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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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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.

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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.

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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?"

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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?"

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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.

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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.

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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".

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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..."

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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?

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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!"

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This story was told to me by my advisor.

A Ph.D. student in logic was having an extremely difficult time finishing his thesis and was starting to succumb to hopelessness. Every evening he would trudge home, open a beer, and sit down in front of the television. This was the 1960’s. Evidently there was a running show called Whiz Kids that showcased the achievments of child prodigies; I’m imagining something of a Johnny Carson style setting involving banter with an unctuous host before a studio audience. One week, the young Harvey Friedman was on the show. The host asked Harvey what he had been up to recently, to which the latter responded that he had proved that "every end extension of a model of standard arithmetic has an elementary submodel such that..." and on to the technical details, much to the amusement of the studio audience. The student watching home at that moment realized: that closes precisely the gap I need to finish my thesis!

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This is a story that I heard from one of the postdocs from my university, which in turn heard it from one of the professor at the university (I didn’t bother to verify with him as the source seems relatively reliable).

The said professor was a postdoc in some university in the USA a few decades ago, and he was teaching a basic course on group theory. One of the homework assignments had a question of the form: "Let $G_1$ be the group $\ldots$, and $G_2$ be the group $\ldots$ Prove that $G_1$ and $G_2$ are isomorphic."

One of the papers submitted had an answer "We will show that $G_1$ is isomorphic..." and some nonsense, followed by "Now we’ll show that $G_2$ is isomorphic..." and more nonsense.

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Here is another scary example (known to be true, by way of two of the participants): a (then) young postdoc approached R. Langlands and A. Borel (this was in the late seventies), in the IAS tea room, and the following conversation ensued:

Postdoc: Do you guys know anything about automorphic forms?

B&L: Maybe

Postdoc: Well, can I ask a stupid question?

B&L: Well, you have already asked one.

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At the Hebrew University, during a complex analysis course, the professor states and proves the famous "Liouville’s theorem", that every entire bounded function is constant. One confused student, trying to get some general clarification, asks "maybe you can give an example?". The professor without hesitation answers "yes, Of course. 7" and continues... we all sat still trying not to laugh so that the confused student wan’t be embarrassed, but he was still quite embarrassed though...

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This one happened - I was there (as an observer, not a principal). Only the names have been changed.

X was Professor A’s first doctoral student, and their relations weren’t good. Rumor had it that the first time A saw most of X’s thesis was when X handed in the final draft.

By the rules, there had to be a non-mathematician on the thesis defense committee - let’s call him Professor H. Professor H made a valiant effort to read the thesis, understandably didn’t get very far, but decided he was going to ask a question at the defense, to justify his being there in the first place. So he says to X, I notice you didn’t provide a proof of your Lemma 2.3.1 - how does it go? X says, well, 2.3.1 isn’t my work, it’s a well-known result of van der Corput.

This satisfies H, but A says, OK, it’s a result of van der Corput - but, how do you prove it? Well, X was prepared to answer questions on his own work, but hadn’t brushed up on all the previous work that his thesis rested on. He hummed and hawed, started to give a proof, got stuck - at which point A gave him a hint. Using the hint, X got a little farther, but got stuck again - so A gave him another hint. This went on for an excruciating fifteen minutes (which, I’m sure, felt like 15 years to X), until finally Professor N broke the tension by saying, say, just whose thesis defense is this anyway, X’s or van der Corput’s?

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When I was at the University of Oklahoma in the early ’80s, we were all required to write a brief description of our research for the (rather conservative, this being Oklahoma) Board of Regents of the University. An colleague in algebra, perhaps hoping for more state support, wrote that he was studying "annihilating radical left ideals."

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I have heard the following story from a few sources (among them, I think, an MO thread, possibly Terence Tao’s blog, and Richard Lipton’s blog), so it might even be true.

The story goes that once upon a time a student wrote his thesis on Hölder-continuous maps with $\alpha > 1$, since he had only seen the case $\alpha \le 1$ addressed in his books. The student proved many wonderful theorems about these maps and was very excited for his defense.

At his thesis defense, one of the examiners (is that the right word?) asked him to provide a nontrivial example of such a map. The student was flustered. As it turns out, all such maps are constant - no wonder the theorems were so nice.

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A certain Greek professor, let’s call him AF, happened to have attended medical school in the US before becoming a professional mathematician.

He attended a talk by another mathematician, who claimed to have proved in N dimensions a result which AF had struggled to prove for N=2. Disconcerted, he spent the entirety of the talk constructing a counterexample to the speaker’s result.

At the end of the talk, when questions were invited, AF walked up to the board and wrote down his counterexample. He turned around as he heard a loud thump from behind him. The speaker had fainted.

Undeterred, AF used his medical training to revive the speaker before returning to his seat.

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Some time in the early 90s Goro Shimura was giving a lecture course on algebraic number theory at the ENS in Paris. According to someone who was in the audience, one of the lectures started thus.

Let $a$ be a rational number. [Pause; the lecturer writes $a$ on the blackboard.] Is this clear? [Pause.] Do you follow me? [Long pause.]

Ok then. [Pause.] Let $\beta$ be an irrational number. [Pause; the lecturer writes $\beta$ on the blackboard.] Is this clear? [Pause.] Does everyone understand? [Long pause.]

Ok then. So consider a global field of prime characteristic and an automorphic representation of an algebraic group over its adelic ring. Now take the absolute Galois group and the category of perverse l-adic sheaves on ...

[The third phrase here is a random and probably inaccurate reconstruction, but I’m pretty sure the numbers were called $a$ and $\beta$.]

upd: I’ve emailed the person I heard this from and they provided the following version. It seems that I got everything wrong; apologies. Anyhow, the course took place at Jussieu, not ENS and began thus.

Professor Shimura:

Consider alpha algebraic number, writes alpha on the blackboard, pause (on the same line) now theta transcendental number, writes theta, pause (below the first line) f holomorphic function, writes f, pause (on the same second line below theta) g non-holomorphic function, writes g, pause

long silence which I interpreted as "think deeply about the meaning of this square"

Professor Shimura takes a deep breathe and in one sentence restarts:

Let f be a Siegel modular form of weight k and level N ....

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I have no idea whether this one is true - I heard it at Harvard, around 1970. The story goes that a PhD student was so sure no one would ever read his dissertation that he stuck in the middle of it an offer to send fifty dollars to the first five people who asked. Every few years he’d get a letter from someone who stumbled across the offer, and he’d pay out.

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George Mackey is reported to have been overheard saying "I’ll write his thesis for him, but I’ll be damned if I’m going to explain it to him."

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I am not sure where or when this happened, but I still think there may be some truth to the story.

Once someone from the engineering (or physics?) department of some university came to see Joseph Bernstein and asked if he knew a formula for a conformal mapping of the interior of a regular $n$-gon to the upper half-plane. Bernstein knew the formula, but decided to first ask what the person needed it for. The reply was: "Well, you see, what I really need is a formula for the unit disk, but that’s probably too complicated, so I decided to find out the formula for the $n$-gon first and then take the limit."

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Not an urban legend: I was there.

Abhyankar was speaking at Mumford’s seminar, so Zariski, though long-retired, came to hear his former student speak. Abhyankar began his talk by stating that he would only be working in characteristic 0.

Zariski interrupted to ask "Are there any additional difficulties in characteristic p?"

Abhyankar smiled and said "Only psychological difficulties."

Zariski turned to the audience and stated, most forcefully, "I have NEVER had psychological difficulties."

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Here’s something that keeps me up at night:

During the Russian revolution, there is a story of a mathematician (I’ve heard Igor Tamm may be the one) who was mistaken by rebels to be a communist spy. He was promptly captured by a local gang and interrogated. When he said that he is a mathematician, the gang leader asked him to back up his claim by deriving the formula for the Taylor Remainder Theorem. He was warned that if he failed, he would be shot on the spot. After some sweating the mathematician finally derived the result. The gang leader was satisfied with the proof and let him go.

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Here is another story from Krantz’s Mathematical Apocrypha Redux which I thought was quite funny.

One of the most common and popular Norbert Wiener (1894-1964) stories is of a student coming to Wiener after class and saying, "I really don’t understand this problem that you discussed in class. Can you explain to me how to do it?" Wiener thought a moment, and wrote the answer (and only that) on the board. "Yes," said the student, "but I would really like to master the technique. Can you tell me the details?" Wiener bowed his head in thought, and again he wrote the answer on the board. In some torment, the student said, "But Professor Wiener, can’t you show me how the problem is done?" To which Wiener is reputed to have replied, "But I’ve already shown you how to do the problem in two ways!"

Dick Swenson, who was at MIT in those days, tells this variant of the story: Wiener showed the kid the answer twice, as just indicated. Then the student said, "Oh, you mean...," and he wrote the answer (and only the answer) on the board. Wiener then said, "Ah, very nice. I hadn’t thought of that approach."

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Oral maths exam for engineers, 1960s, Budapest. To prove: there are infinitely many prime numbers. Candidate shuffles in his chair, has no idea really. Professor tries to help: let’s recall the definition of prime numbers. Let’s talk about some examples. Etc etc. After 15 excruciating minutes, candidate summarizes progress thus: Professor, I now understand that all odd numbers are prime. But I still don’t see why are there infinitely many...

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Since the OP gave a physics example, here is another one, also at Princeton. Why are they always at Princeton? Student finishes his presentation on very mathematical aspects of string theory. An experimentalist on the committee asks him what he knows about the Higgs boson. He hems and haws and finally says "well, it was discovered a few years ago at Fermilab", Experimentalist: "Can you tell me the mass?" Student: "I think around 40 GeV."

This was more than 20 years ago and actually happened. I was there. The student passed, but the next year all Ph.D students working on string theory were required to take a course on the phenomenology of particle physics.

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Somebody posted the following:

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.

Now I have never taught linear algebra at Chicago, since as somebody else pointed out we have no undergraduate linear algebra courses, but in the 1960’s and 1970’s I did in fact drive to and from seminars and conferences at Northwestern seminars without defrosting the windshield in order to have a convenient blackboard. I recall that it worked very well.

Peter May

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Not a horror story. On the much nicer end of the spectrum, there is a well-known urban legend about a student unwittingly solving an open problem, thinking it was homework. Though details of the tale may vary, there is at least one instance where the urban legend is true, George Dantzig in 1939. The funniest part of the story is when Don Knuth apparently came to learn of this story through a sermon by an Indiana pastor!

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Once during a mathematical conversation with a student, Alexander Grothendieck was asked to consider an example of a prime number.

"You mean an actual prime number?" The student replied, "Yes, an actual prime."

Grothendieck then said, "Alright then, take $57$".

-Taken from the Comme Appelé du Neant article in the Notices of the AMS

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