovery of a new class of incredible Riemannian symmetric spaces with incredible properties, missed by Elie Cartan. The undergrads have no idea what he is on about; but the faculty all get very excited, and start sitting in on his Linear Algebra course. Ignoring the syllabus, Prof. Maschler begins to give lecture upon lecture about the new incredible symmetric spaces which he discovered. The excitement builds. Will he win a prize? Will he win the Fields Medal?...
    And then, 3 lectures in, a student (some say it was Avinoam Mann, about whom many stories are told) gets up and asks, "Excuse me, sir. How can you distinguish your space from a sphere?"
    Maschler turns to answer the "stupid question", but he freezes in mid-motion... Gradually, his face turns white. The lecture hall is so silent you can hear a pin drop. Finally, after what seems like an eternity, Prof. Maschler unfreezes. "By golly, a sphere it is," he murmurs in an undertone. And he picked the Linear Algebra textbook up from his desk, and resumed teaching where he had left off. The subject was never broached again.
    And so, some Hebrew University students of my generation call spheres "Maschler spaces".

  • I've heard that in the earliest days of communist Hungary, Pal Turan was stopped on the street by a patrol. These patrols were charged with collecting a quota of people to be shipped off to Siberia (Stalin was still in charge, and arbitrary punishment is a big part of inducing the Stockholm Syndrome). While being searched and interrogated for his "crimes", the policeman was surprised and impressed (and perhaps a bit intimidated himself) to find a reprint of a paper of Turan's published pre-war in a Soviet journal. Turan was allowed to go free. That day, he wrote a letter to Erdos beginning, "I have discovered a most wonderful new application of number theory..."

    Turan was in forced labor camps during much of the second war. This sounds like an incident I read about that took place in one of those camps. I wonder if we haven't had a conflation of two dictatorships.

    Perhaps we should consider "urban legends" as parasites (or symbiotes) on the mathematical ecosystem. They certainly mutate and (as Gerry indicates) perhaps even have sex.

    I've tracked down the incident I read about. It's in Szego's preface to Hungarian Problem Book I, which was volume 11 in the New Mathematical Library. It's too long to write out here. Szego doesn't give a source, doesn't claim all the details are accurate, and doesn't name the mathematician. In short, X was in a forced labor camp circa 1940, the supervisor recognized his name from Hungarian problem-solving competitions, and gave him more lenient treatment. The story is also quoted in Rosemary Schmalz, Out Of The Mouths Of Mathematicians, MAA 1993

    Now I have found an unimpeachable source for Szego's story; Turan himself wrote it up in "A note of welcome," J Graph Theory 1 (1977) 7-9.

    Here is the text from "A note of welcome". In September 1940 I was called in for the first time to labor-camp service. We were taken to Transylvania to work at railway building. Our main work was carrying railway ties. It was not very difficult work but a spectator could of course easily recognize that most of us-I was no exception-did it rather awkwardly. One of my more expert comrades said this at one occasion quite explicitly, even mentioning my name. An officer was standing nearby, watching our work. When hearing my name, he asked the comrade whether or not I was a mathematician. (Contd)

    It turned out that the officer-Joseph Winkler by name- was an engineer. In his youth he had placed at a mathematical competition; in civilian life he was a proofreader at the printing shop where the periodical of the Third Class of the Academy (Mathematical and Natural Sciences) was printed and had seen some of my manuscripts. He could do no more than assign me to a wood-yard where big logs, necessary to railroad building, were stored, classified according to their diameter; my task was merely to show incoming groups the place where they could find those logs with the prescribed width.

    While there is no mention of the purported letter to Erdos, he does say that in the following few days, while assigned to the wood-yard, he proved Turán's theorem.

  • A wholly different set of "named urban legends" (in order of time):

    Allegedly, Jacobi came to show Gauss his cool results on elliptic functions. Gauss' response was to open a drawer, point at a sheaf of papers, and say: that's great you are doing this! I have actually discovered these results a while ago, but did not think they were good enough to publish... To which Jacobi responded: Funny, you have published a lot worse results.

    When the logician Carnap was immigrating to the US, he had the usual consular interview, where one of the questions was (and still is, I think): "Would you favor the overthrow of the US government by violence, or force of arms?". He thought for a while, and responded: "I would have to say force of arms..."

    Finally, on the graduate experience front, it was rumored at Princeton that Bill Thurston's qualifying exams at Berkeley were held as his wife was in labor with his first child -- the department refused to change the date for such a minor reason! I have just asked him about this, and it's true...

    EDIT A certain (now well-known) mathematician was a postdoc at IHES in the late 1980s. Call him R. R comes to lunch, and finds himself across the table from Misha Gromov. Gromov, very charmingly, asks him what he was working on. R tells him, Gromov has some comments, they have a good conversation, lunch is over. The next day R finds himself across from Gromov again. Misha's first question is: so, what are you working on now?

    I'd never heard the Carnap story. It's reminiscent of Godel's US citizenship test.

    Gauss was some 20 years older than Jacobi and was, well, Gauss. It would have take *great* nerve...

    The Thurston story is recounted in his interview in More Mathematical People.

    Around the Jacobi/Gauß anecdote: the correspondence between Jacobi and Legendre is fascinating and worth reading. It's quite moving to see this old mathematician welcoming with enthusiasm the work of two younger ones (Jacobi and Abel) following the study of his favorite mathematical field. It's almost melodramatic, with episodes of anger against Gauß (Legendre has had his share of paternity disputes with him, for example regarding the quadratic reciprocity law or the law of least squares) or mourning after Abel died...

    ... And as usual with mathematical texts of this time, the elegance of the language is baffling. Particularly so if one remembers that Jacobi doesn't write in his mother tongue.

    Reiman István's book *A geometria és határterületei* tells a similar story about Gauss. In that one, the other party is Bolyai János, the discovery concerned is hyperbolic geometry, correspondence is through mail only (not in person), and Gauss gives a different reason why he didn't yet publish about the topic.

  • Another urban legend, which I've heard told about various mathematicians, and which Misha Polyak self-effacingly tells about himself (and therefore might even be true), is the following:

    As a young postdoc, Misha was giving a talk at a prestigious US university about his new diagrammatic formula for a certain finite type invariant, which had 158 terms. A famous (but unnamed) mathematician was sitting, sleeping, in the front row. "Oh dear, he doesn't like my talk," thought Misha.
    But then, just as Misha's talk was coming to a close, the famous professor wakes with a start. Like a man possessed, the famous professor leaps up out of his chair, and cries, "By golly! That looks exactly like the Grothendieck-Riemann-Roch Theorem!!!"
    Misha didn't know what to say. Perhaps, in his sleep, this great professor had simplified Misha's 158 term diagrammatic formula for a topological invariant, and had discovered a deep mathematical connection with algebraic geometry? It was, after all, not impossible. Misha paced in front of the board silently, not knowing quite how to respond. Should he feign understanding, or admit his own ignorance? Finally, because the tension had become too great to bear, Misha asked in an undertone, "How so, sir?"
    "Well," explained the famous professor grandly. "There's a left hand side to your formula on the left."
    "Yes," agreed Misha meekly.
    "And a right hand side to your formula on the right."
    "Indeed," agreed Misha.
    "And you claim that they are equal!" concluded the great professor. "Just like the Grothendieck-Riemann-Roch Theorem!"

    This sounds like a whole generation of French-educated algebraists has never seen an equation in their whole life :D

    I guess nowadays one has the analogous "But that is a special case of the (coarse) Baum-Connes conjecture for (quantum) group(oid)s" ...

    If my memory serves me correctly, Victor Ginzburg once said something like: "that's a Langlands correspondence, because it's a correspondence between two sets" in a seminar.

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