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.
 A: 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.
A: 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". 
A: 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..."
A: 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?
A: 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!"
A: 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!
A: 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?")
:-)
A: 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.
A: 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.
A: 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... 
A: 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? 
A: 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."
A: 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.
A: 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 ...
A: 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.
A: 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 ....
A: 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. 
A: 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."
A: 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."
A: 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."
A: 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. 
A: 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."

A: 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...
A: Prof. M eventually says to his student: "There's nothing more I can teach you on this subject, I'll have to send you to Prof. B". So the student goes to study with Prof. B. A few months later he returns to Prof. M, a whimpering wreck. Prof. M calls up Prof. B. "What did you do to my student?" "You told me to stretch him." "Yes, but not on the rack.".
OTOH Another Prof. B told me that this was exactly the kind of story that the original Prof. B used to calculatedly spread about himself to give himself a certain reputation.
A: I've heard stories about von Neumann chatting up recent Ph.D.'s and solving their thesis problems in his head.  An incident along those lines is recounted in Sylvia Nasar's biography of John Nash.  Perhaps someone here can shed some light on which von Neumann stories are purely mythological?
A: This actually happened in one of the initial lectures in an introductory course in linear algebra. This was an altogether new experience for us to get acquainted with abstract way of thinking. So the teacher said -"Let a, b and c be three linearly independent vectors in the vector space R^n." A guy interjected -" Sir, can you be more concrete?" "Ok"-said the professor and continued - ""Let alpha, beta and gamma be three linearly independent vectors in R^n".
A: Allegedly, when Peter Lax was receiving his National Medal of Science, everyone had to describe their more notable accomplishments, and being somewhat cowed by the tales of curing cancer, turning water into wine, etc, etc, when the turn came to him, his description of his accomplishments was:
I integrated by parts...
A: 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.
A: 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
A: 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!
A: 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
A: 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."
A: 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."
A: 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?"
A: 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.
A: (A rather sad story)
For obvious reasons, I won't give the place and/or names.
On a thesis defence (we have here a procedure very different from Europe or US; for instance, the committee is more or less fixed) one member of the committee rose and asked to vote against the thesis because of plagiarism: the thesis contained (almost verbatim!) definitions from a book X.
A: When I was a grad student, I lived across the street from an electronics store. The owner of the store  had done some graduate work (in some sort of Engineering, I think). He ran a weekly ad in the local newspaper, and placed at the bottom of the ad a relatively hard math problem. And he gave anyone who could solve the problem the choice of a free radio or telephone (each of which would retail for less than ten dollars). After solving one of his problems, and speaking with him for a bit, I convinced him to place a problem from group theory (it was about equations over the group $Z_2$) in the next ad.
The next week the ad came out. My problem was there in print. My first publication. A few days later I went to talk to the owner of the store. He was furious. A whole bunch of people had come in with solutions and he gave away a lot of free radios and telephones.
The sad thing was that none of the solutions were correct.
EDIT- now for the mathematical urban legend... 
A well-known topologist, let's call him X, told me that this had happened to him. He had been in a seminar for graduate students. The student speaker was proving one of X's theorems. X found this boring and fell asleep. And he started snoring.
They had to wake him up because no one could hear the speaker.
A: 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.
A: 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.
A: 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?"
A: 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.
A: 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.
A: 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.
A: 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.

A: 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."
A: 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."
A: 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.
A: 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.
A: 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.
A: 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 ?'.
A: 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'."
A: 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".
A: 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."
A: 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.
A: 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!
A: 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.
A: 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: 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 :-)
A: 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.
A: 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.
A: 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.
A: 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!"
A: 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.
A: 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?"
A: 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.
A: 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?"
A: There is also the story of the young mathematician which speaks in a conference, and at the end of his talk a (famous) mathematician provides a counterexample to his main theorem. This is just another variation to the story posted by WW.
I head this story from couple people (they claim they were there), but, if I recall right, I also read it in "The Puzzling adventures of Dr. Ecco". 
A: Was the Chicago professor M. H. Stone of Stone-Weierstrass fame?
Prof. Abhyankar has several times recollected his lectures at tata institute,
which he attended as an undergraduate student, starting with : let X be a Hilbert space, over reals or complex nos or quaternions.
A: In a fabulous French comedy La moutarde me monte au nez Pierre Richard's character is a maths teacher at a local girls' college. One day he accidentally ends up in a mansion that belongs to an American movie star (Jane Birkin). Frighened of a potentially dangerous intruder, she sets her pet cheetah on him, and the academic promptly jumps on a huge chandelier. 
To prove to her that he was indeed a maths lecturer, he had to answer a few questions. First - to expand $(ax+b)^2$ (which he did) and then - to integrate $\sin(ax+b)$ (again, success). All that - hanging from a chandelier, with a cheetah pacing below. 
Only after that she allows him to climb down to the floor... and then they have a romantic dinner, needless to say. 
