Terence Tao's Raw Notes After His Princeton Comprehensive Exams (1999) | Hacker News Reader

Terence Tao's Raw Notes After His Princeton Comprehensive Exams (1999)

web.math.princeton.edu

240 pointsbreck89 comments

I guess it has been reposted because of his latest blog post. The source that sparked it is really interesting :

https://www.ams.org/about-us/LivingProof.pdf#%5B%7B%22num%22...

To boost this, I do recommend checking out Living Proof, the book linked above, if you're interested in the mathematical life. Today's mathematicians discuss their challenges, successes, life, art, attitudes, etc. Especially if you're a student, check this out! You'll find some story that resonates with you.

This caught my attention:

"""

Then they asked how Dirichlet got an explicit formula for this when \chi was a real character. I was going to write a messy (but finite) expression involving sines and logs, but then I realized that they were talking about the class number formula. (I said carelessly though that "this was a disgusting way to do it", since I was still thinking about the sine-log formulas. Then they made a comment that "This would put thousands of people out of work", or something like that.)

"""

Can someone explain the comment made by the examiners?

> "This would put thousands of people out of work".

Thanks!

I read this as a joking reference to the fact that class field theory is an important area of contemporary mathematics, which presumably employs many people (though perhaps not literally thousands).

I’d be curious to know if anyone reads it differently.

My Mum attended one maths course at university (as a mature age student) with Tao, and I distinctly remember her coming home and complaining about some 'smart-arse kid who wouldn't stop challenging the lecturer'. I guess it can be tough being very smart and capable and also young at the same time.

Whats the relation of someone complaining and tough in this context ?

It's tough to be smart, young, and capable. Because people will call you smart arse and not like you.

If you are smart then you are smart enough to ignore that.

If this were true, many many smart people would have much less anxiety and self-doubt.

A lesson which many highly intelligent persons never learn as long as they live is that human beings in general are incorrigibly very different from themselves in thought, action, and desire. Many a reformer has died at the hands of a mob which he was trying to improve. The highly intelligent child must learn to suffer fools gladly--not sneeringly, not angrily, not despairingly, not weepingly--but gladly, if personal development is to proceed successfully in the world as it is. Failure to learn how to tolerate in a reasonable fashion the foolishness of others less gifted leads to bitterness, disillusionment, and misanthropy, which are the ruin of potential leaders.

As a form of failure to suffer fools gladly, negativism may develop. The foolish teacher who hates to be corrected by a child is unsuited to these children. Too many children of IQ 170 are being taught by teachers of IQ 120. Into this important matter of the selection of the teacher we cannot enter, except to illustrate the difficulty from recent conversation with a ten-year-old boy of IQ 165. This boy was referred to us as a school problem: "Not interested in the school work. Very impudent. A liar." The following is a fragment of conversation with this boy:

   What seems to be your *main* problem in school?

   Several of them.

   Name *one*.

   Well, I will name the teachers. Oh, boy! It is bad enough when
   the *pupils* make mistakes, but when the *teachers* make
   mistakes, oh, boy!

   Mention a few mistakes the teachers made.

   For instance I was sitting in 5A and the teacher was teaching
   5B. She was telling those children that the Germans discovered
   printing, that Gutenberg was the first discoverer of it, mind
   you. After a few minutes I couldn't stand it. I am not supposed
   to recite in that class, you see, but I got up. I said, "No; the
   Chinese *invented*, not discovered, printing, before the time
   of Gutenberg--while the Germans were still barbarians."

   Then the teacher said, "Sit down. You are entirely too fresh."
   Later on she gave me a raking-over before the whole class. Oh,
   boy! What teaching!

It seemed to me that one should begin at once in this case the lesson about suffering fools gladly. So I said, "Ned, that teacher is foolish, but one of the very first things to learn in the world is to suffer fools gladly. The child was so filled with resentment that he heard only the word "suffer."

"Yes, that's it. That's what I say! Make 'em suffer. Roll a rock on 'em."

I quote this to suggest how negativistic rebels may seize on the wrong idea. Before we finished the conversation Ned was straightened out on the subject of who was to do the suffering. He agreed to do it himself.

I will cite another conversation, this time with a nine-year-old, of IQ 183.

   What seems to be the *main* trouble with you at school?

   The teacher can't pronounce.

   Can't pronounce *what*?

   Oh, lots of things. The teacher said "Magdalen College"--at
   Oxford, you know. I said, "In England they call it Môdlin
   College." The teacher wrote a note home to say I am rude and
   disorderly. She does not like me.

-- Leta Hollingworth, "Children Above 180 IQ Stanford-Binet"

"To suffer fools gladly" is an interesting way to put it, but it crystalizes what I've observed about the missing piece of understanding in how "smart-ass" children behave.

It's hard to blame the kid for not knowing how to maturely handle actually being right when the adult/teacher is wrong and being punished for it, because we expect the process of growing up to teach that. But when does the kid actually get taught that? By the time they're in middle school (I have one specific kid in mind who I observed as a math coach), everyone else just hates them for being that annoying kid nobody likes.

This serves as a pretty good example of why "smart" kids (or adults) can be annoying. Useless pedantry is not an enjoyable thing for most people. Yes, the Chinese had printing, but it was done by hand. Movable type (which, yes the Chinese also had) plus the mechanical press (which they did not) is what allowed printed material to flourish and change the course of history.

It's like saying "James Watt invented the steam engine" and the kid in the back chiming in with "Well actually, the Romans had an Aeolipile". Sure, they did, and no one cares.

The real problem is that the teachers ought to know what you just said, but they don't. So instead of teaching the smart kid -- and the rest of the class -- about the advantages of the mechanical press, they call him a smart-ass.

Well, at least the kid in the back cares and maybe someone else too.

If you disagree with that kid you can always refute back.

You can say its annoying but the it maybe also annoying to the kids in the back for you to make inaccurate statement (at least according to the kid perspective).

There is little to no correlation between intelligence and social resilience.

It's tough, because you are a pain in the ass, and not even realize it? (especially if coupled by arrogance).

You don't have to please anyone, regardless what you do there are always going to be someone somewhere who think you are pain in the ass.

If I'm smart and capable and someone find me pain in the ass, then well too bad, thats your problem.

If you're smart and capable then you should also have sufficient social skills for people not to think you're a pain in the ass.

Regardless how much social skill you have, there are going to people who think you are pain in the ass.

If they think I'm in pain in the ass they are the one who has problem.

There'll always be someone who doesn't like you, but you can minimize that number by your own actions.

If everyone dislikes you, then they'll make it your problem. You can gripe about it, or you can try to change what you can.

>but you can minimize that number by your own actions

you can but you don't necessarily have to.

>If everyone dislikes you, then they'll make it your problem

its unlikely that everyone hates me, even hitler has people who like him or I don't gripe in the first place and then still depends on what they actually do.

> >If everyone dislikes you, then they'll make it your problem

> its unlikely that everyone hates me, even hitler has people who like him

Maybe I'm too tired to pick up on the satire, but this is just needlessly pedantic and does nothing to contribute to the conversation.

I see a series of people pointing something out to you, and you continually pushing back on why they are wrong. I guess the question to ask is how are you so sure you're right when everyone else is telling you that you're wrong?

Something that has gone a long way for me to better myself throughout my life is to recognize that when everyone is telling me I'm wrong, it's almost certain they are right and I am the one who needs to change. And if I still think I'm right, I should be able to communicate it effectively enough that others will agree with me. If not, then it literally is me against the world and even if I'm right what fucking good is that?

>Something that has gone a long way for me to better myself throughout my life is to recognize that when everyone is telling me I'm wrong, it's almost certain they are right

Maybe but its not always. Isn't there many people who become famous inventor or great discovery made because they goes againts the commonly held idea.

> Isn't there many people who become famous inventor or great discovery made because they goes againts the commonly held idea.

No. There are a few - probably something like 1 in 10,000,000 or so. Which echoes the most important part of my comment - what makes you so sure you're right when you've failed to convince anyone in this comment thread of anything other than you're a difficult individual?

Maybe you're the next Isaac Newton, but I'd literally take 1,000,000:1 odds you suffer from some combination of a superiority complex and delusional thinking.

>No. There are a few - probably something like 1 in 10,000,000 or so

Using your number, there are 7 billion people in the world, then there are 700 such people. For me , it still quite a lot.

>Which echoes the most important part of my comment - what makes you so sure you're right when you've failed to convince anyone in this comment thread of anything other than you're a difficult individual?

I didn't claim I'm right or wrong, I'm questioning and making argument. Whether you are convinced or not is up to you.

That's quite emotionally immature really.

Yes, you can choose to be an asshole, regardless of your intelligence, and others can choose to reciprocate in kind. Worst case you're lynched and gain some sort of martyrdom people might or might not care about. Best case you drive everyone away and end up abandoned and ignored, left to your own bitterness.

Or you could try communicating your genius idea, improving people's lives in the process, if its a worthy one. But that only makes sense if it was about the idea in the first place. If it was about proving everyone else wrong you might want to reconsider your motivations.

Having a different idea can potentially carry risk but it is what it is.

It is about idea in the first place.

You don’t have to please anyone.

You don’t have to make anything, go anywhere, or discover anything, either.

Seems like you’ll be happier in the long run if you do, though.

Perhaps, but if everyone finds you a pain in the ass, it’s your problem — assuming you want to work with people, have friends, engage in intimate relationships, and so on.

yes, assuming I want to work with the specific people/group.

but its unlikely that everyone hates me, even hitler has people who like him

Without more context in social situations (I wouldn’t assume a large college lecture to have much context on fellow students), it can be hard to determine whether or not someone is actually a genius or is suffering from Dunning-Kruger.

The most impressive thing about really smart people, to me, is how much they actually write down! I feel like writing is an underappreciated thinking tool. I should start doing it more.

"Reading maketh a full man, conference a ready man, and writing an exact man", said Mr. Knowledge is Power, "And, therefore, if a man writes little, he had needed have a great memory; if he confers little, he had need of a ready wit; and if he read little, he had need of much cunning to seem to know that he knoweth not”

Is this a quote from something?

Francis Bacon: http://www.psy.gla.ac.uk/~steve/best/BaconJohnson.pdf

The greatest advantage of writing is that it exposes your own thinking and ideas as sloppy and confused.

I feel personally attacked by this. I should probably write more

This is part of the Princeton math department Graduate Students' Guide to Generals: https://web.math.princeton.edu/generals/

There you can see similar notes made by other students; the standard is really high. E.g. here are notes by Manjul Bhargava: https://web.math.princeton.edu/generals/bhargava_manjul

I can only comment on the first two subjects in the second link, but they look like standard graduate level coursework questions. The standard here and the standard of special topics questions Terrance Tao was asked seem quite different.

Very interesting. Does such a detailed list of accounts like this exist for other fields/departments/at other institutions?

I would like to see for graduate CS departments.

Looks like the year is not correct.

The text has clearly been written right after Terence Tao sat the exam, and given that he received his Ph.D. in 1996, most likely the notes are from 1993 or 1994.

Thanks. I found it in a comment that said “20 years ago”, but wasn’t sure if that was an estimate or exact.

A thread from 2011: https://news.ycombinator.com/item?id=2771031

In that thread someone says that Tao created his own lecture notes for a linear algebra course he was doing[1]. But that page is marked as coming from "2002", which would make him 27. But surely Tao didn't first take linear algebra at 27?? Can anyone make sense of this for me?

[1] https://news.ycombinator.com/item?id=2772019

He was probably teaching the linear algebra course.

The post you linked says that Tao was teaching the class and made notes for it since he didn't like the book

I don't know much pure math. If I pretend I had never heard of Tao, then I would have completely believed these notes if they had ended with "And that's how I failed my comprehensive exam," instead of, "After this, they decided to pass me."

It's funny to me that just going by the tone (without having any domain knowledge), it's hard to tell how well he was doing.

These exams often work by having the examiners ask questions until one reaches the boundary of the student's boundary, partly to see where that boundary is and partly to see how they handle things they do not know. Ideally, anyway. The oral examination format can be very intimidating, and committees often don't care to (or don't know how) to put students at ease. So how well performance correlates with actual ability really varies.

It feels like a common idea that the worse you felt you did in Oxbridge entrance interviews, the better you probably did because the interviewers were trying to stretch you.

For what it's worth, I'm nowhere near as bright as Terence (which puts me in copious company, of course) but notes of my comprehensive exams would have looked kind of similar in tone if not content.

Having given oral exams as well - often you are trying to find the boundary where certainty breaks down to "on the fly" thinking for the candidate. You can get a pretty accurate view of how well someone knows the material quite quickly this way, but you certainly have to account for "nerves" also. I remember being asked a question and just having no idea how to answer it - another examiner jumped in with a `different' question which I answered, then the 1st came back with "can you show how that is equivalent to what I asked" and it took me two seconds to realize they were basically the same question. That stuff can really throw you off.

A friend from my Caltech undergrad did his PhD with Terence Tao. In harmonic analysis :-)

Any stories?

Not a great one, but perhaps one useful to aspiring mathematicians! My friend, who is now an assistant prof at a major research university, said he could only do ~4 hrs of hard mathematical thinking per day. His colleagues and undergrad advisors all experienced the "four hour threshold." However, he said that Terence Tao was able to stay switched on for most of his waking hours, and that he'd structured his life - with nannies and a house close to UCLA - to take advantage of that. I guess the moral of the story is that normal mathematicians can be productive and quite happy with just a few hours of work per day, but that generation defining mathematicians have to put in longer hours :-)

I think it's a great one. Thank you for sharing! :)

When you have a genius-prodigy guy like this being 'examined' -- the professors ask him questions, some (many?) he doesn't know the answer to, or perhaps an incomplete answer to, while they (presumably) do know. Is this because they're equally genius as him? or are 'merely' much-better read than him, although maybe not as gifted in whatever other magical ways he is?

Mathematics is not only brain teasers.

I remember my first lecture in topology. Our professor said that while we may be smarter and beat our teachers in things like group theory and algebra, in this subject no one was going to be better than him. Experience also matters a lot in some subjects.

Of course the professors were better read and had much more experience and knowledge. But you con infer from that report (even if the author tries to hide it) that the professors were extremely impressed by the young candidate.

Terrence's own recounting of the event [1] seems to contradict you:

"After many nerve-wracking minutes of closed-door deliberation, the examiners did decide to (barely) pass me; however, my advisor gently explained his disappointment at my performance, and how I needed to do better in the future."

[1] https://www.ams.org/about-us/LivingProof.pdf#%5B%7B%22num%22... (found elsewhere in the thread)

The sentence before the one you quote is this:

"The rest of the exam then went fairly quickly as none of the examiners had prepared any truly challenging algebra questions."

Thus, he emptied the pool of questions prepared by the professors.

Notice that your quote does not really contradict a good impression upon the examiners. The nerve wracking is self reported, as is the word ``barely''. The later comments by the examiners may be also a generic sentence they say to everybody. Regardless of your level, you can always benefit for being told to do better.

Sure, if you ask these professors today they will probably say that they were very impressed by the young genius... but this does not really say much.

There are smart kids in grad school all over. Half of them flame out 'cause they never figure out how to concentrate, half of the remainder figure out how to work but then get jobs in industry rather than academia, and then of the remaining 25%, some of them become amazing professors who then ask questions of the next genius-prodigy.

Tao is certainly unique, but there's a lot of genius floating around and it can come in very different configurations.

The thing that puts Tao in that upper echelon of mathematicians is his ability to solve unsolved problems and create new mathematics. Those aren't really the skills being tested on a qualifying exam, and you're right that it's just a matter of being more well-read.

Reminds me of George Dantzig, who solved two unsolved problems in statistics because he mistook them for homework.

https://www.snopes.com/fact-check/the-unsolvable-math-proble...

I wonder how many other breakthroughs have occurred in a 'hard' problem that was thought to be 'easy'.

There's an urban legend in the Princeton physics department of professors including questions in exams that they don't know the answer to in hopes that some student might crack it.

Probably happens from time to time.

I never saw it myself since I quickly realized I wouldn't cut it as a physics major there.

When I was a math major at U. Chicago, there was at least one professor who did this.

The math pirate!

I'm not familiar with Rudnick, but Stein (RIP) and Klainerman are giants in analysis/PDEs. They contributed much to the theories on which they were quizzing him. These are professors at one of the top mathematics institutes in the world.

The examiners don't need to know the answers to all the questions they ask. Certainly I was asked questions in my qualifying assessment that the committee members later admitted to not knowing the answer to.

I don't think you need to be "much" better read than someone to pull this off. I think I could lead a much better and smarter programmer than me through -- for example -- monads, and stretch their brain a bit if they hadn't encountered them before, as part of an examination

> Talk about primes in arithmetic progressions.

Tao would later go in to prove that there are arbitrarily long arithmetic progressions of primes in his most famous work to date, The Green-Tao Theorem.

https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem

Funny because it recalls an exam that he essentially aced as if it was a great failure.

He was also only 16-17 years old at the time, completing an exam for university graduates.

EDIT: This was his second year, so he was 17-18 years old.

Also note this qoute: All in all, I probably only did about two weeks’ worth of preparation for the generals, while my fellow classmates had devoted months. Nevertheless, I felt quite confident going into the exam.

I remember doing something like this for my comps. I had left one school with a masters, looking for computational physics stuff I was interested in. Found another and an advisor who was interested in this work.

I applied, got in, and started, while working full time. Graduate advisor called me up 2.5 weeks before starting and said "we want you to take the comprehensive exam in 2 weeks."

After much swearing and cursing under my breath, I said "sure".

I was told I had one of the highest scores in the written part. The oral part was just like this ... people asking me questions with vague definitions of various things. The example that sticks out to me was this one.

"Is the atomic radius of an neutral atom a strong function of Z".

Prof got annoyed when I asked them what "strong function" meant in this context; monotonically increasing/decreasing? Something else?

I do remember being asked a few questions I had no idea how to answer, so I basically started from first principles and hashed out approximations/calculations very quickly.

That was 29 years ago for me.

lol, what is a "strong function"? It is not the first time I hear physicist use this expression and I have no idea what it means.

It's asking whether Z dominates in terms of contribution to the result. AKA "Does knowing Z allow you to estimate the radius to first-order?"

That's how mathematics and baseball batting works. Far more misses than hits, even for the greats.

if you're batting, say, .500 in mathematics, you're not doing things that are hard enough. (Actually this is probably true even for .250, which would be a decent batting average in baseball.)

What makes you think he aced it? Reading through, he muddled through it -- couldn't say much specific, messed up some Galois theory, failed to define H^1 (Hardy space) correctly and instead confusing it with L^1. Nothing catastrophic, but not great at all.

I seem to remember hearing a story how he failed the first time and spent a while cooped up with Stein afterwards going over everything but maybe that was a garbled version of this?Or there was a written part?

Anyone know of a decent documentary on Terence Tao? Not finding much on Youtube.

He's a middle aged man. Doing a docu on him today isn't likely a good idea, he's still got a lot of life left to live. Numberphile on YouTube has some good stuff on him though.

https://www.youtube.com/results?search_query=numberphile+tao

Tao is the reason I recently started taking math classes again. I wanted to be able to at least understand some of what he talks about. :-).

Is this a type-up of handwritten notes, or did he type them originally?

This is part of the Princeton notes on the general exams, just an effort by grad students to support each other by sharing knowledge. It's not just helpful for them -- when I did my exams, I looked up every question they posed in algebraic geometry & my minor topic area and used it as a question bank, despite not going to Princeton :)

https://web.math.princeton.edu/generals/

In particular this would probably have been written up a few days after the exam - these are oral exams.

I took a similar exam at a similar program. We also had a repository of exam writeups like this. But I never got around to writing mine up - there were so many things I had put off until "after orals" that the last thing I wanted to do was relive them.

(Like Tao, I passed; like Tao, I muddled through large portions of my exam.)

This teenager's document is a great counter point to Tao's naive claim that he is not a native genius and that anyone can do what he does if they merely study as much as he did growing up.

Where does Tao claim that "anyone can do what he does if they merely study as much as he did growing up" ? I find that hard to believe.

Unless you have an actual citation, it's much more plausible to me that he said hard work is important in addition to raw talent.

https://terrytao.wordpress.com/career-advice/does-one-have-t...

Not exactly "anyone can do this," but the first paragraph in the above link:

"The answer is an emphatic NO. In order to make good and useful contributions to mathematics, one does need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other mathematicians, and think about the “big picture”. And yes, a reasonable amount of intelligence, patience, and maturity is also required. But one does not need some sort of magic “genius gene” that spontaneously generates ex nihilo deep insights, unexpected solutions to problems, or other supernatural abilities."

This is about whether any intelligent person can "do mathematics", not about whether anyone can be like Terence Tao, one of the most brilliant mathematicians in the world.

Nowhere in there does he claim he is not a native genius. Nor does he say that anyone can do what he does if they merely study as much as he did.

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