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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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Since every finite CW complex is weakly homotopically equivalent to a finite topological space, that does not sound so bad... :) –  Mariano Suárez-Alvarez 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

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

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.

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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

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.

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I am extremely curious as to what this inconsistency is. –  Qiaochu Yuan Apr 24 '11 at 7:42
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Perhaps these guys are unable to use Google for some reason... blog.plover.com/law/Godel-dictatorship-2.html –  Gerald Edgar May 23 '11 at 21:16

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-Alvarez May 24 '11 at 1:54

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".

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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

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...

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Denis Serre has unfortunately already posted this... –  Qiaochu Yuan May 23 '11 at 19:35
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Not so unfortunate, but I should pay more attention:) –  Igor Rivin May 23 '11 at 20:08

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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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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This pattern of behavior occurs with surprising frequency. I have always found it very interesting. –  Mariano Suárez-Alvarez Jan 28 '11 at 22:18
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Sorry; (as you admit) automorphic representations and (especially) perverse $\ell$-adic sheaves does not sound like the Shimura I know, love and fear. I have two colleagues who are former students of the master: perhaps I should ask them to suggest something more plausible? –  Pete L. Clark Jan 29 '11 at 0:25
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Pete -- yes, by all means! The only thing I'm sure about, apart from the $a$ and the $\beta$, is that the level of difficulty increased rather steeply after the first two phrases. I've tried to convey this impression while adding some details for dramatic purposes (or else this wouldn't be much of an urban legend, would it?). On the other hand I can ask the person I heard it from (and who was sitting in that lecture) for a more accurate account. –  algori Jan 29 '11 at 0:52
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@algori: well, I didn't see your response until just now, so it's probably too late. But I forgot to mention that -- like any good urban legend -- in spirit your story rings true. As an undergraduate I remember marveling at how much time certain professors spent explaining the most trivial things, only to race through the hard stuff in a big ball of frenetic activity. And now, as an instructor, I see myself doing the same thing at times! I guess we think, "Well, I really don't want to lose anyone on the first day" and at some point we think that we've lost the people we're going to lose... –  Pete L. Clark Apr 17 '11 at 8:31

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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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

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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Pete -- I should have been more explicit: I've heard this story several years ago during a late night conversation in Bonn and I can not really vouch for the details. It may have been biology or chemistry or something else. The moral however is, if I may say so, that one should be careful when speaking to someone who is about to apply mathematics. That is, if this person says they need to be able to solve a particular (difficult or hopeless) problem, it may be worthwhile to ask again. –  algori Apr 17 '11 at 9:21
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@Pete: Engineers are smart, but you can't expect them to know all the math out there. In polynomial system solving, I've heard stories of engineers adding extra variables to lower the degrees of their equations. Anyone who is familiar with the complexity of Grobner bases will realize that this is not the way to go. But I'm sure it seemed to make sense at the time. –  Thierry Zell Apr 27 '11 at 15:34

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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I did say 1960s (-: –  Balazs May 25 '11 at 6:27

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'."

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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.

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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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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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Another from the "I was there" file, Joel Hamkins may remember this from grad school: one Prof. (not J.) H. was lecturing on (related to Shelah's classification) something like omitting types, one of which was named p. H. said "but we can't do this, else it loses its p-ness and can't fork anymore". The class, not a mixed group, all laughed at this, and Prof. H. looked surprised. Some of us (Joel too, I think) later thought H. had worked that phrase in; it would be in character for H. (Joel, feel free to tell your view of it.) Gerhard "Don't Quote Me On This" Paseman, 2011.04.13 –  Gerhard Paseman Apr 13 '11 at 16:25
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In a variation of the old "integral scheme of finite type over a field" story, I have actually seen Peter May answer the question "What's a ring?" with "Oh, a ring is just a Z-algebra." –  Harry Altman Apr 13 '11 at 22:26

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.

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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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why, 57 is prime, up to multiplicative constants –  Pietro Majer Feb 6 '12 at 9:11

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."

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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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I've heard the non-defrosting story told about R H Bing, who did this during heavy snow while driving people to Chicago. –  Richard Kent Feb 6 '11 at 3:44
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Ooooh. Given the pretty drawings i've seen accompanying some of Bing's writing, it is probably a pity no one saved at least some of those windshields in a freezer! Welcome to MO, by the way, Prof. May :) –  Mariano Suárez-Alvarez Feb 6 '11 at 3:49

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.

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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."

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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?"

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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 ?'.

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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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Who cares if it's true or false? That's what makes urban legends so fun! It's shocking, and yet we feel sure that somewhere some similar incident must have happened... –  Thierry Zell Jan 31 '11 at 20:58
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Andre Weil's law of university hiring (according to Wikipedia, undocumented): "First rate people hire other first rate people. Second rate people hire third rate people." This always left me wondering, who hires the second rate people? Maybe Igor's story answers my question. –  Gerry Myerson Jan 31 '11 at 22:34
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"And third rate people hire fifth rate people." –  Nate Eldredge Feb 3 '11 at 15:40
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Ah, I understand. Nth-rate people are hired by mth-rate people, where m is the closest integer to n/phi. –  Tanner Swett May 25 '11 at 2:26
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From the Chronicle of Higher Education: chronicle.com/article/You-Were-Too-Good-for-Us/46833 –  Philip Brooker Jun 3 '11 at 0:29

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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I gave a homework problem, "Let $G_1$ be the group $\dots$, let $G_2$ be the group $\dots$. Are $G_1$ and $G_2$ isomorphic?" and was astonished to get the response, "$G_1$ is, but $G_2$ isn't." Are Asaf's story and mine isomorphic? –  Gerry Myerson Jan 31 '11 at 22:39
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@Gerry: Asaf's is, but yours isn't. –  Nate Eldredge Feb 1 '11 at 1:20
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You've been a lovely audience. Nate and I will be here all week. –  Gerry Myerson Feb 1 '11 at 11:41
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This is the same syntax as the joke "Oh Harry, if only we were married!" "We are, Sally... Oh, did you mean to each other?" –  David Speyer Feb 1 '11 at 14:30
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Don't forget to tip your server! –  Nate Eldredge Feb 3 '11 at 15:41

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.

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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".

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I don't really get it...The joke is that some professor thinks that "$\alpha$" is more concrete than "a"? To me that's just weird. –  Pete L. Clark Apr 17 '11 at 8:50

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