
Terence Tao's Raw Notes After His Princeton Comprehensive Exams (1999) - breck
https://web.math.princeton.edu/generals/tao_terence
======
hycaria
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...](https://www.ams.org/about-
us/LivingProof.pdf#%5B%7B%22num%22%3A351%2C%22gen%22%3A0%7D%2C%7B%22name%22%3A%22FitH%22%7D%2C648%5D)

~~~
kaitai
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.

------
dpflan
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!

~~~
robinhouston
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.

------
mb_72
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.

~~~
matz1
Whats the relation of someone complaining and tough in this context ?

~~~
hojjat12000
It's tough to be smart, young, and capable. Because people will call you smart
arse and not like you.

~~~
matz1
If you are smart then you are smart enough to ignore that.

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

~~~
waterhouse
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"

~~~
defen
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.

~~~
apricot
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.

------
nerdponx
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.

~~~
kodz4
"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”

~~~
nerdponx
Is this a quote from something?

~~~
majjam
Francis Bacon:
[http://www.psy.gla.ac.uk/~steve/best/BaconJohnson.pdf](http://www.psy.gla.ac.uk/~steve/best/BaconJohnson.pdf)

------
svat
This is part of the Princeton math department Graduate Students' Guide to
Generals:
[https://web.math.princeton.edu/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](https://web.math.princeton.edu/generals/bhargava_manjul)

~~~
urmish
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.

------
nisuni
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.

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

------
dang
A thread from 2011:
[https://news.ycombinator.com/item?id=2771031](https://news.ycombinator.com/item?id=2771031)

~~~
tw1010
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](https://news.ycombinator.com/item?id=2772019)

~~~
j7ake
He was probably teaching the linear algebra course.

------
7402
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.

~~~
ska
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.

------
a1pulley
A friend from my Caltech undergrad did his PhD with Terence Tao. In harmonic
analysis :-)

~~~
iamcreasy
Any stories?

~~~
a1pulley
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 :-)

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

------
audiometry
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?

~~~
ljcn
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...](https://www.snopes.com/fact-check/the-unsolvable-math-problem/)

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

~~~
hkmurakami
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.

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

~~~
closeparen
The math pirate!

------
jdoliner
> 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](https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem)

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

~~~
MRD85
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.

~~~
hpcjoe
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.

~~~
enriquto
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.

~~~
qwhelan
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?"

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

~~~
Balgair
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](https://www.youtube.com/results?search_query=numberphile+tao)

------
jason_slack
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. :-).

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

~~~
kaitai
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/](https://web.math.princeton.edu/generals/)

------
lonelappde
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.

~~~
umanwizard
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.

~~~
amch
[https://terrytao.wordpress.com/career-advice/does-one-
have-t...](https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-
genius-to-do-maths/)

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."

~~~
BeetleB
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.

