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.

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    Since every finite CW complex is weakly homotopically equivalent to a finite topological space, that does not sound so bad... :) – Mariano Suárez-Álvarez Jan 24 '11 at 20:54
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    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!" – Nick Salter Jan 24 '11 at 21:04
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    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. – Gerry Myerson Jan 24 '11 at 23:18
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    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 ... – Martin Brandenburg Jan 25 '11 at 8:54
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    I have to agree with Martin. This is a very entertaining thread but it seems quite outside the mandate of MO. – Ryan Budney Jan 26 '11 at 16:30

69 Answers 69

As an undergraduate at Yale in the '70s I heard a variation on the basic legend, which I'll spell out a little since it has a slightly different moral from any others above.

Student goes to advisor saying I'd like to do a thesis generalizing the results in article X. Advisor (and I think I heard it with Milnor as the advisor) says, "I don't recommend that because I don't think that's a very good article." Student persists, writes thesis, states theorem at the defense and at that point the advisor rises to say "consider the following counterexample..."

I also heard a variation on "functions which turn out to be constant" legend. But the version I heard has the thesis getting accepted, the vacuity of it contents going unnoticed for several years until an undergraduate supplies a one-line proof.

John Myhill told me about junior faculty at the University of Chicago about to grade qualifying exams in their legendarily ruthless way. André Weil pops his head in the door and says "Pass them all, they're no worse than you are."

  • That Weil anecdote is new to me, and highly amusing (not to mention salutary) – Yemon Choi Jan 25 '11 at 1:39
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    @Yemon Graduate student that I was, the Myhill's story painted Weil for me as a hero. Myhill, a very seasoned faculty member by then thought Weil came off as a monster. A question of perspective I suppose. Myhill did say that that year all the students passed. – David Feldman Jan 25 '11 at 2:28
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    "Experience confirms that severity towards others and self-indulgence are one and the same vice" - La Bruyere (trans. Choi, probably badly) – Yemon Choi Jan 25 '11 at 5:29
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    @Yemon Choi : translation is ok : for completeness the original is : "L'expérience confirme que la mollesse ou l'indulgence pour soi, Et la dureté pour les autres n'est qu'un seul et même vice." Citation de Jean de La Bruyère ; Les Caractères, Du cœur - 1688. – Jérôme JEAN-CHARLES Feb 3 '11 at 0:22
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    @Feldman: The story would certainly paint Weil for me as a hero if he had said, "Pass them all, they're no worse than I am." – Timothy Chow Apr 15 '11 at 1:37

As A.N.Whitehead, of PM fame, was still lecturing on mathematics at Cambrdge, he later became a philosopher in America, he arrived somewhat early in the lecture room one day. To fill in the time he started working on a problem from his research on the blackboard. As the students arrived he was still absorbed in his work so they sat down and waited for him to start the lecture. At the end of the allotted time he was still working on his problem and so the students got up and left. Somewhat later he finished his work, packed up his things and went home. Arriving home he said to his wife, "You know a rather strange thing happened at the university today, nobody came to my lecture."

I have no details to provide, but it is said that Ofer Gabber has derailed more than one talk at IHES after the speaker presents a definition by asking, "But what about the empty set?"

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    He has actually derailed many Sem. Bourbaki by asking a stream of questions in English (the official language was, and presumably still is, French), until the speaker would start speaking in English. – Igor Rivin Jan 25 '11 at 3:57
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    I don't think the Séminaire Bourbaki has an official language. Like any lecture in France (or, I guess, in a French-speaking country), it's just convenient to do it in French unless the speaker isn't francophone. – Maxime Bourrigan Jan 25 '11 at 22:26
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    I recall a seminaire bourbaki in which Gabber persistently questioned Deligne in English, who answered just as persistently in French. – roy smith Apr 13 '11 at 17:35

One time Henri Berestycki was riding the Paris subway on the way to work and doing some calculations. All of a sudden, an elderly lady sitting across from him said: "Why don't you multiply by alpha and integrate by parts?" This did not solve his problem, but it was a reasonable thing to do.

It turned out the old lady had once worked with Lebesgue. She remembered J.L. Lions as a "clever lad."

I heard this story from my advisor Klaus Kirchgaessner who had heard it from Berestycki himself.

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    In the same spirit, I was sitting in a train, doing some mathematics. The person in front of me interupted me: your formula for the derivative of a product is false. Of course, he couldn't know that such a strange animal as a convolution product existed... – Denis Serre Jun 7 '11 at 5:50

I have heard (from two sources) that at the University of Chicago a senior faculty member was temporarily banned from teaching undergraduate courses. The reason is that during a first semester undergraduate linear algebra course he did everything over the Quaternions.

This one isn't so much academically scary, but my advisor told me that it was always interesting riding to conferences with the above professor because he would refuse to defrost the windshield so that he could draw diagrams on it and do math while he was driving.

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    Actually, that is not that bad of an idea! I have seen the face of my students when I tell them «you should go through your linear algebra notes to see how much of it carries over to the case of skew-fields» right before proceeding to pick a basis for an $\mathbb H$-module, say... – Mariano Suárez-Álvarez Jan 25 '11 at 5:57
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    As an undergraduate I heard a secondhand story about a knot theorist teaching an introductory calculus class. The first question on the final was basic calculus; the rest involved knot theory. – Steve Huntsman Jan 25 '11 at 7:01
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    In Germany many professors would be happy to get banned from teaching undergraduate courses (and behave accordingly). There used to be payment by number of students some years ago, but now it has been levelled, and teaching undergraduate courses has become nothing more than a chore people want to get rid of. – darij grinberg Jan 25 '11 at 8:24
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    This story sounds strange to me because (at least the past few years, when I was there) the University of Chicago, SFAIK, doesn't have a straight-up linear algebra class for math majors. The easier stuff you're basically expected to just up, the harder stuff gets stuffed into the general "algebra" sequence. – Harry Altman Jan 25 '11 at 10:09
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    Concerning quaternions, there is also a story, which has happend in Cambridge as my brother told me: A professor asks in a lecture: "Is here somebody who does not know everything about quaternions?" A single student raises slowly her hand. "What?? Then learn it until tomorrow!" - it goes without saying that there were students in the class who did not raise their hand and did not even know what quaternions are... – Lennart Meier Feb 4 '11 at 20:48

I've heard the following story (I don't know if it is true). A math professor gave his PhD student this journal paper, and asked him what consequences he could derive from it. The student started proving more and more interesting results based on this paper, until finally he proved a result that the professor knew was false. This led them to look more closely at the original journal paper, and upon close inspection, they discovered that it was wrong, rendering all the research the student had done so far worthless.

  • What is the worst known example of such a chain? Such as a 10 year old theory which based on completely false research? – Martin Brandenburg Jan 26 '11 at 22:26
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    I've actually seen something like this happen. – H A Helfgott Jan 29 '11 at 22:17
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    That is a decisive argument for the motto "Give examples!". Examples are mental crutches, guides and railings too (as in this case) . - – Jérôme JEAN-CHARLES Feb 3 '11 at 0:38

This story, according to the person I heard it from, happened some time in the 80s. It was about 10 years after Deligne's Hodge theory came out, but before Saito. It was not very clear how to define the mixed Hodge structure in non-constant cohomology. However, many people were convinced that such a thing existed (as turned out to be the case) and a number of competing proposals circulated. One such proposal was presented in a seminar talk where it was claimed that something was the "right Hodge filtration". At this moment Ofer Gabber (someone known, among other things, for giving hard time to speakers) intervened saying "What do you mean, the right Hodge filtration? What's the left Hodge filtration?"

  • Of course, if the talk were given in French, such a problem wouldn't have happened. (See Igor Rivin's comment below: mathoverflow.net/questions/53122/mathematical-urban-legends/… ) – Willie Wong Jan 25 '11 at 20:53
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    Ofer is a living legend, and stories about him as a graduate student at Harvard and permanent member of IHES abound. He is brilliant but demands a level of logical rigor and precision that even other mathematicians have difficulty providing. My understanding is that his name should be on many important papers, but he demanded that his name be removed because he was not comfortable with every detail stated in each paper. – Deane Yang Jan 25 '11 at 21:13
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    I'd love to hear more of his stories! – Martin Brandenburg Jan 26 '11 at 22:33
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    My undergraduate career overlapped with Gabber's graduate student career. (He was a few years younger.) Once I had the satisfaction of offering a neat proof of some statement that came up in a differential geometry course we were both attending. My pleasure was not really dimmed by Ofer's comment that my proof did not work in characteristic $p$. – Tom Goodwillie Jan 29 '11 at 3:56
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    @Jean-Charles: much more. – mmm Feb 4 '11 at 17:58

When Peter Lax went to receive the national medal of science, he was asked by the other recipients about his merits. His answer was (apocryph) I integrated by parts.

I heard the following story told about R. L. Moore.

It seems he was teaching a class in which several of the students were obnoxious and unruly. So one day he walked into the lecture hall, opened his briefcase, took out a pistol, set it on the table in front of him, and then began to lecture as usual. He had no further trouble with the rowdy students.

I have no particular reason to believe this is true, but it makes a good story. I think I have seen other references to firearms in the math department at the University of Texas, though.

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    But I thought Moore didn't lecture! – JSE Jan 25 '11 at 6:13
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    As in all the best urban legends, it does not really matter if it's true or not: it does sound like something Moore would do. – Thierry Zell Jan 25 '11 at 16:23
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    I think Moore gave lectures in some of his classes at some points of his career. – Pete L. Clark Jan 26 '11 at 8:35
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    A friend of the family claims that in first grade, his teacher had a glass eye. The students didn't know until he had to leave the room to go to the bathroom. As he got up to go, he took out his eye and placed it on the table saying "Be good while I am gone - I'm watching you". These are 6 year olds... – Steven Gubkin Jan 29 '11 at 1:47
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    What Pete said is correct. Mary Ellen Rudin recounts in her interview for More Mathematical People that Moore gave lectures for calculus classes, and Halmos in his automathography recalls being permitted to sit in on one of Moore's calculus lectures. I think nevertheless that he would send students to the board in such classes. – Todd Trimble May 23 '11 at 20:53

From the article "A credo of sorts" by Vaughan Jones, in the book "Truth in Mathematics":

Once, at a seminar, one of the world's best low-dimensional topologists was presenting a major result. At a certain point another distinguished topologist in the audience intervened to say he did not understand how the speaker did a certain thing. The speaker gave an anguished look and gazed at the ceiling for at least a minute. The member of the audience then affirmed "Oh yes, I hadn't thought of that!" Visibly relieved, the speaker went on with his talk, glad to have communicated this point to the audience.

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    We had a lecturer in Bielefeld whose proofs were sometimes a bit terse. From time to time he had spend a minute or a few after writing down the proof at the blackboard before he remembered (silently) the point and drew the square. – Lennart Meier Nov 19 '12 at 15:59

There is this story set at Harvard. During the Vietnam War there was a student strike. One math professor goes to his graduate course and finds the room empty. But he delivers his lecture anyway as usual. When he gets back to his office and tells someone about it, they ask him why he did that. He replies, "So I'll know where to start next time."

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    Isn't this one already in the list? – Mariano Suárez-Álvarez May 24 '11 at 1:54
  • an unfortunate feature of "big list" questions that revive after many months – Gerald Edgar May 24 '11 at 12:54

I heard this story a couple of years back (not sure though if it is true): A young Japanese mathematician was giving a talk based on his results at Courant Institute. His work was built on the work of S.R.S Varadhan. But apparently during the talk Varadhan had his eyes closed and the speaker mistook it for him sleeping. He made a joke by saying somthing like "hopefully not everybody is sleeping". A few minutes later Varadhan open his eyes and said "consider this counterexample". But Varadhan liked the speaker's idea and invited him to spent some time at Courant institute. The correct result is now known as 'Speaker'-Varadhan theorem.

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    I suppose you mean S.R.S.Varadhan? – fherzig Jan 25 '11 at 19:38
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    Haha. Last August at the ICM, when he was chairing a session, even I thought he was sleeping. :) – Koundinya Vajjha Mar 8 '11 at 16:29

A Japanese professor writes a letter to his American colleague, asking to send a preprint. The letter (very long and polite) is finished with the sentence:

"Please forgive me my shameless desire."

Here is a story I heard when I was student.

Professor S.'s student had finished his dissertation to everybody's satisfaction. All that was pending was his advisor's signature. S. agreed to sign on one (half joking?) condition: The student had to defeat S. in a jalapeño-eating contest.

For some reason the student agreed. (Hopefully this is not just a plot device. If the story is true, I would like one day to ask the student what he was thinking.)

They went to S.'s favorite Thai restaurant. He explained to the staff the contest. They set up a table for them, and brought them jalapeños, they would eat them, new (hotter) ones would be brought, etc. The whole staff was watching and having a great time.

The poor student, of course, was suffering, really worried that perhaps S. was serious, and he would never get his degree, since it soon became clear S. was going to defeat the student without difficulties. S. would grab the jalapeños and eat them while explaining where they were from and what the ideal way to prepare them was.

At some point, a drop of sweat from S.'s brow was threatening to fall into his eye, and without realizing what he was doing, S. passed his finger through his eye to remove the sweat.

Apparently the pain was agonizing, and the student got his dissertation signed.

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    @Andres: I like this as a story, but not as an answer to this question. Most of these stories are interesting because they tell us something that we recognize (or enunciate things we fear) about the math profession. But where's the math in your story? Are mathematicians notorious jalapeno poppers? [Sorry to be a stick in the mud.] – Pete L. Clark Jan 26 '11 at 8:33
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    The fear isn't mathematical but academic: to get one's thesis approved there are dozens of (to us inconsequential) hurdles, including margins, fonts, citation styles, microfiche fees and, for one unlucky acolyte, jalapenos. – Kevin O'Bryant Jan 28 '11 at 14:25
  • @Kevin: I have my neuroses (and so do they!), but I confess that fear of jalapenos does not hit very close to home. – Pete L. Clark Apr 17 '11 at 8:42

There's a bar in Bonn, which has the name 'Blow up' and closes only very late at night. At some occasion, an algebraic geometer A was in this bar well beyond midnight and was getting quite drunk. After some time, he decided it would be a very good idea to explain to some person B in the bar he only met this night what a blow up is in mathematics. And so he starts to explain until B interrupts him: "Hey, I know all this stuff. I've done my diploma thesis in Estonia in complex geometry."

Since I know A (although I heard the story fromy someone else), I suppose this has happened essentially this way.

In the early eighties, fleeing from Romania, C. Foias got a professorship position in Orsay. He gave a graduate course on 'Contractions et dilatations' (Contractions and dilations). Someone handwrote on the annoucement 'Is this a course on Obstetrics ?'.

The wikipedia entry for Borel summation narrates the following recollection by Mark Kac, about an encounter between Emile Borel and Mittag-Leffler. This is one of my favourites.

"Borel, then an unknown young man, discovered that his summation method gave the 'right' answer for many classical divergent series. He decided to make a pilgrimage to Stockholm to see Mittag-Leffler, who was the recognized lord of complex analysis. Mittag-Leffler listened politely to what Borel had to say and then, placing his hand upon the complete works by Weierstrass, his teacher, he said in Latin, 'The Master forbids it'."

Heard from Carsten Thomassen:

He was giving a lecture on matchings in graph theory, and presented a game where two players would alternately pick some edge in a graph, and at the end one person would win (i do not remember the exact rules of the game). Then Carsten asked the students, which player would win this game. A student raised his hand and replied "You will".

I've heard the following story (don't know if it was true, or who was supposedly involved):

As is well-known, at a certain big-name university the advisor defends the student's thesis. A student worked with a certain big-shot for five years and produced what many looked at as a fine dissertation. The day of the defence came. The advisor got up to the board, gave a quick introduction, and embarked on stating the main theorem in the dissertation. Half way through writing it, he put down the chalk, and paced around a bit. He then turned, apologized, and said, "I'm sorry, but I think I've found a counterexample."

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    If we're thinking of the same big-name university, then I heard that the impetus for the whole "advisor defends the student's thesis" system arose from an advisor making a quip along the lines of "this is the best thesis I ever wrote" in a defense prior to this system. It's not clear why this would have been an effective solution to this sort of problem, so this is probably pure apocrypha. Amusing nonetheless! – Ramsey Jan 24 '11 at 22:39
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    I have heard the same story as Ramsey, about the same university -- but the quip is supposed to be "OK, so it's not the best thesis I ever wrote!" – JSE Jan 24 '11 at 22:57
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    When I was in grad school, a senior mathematician told me: "I never minded writing a student's dissertation, but I draw the line at having to explain it to him or her". – Thierry Zell Jan 25 '11 at 0:08
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    There's an old joke-definition of a dissertation, something like, "a research paper written by a senior academic under the most trying circumstances." – Gerry Myerson Jan 25 '11 at 2:27
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    This all somehow reminds me of a quote (supposedly from Haydn) "Don't make a sour face when listening to opening notes of a sonata written by some grand duke: you never know who actually composed it". – fedja Jan 25 '11 at 4:01

Apparently a postdoc at IHES cornered Dennis Sullivan back in the eighties, and asked him a long and involved question concerning the stuff the postdoc was studying. Dennis' response was:

That's a good question! I think you should work on it!

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    I'm surprised there aren't more Dennis Sullivan stories here. He's definitely one of the more colorful mathematicians of our time. – Deane Yang Apr 23 '11 at 15:46

When I took analysis from Paul Sally, he claimed that a student once asked him in class, "Professor Sally, why is it called the p-adic norm? If it's a norm, what does it measure?" Without thinking, Paul loudly replied, "Well, it measures the p-ness of a number."

I suspect that he just substituted himself into an existing urban legend, yet I would not be surprised if it were true.

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    The related story that I've heard (from people who were there, I believe) is that in the early 1970s in the Ohio State summer math program for high school kids an elderly female European giving a lecture about finite groups once innocently said, in coming to a key step in a proof: "But we still haven't used the $p$-ness of the group." – Tom Goodwillie May 24 '11 at 21:45

I have a story of this kind. My thesis advisor J.-M. Souriau used to talk this story about one of his close friend (I'll keep quiet the name) : "Avant de devenir directeur de l'école normale supérieure (Ulm) il a passé la moitié de sa vie mathématique à définir le nombre de [[put his name here]] et l'autre moité à démontrer qu'il était égal à 1." I don't know if it is true, I doubt but not that much :-)

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    Is this poor soul's number a candidate for mathoverflow.net/questions/32967/… ? – Willie Wong Jan 25 '11 at 0:39
  • OK... So how many mathematicians have been directors of "L'Ecole" recently?... The identity of the poor soul should be easy t work out. – Thierry Zell May 7 '11 at 13:56
  • Well, I guess there are two candidates, actually. One more than I expected... – Thierry Zell May 7 '11 at 13:58

Here's another story not particularly relevant to the original question: When I was a graduate student at Harvard, there was a much older Greek graduate student (whose name I forget) who was viewed by at least some of my classmates as being one of if not the smartest graduate students there. I was told that he was responsible for providing the critical ideas for least two classmates' Ph.D. theses. But he never completed a thesis himself and, as I recall, found a good career working for the European Community in Brussels.

This may be an urban legend, but it's true as far as I know.

During R. L. Moore's reign at University of Texas, sometimes a grad student would be awarded a PhD for work that was original for the student even if it had been done before.

Moore insisted that students reproduce everything from scratch (though guided with Socratic questions). This produced outstanding students, at first. But it got to be a tragedy by the time Moore was put out to pasture. The gap between what students graduated knowing and the vanguard of research became insurmountable.

This was before my time, but I did speak to someone who said that he recused himself from a PhD committee shortly after coming to UT because he could not sign off on a dissertation whose results he knew were not original.

Ed Dean linked to this story in a comment, but I think it is too nice to stay hidden there:

On December 5, 1947, Einstein and Morgenstern accompanied Gödel to his U.S. citizenship exam, where they acted as witnesses. Gödel had confided in them that he had discovered an inconsistency in the U.S. Constitution, one that would allow the U.S. to become a dictatorship. Einstein and Morgenstern were concerned that their friend's unpredictable behavior might jeopardize his chances. Fortunately, the judge turned out to be Phillip Forman. Forman knew Einstein and had administered the oath at Einstein's own citizenship hearing. Everything went smoothly until Forman happened to ask Gödel if he thought a dictatorship like the Nazi regime could happen in the U.S. Gödel then started to explain his discovery to Forman. Forman understood what was going on, cut Gödel off, and moved the hearing on to other questions and a routine conclusion.

(cited from wikipedia)

EDIT: Thanks to Gerald Edgar (and Google) you can find the answer to what the loophole in the US Constitution is here.

One urban legend I remember was of a student who just wanted to schedule a language exam, but the professor opened a text to the introduction and asked him to translate it. The student asked to switch to the mathematics, saying, "I don't know any verbs!"

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    That is not an urban legend, well, at least not the first half. My (French) language examiner at Princeton was disappointed that I brought a mathematics textbook (in French) for the exam. After looking around and couldn't find a French-language roman handy in his office, he begrudgingly passed me after I translated the Preface and Acknowledgements (in addition to several mathematics-laden passages from the middle of the book). – Willie Wong Jan 24 '11 at 22:05
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    I once took a math course for which the textbook was in French although the course was otherwise taught in English. You don't need to know much French to understand that sort of thing. – Michael Hardy Jan 25 '11 at 2:54
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    I heard from an older student in college that Nick Katz taught a course on local class field theory using Serre's Corps Locaux and that student didn't realize the book had been translated into English until after the course was over. – KConrad Jan 25 '11 at 6:13
  • @Michael: Some fellow student of mine went to a course (in the US) where all literature was Chinese, and the lecturer being surprised about him showing up had a hard time giving the lecture in English. My friend didn't stay long in that course... – Someone Jan 25 '11 at 9:09
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    @Someone: I actually dragged a Serbian friend of mine to seminar just for that. Otherwise they'd look at me and give the discussion in Chinese, and while it is my native language, it is by far not my native mathematical language. – Willie Wong Jan 25 '11 at 12:49

Bob Stong told me that a Ph.D. candidate once presented his thesis in topology without any examples. One of the committee asked for any space for which the work was true. The student said that he had yet to think of one. He was failed in short order. I seem to remember that the story was from University of Chicago, but I could be wrong. Whether Professor Stong was pulling my leg or not is not known.

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    I was actually present for something like this. This was in Oregon, maybe 15 years ago. The speaker lectured for 75 minutes on the cohomoolyg of a certain class of spaces. I was a beginning graduate student at the time, so I didn't really understand the talk, but, apparently, the class of spaces had very unusual properties. Someone asked - as in your story - at the end of the talk if the speaker could give an accessible example of this class. The speaker said, unabashed, that he couldn't. In fact, he strongly suspected that there were \textit{no} examples at all! – John Iskra Jan 28 '11 at 17:31
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    I don't think this is so unreasonable. I am sure many people try to characterize, eg, $\{x | \zeta(x)=0, \Re(x) \neq \frac12\}.$ – Igor Rivin Jan 28 '11 at 23:04
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    I heard once that Cauchy wrote a paper about "Bounded Entire Functions". From what I know, he later proved the Liouville Theorem. – Nick S Feb 6 '11 at 17:45
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    The paper at annals.math.princeton.edu/2010/171-1/p10 which is based on a PhD thesis, makes Igor's point even better than his example does. – Dan Fox Apr 13 '11 at 16:50
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    [Igor probably meant to include a hypothesis such as $x \notin \bf Z$ :-)] – Noam D. Elkies Jun 5 '11 at 23:35

I heard this in Oxford in 1970. I can't believe it:

A PhD student decides to see what happens if he assumes the inverse of the triangle inequality. He finds he can prove that there are various interesting consequences - for instance, certain sets of points must be collinear. He eventually writes it all up as his thesis. His examiner starts with the question, "are you aware that such a space can only contain one point?"

Lacking sufficient reputation on this site to comment on posts, I'm going to make this an answer, and you know it's a true urban legend if you read the posts above. Once, at a Princeton physics exam, a group of the senior Princeton physics faculty were trying to figure out why, when you shake a bunch of rods in a container with certain asymmetries in the geometry of the container, the rods assume a "more ordered" state (they tended to concentrate on one side), and they couldn't figure out why this did not contradict the second law of thermodynamics! In an amazing twist, they speculated that the result had to do with finite size effects.....

(for those who aren't in on why this is just too crazy to believe: the shaken container is not a closed system, so the second law doesn't apply. Further, the forces on the rods, a combination of shaking and frictional forces, do not correspond to thermal noise and dissipation, so there is no reason for the system to go to thermal equilibrium. It's like asking why, when there is a baited mousetrap and a live mouse in the room at time t=0, is it the case that after a certain amount of time the entropy decreases in that the mouse is more likely to be in the mousetrap than not. I think the most clever answer for the student is: "I notice that this exam has gone on for 30 minutes already and you are still walking and talking. Why are you not relaxing to thermal equilibrium? Perhaps the food you ate this morning is helping keep you out of equilibrium?")


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    I was one of those confused professors (although not senior) and you may well be right that finite size effects have nothing to do with it, but I don't find your analogy terribly convincing either. Do you think the behavior would not have been observed if the system was just coupled to a heat bath rather than shaken? I'd really enjoy hearing a more detailed analysis of what is going on. – Jeff Harvey Jan 25 '11 at 20:33
  • If the system was in thermal equilibrium with a heat bath, then indeed this behavior would not happen. Two caveats: first (a technical point): of course, in true thermal equilibrium the rods would combust with the oxygen in the air, etc... but at intermediate time scales we can ignore that and consider an ensemble of rods being equally likely to be distributed anywhere. Second, even in thermal equilibrium there can be some interesting entropic effects near a boundary; basically, yes, there can be some finite size effects due to, say, more available orientations near one side of the boundary. – user12494 Jan 25 '11 at 21:11
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    it is a condensed matter analogue of the "anything not forbidden is compulsory" in particle physics. Consider a particle at point x in a potential U(x) with some non-thermal noise and a damping (so force=U'-eta v+noise, where eta is friction and noise has non-thermal spectrum). Suppose the potential has a sawtooth shape. This shape destroys reflection symmetry. The non-thermal noise then means detailed balance is broken, and so nothing forbids a current. If you pick a generic potential and non-thermal noise an dissipation and run it on a computer, odds are you will see the current. – user12494 Jan 25 '11 at 21:17
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    Thanks for the explanation. I talked to my local condensed matter guru and he also emphasized the role of dissipation, although in this system it is not clear without more analysis whether dissipation in collisions between the rods or in the rod-boundary collisions is more important in driving the system. Amusingly you can replace the rods by spheres and make the barrier symmetric and you will still get an accumulation on one side, but which side it is will be random. Dissipation from collisions increases at higher density and cool the system, so a fluctuation towards higher density grows. – Jeff Harvey Jan 25 '11 at 21:35
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    Yes, Chicago has some real experts on this kind of thing! My favorite absolutely bizarre thing I learned about in Chicago has to do with the "brazil nut" effect, that if you shake a jar of mixed nuts, the brazil nuts (the bigger ones) tend to wind up on top. Well, I'd heard about that effect before visiting Chicago, and you can try to puzzle out what happens, why exactly the Brazil nuts wind up on top. So, the crazy thing I learned is that if you repeat the experiment in a vacuum (or maybe it was just in a different liquid, I forget the exact details), the effect is reversed! – user12494 Jan 25 '11 at 21:42

Professor A at Harvard told the following story, supposedly a first hand account of his student days at Chicago, though it never struck me as remotely plausible. (I think he just told it so that he would seem like a teddy bear in comparison.) But I wonder if anyone else has heard variants of this.

At the beginning of a course, Professor X would start asking some reasonable questions, the answers to which students taking the course could be expected to already know. Finally, he would ask one unfortunate student a question which no one taking the course would be able to answer. Upon the student's failure to answer correctly, Professor X wouldn't explain that the student's ignorance was justified, instead letting this event undermine the student's confidence about taking the course. Professor X would continue to single out this poor target for humiliation until he or she finally dropped the course. Professor A claimed to believe that Professor X's motivation was to have lit a fire under the remaining students and make them band together.

Again, it's a rather unlikely tale of abuse. Though perhaps with a plant "student" playing along as the victim it could be an effective ploy ...

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    A version of this happened at my undergraduate institution, but with a plant student. At the very first lecture, the professor sent the plant student to the blackboard and assigned a difficult exercise. The plant student does a nice job (one much better than any other students could do presumably), but the professor gives him a really hard time, harping on any flaw and concluding that this was "barely the level required to survive this lecture". The next day, six students had given up the course, so that the professor had to make sure words reached them that it was all a joke. – Olivier Jan 25 '11 at 10:00
  • Example of a plant student: youtube.com/watch?v=hut3VRL5XRE – timur Apr 24 '11 at 0:25

protected by François G. Dorais Oct 15 '13 at 2:41

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