What I have here is an account of an exam taken by Terence Tao,
one of the world's best mathematicians, while he was in graduate school.
What's interesting about this exam is that Terence Tao nearly failed. This Close call could have
thwarted his math career. But instead, he says that this near failure is what propelled him
to succeed. It was the results of this exam that made Terence Tao stop playing so many video games,
and knuckle down and try harder. It was the Princeton general qualifying exams,
which Terence took at age 18 after a year of graduate work. At the time, around 1993,
the Princeton graduate classes didn't have any homework or tests. And this exam was the one
major thing that you needed to pass in order to proceed with your PhD. It was an oral exam,
kind of like an interview where three professors directly asked questions.
Terence Tao is one of today's greatest mathematicians and has won many awards
including a Fields Medal. On this channel before I have covered an exam that he aced
as a child prodigy, so I was a little surprised to hear that he had ever struggled with an exam.
We know what questions Terence was asked, thanks to a tradition at Princeton of the
students writing up their memory of the exam afterwards. This serves as a useful tool for
other students wanting to know what to expect. This is what Terence wrote after his exam.
My examiners were Stein, Klainerman and Rudnick. Klainerman came in about five
minutes late. The first questions they asked were about harmonic analysis. They
asked me what I knew. And I said, basically singular integrals and functional analysis,
no real harmonic theory at all. So they examined me on singular integrals instead.
For every math student, the general exam will draw questions from five topics,
real analysis, complex analysis, algebra, and then two topics of the students choice.
Probably what they want to pursue their PhD in. Terence's special topics were harmonic analysis
and analytic number theory. Give examples of singular integrals (Calderon-Zygmund),
what theorems do you know about them? And how do you prove boundedness? I stated a T(b) theorem,
which nobody seemed very acquainted with, and said some brief explanation that knowing
boundedness on L2 was sufficient to prove it for LP for P between one and infinity.
Harmonic analysis is the representation of functions as waves, and includes Fourier
transforms. The questions that Terence was asked right off the bat are advanced. And I
don't understand the answers, but that's because they're on very specific topics.
You could make it all the way through a math undergrad and PhD without ever encountering
Calderon-Zygmund results of singular integrals. These topics are what Terence
has told the examiners he is prepared for, and so that's where they're guiding the questions.
Since my T(b) theorem was martingale based, Stein asked me to give a definition of martingales. I
gave the sigma algebra purely aeasure theoretic version, but stumbled a bit on the definition
of conditional expectation. I think Stein was satisfied by my explanation at the end,
though. Stein then asked me what other boundedness theorems I knew, I couldn't think of
any concrete ones offhand. But I mumbled about a convolution operator being bounded if its Fourier
transform was bounded, and had some smoothness condition. But I couldn't recall any details.
It's here that we see things starting to unravel for Terence,
and each step going forwards, he's being asked to demonstrate a complex idea,
but chokes up on the details and in providing concrete examples. Terence had studied harmonic
analysis for his master's thesis, and later said that since he felt it was his strong suit,
he only spent a few days reviewing analysis. This T(b) theorem was part of his master's thesis,
so he does okay with questions about it. But as soon as they move away from it,
his lack of preparation is starting to show. There were some questions about the boundedness
of the Hilbert operator and an easy way of proof. I said the Fourier transform was bounded, and that
should do it for L2. They seem to be hinting at generalisations and eventually I clicked on
to Riesz potentials, but again, I didn't know enough results, only that they were bounded.
With this being an oral exam. We see how different it is to a printed one. There are no set questions
and the examiners are coming up with questions that respond to Terence's answers. This allows
them to really find the depth of his knowledge in topics that he's supposed to be prepared for.
The format can be pretty intimidating. At age 18, Terence would have been a few years younger
than his fellow students. and had found math easy ever since he was a young kid. With his natural
interest and talent for the subject, he had never had to try too hard studying for an exam in the
past. And in earlier studies could get by with cramming the textbook a few days before the exam,
and spending the rest of his time just playing with the ideas that he found most interesting.
According to an article he wrote years later, looking back at this exam, he said
that he had made it all the way to Princeton, without developing systematic study habits.
I was asked for a couple of definitions of H1, I got the atomic definition, right, and the Riesz
potentials, I've got the limit of harmonic functions definition wrong, and I had no idea
how to prove that any of them were equivalent. So they abandoned this after a while, but not until
they made me realise that the latter definition I had given was actually that of L1, not H1.
Instead of treating the qualifying exam as a top priority. Terence said that he spent
an embarrassingly large fraction of his time as a graduate student messing around online,
which might be the most relatable thing I've read about him yet. He discovered the World Wide
Web in his first year, and played computer games until late at night in the Graduate
Student Dormitories computer room. While his fellow classmates did months of preparation,
Terence said that he only did about two weeks worth, he might have been used to getting away
with doing less work than those around him. But natural talent can only get you so far.
Then they asked me about fundamental solutions, I sloughed the exponents in
the denominator a few times. So they began asking me about homogeneity. Eventually,
with a lot of prodding, I got the exponents of the Newtonian and Cauchy potentials correctly.
Here, the examiners are asking easier and easier questions to try and get him to a point where he's
able to give a satisfactory answer. And they're walking him through a derivation of finding the
fundamental solutions of the Laplacian. Terence later described this as painful.
To be clear, for me, none of this is easy, and barely any of the questions are even
comprehensible. But I suppose it's considered easy for someone who has taken the right coursework.
Klainerman asked me about the connections between PDEs and singular integrals, but I didn't know
any more than vague generalities. Terence later said that he had enjoyed playing with harmonic
analysis for its own sake, but had never paid much attention to how it was used in other fields,
such as PDEs. This was the all is lost moment. But Terence's luck was about to change.
Then they switch to analytic number theory. Surprisingly, I fared much better on this topic.
In a stroke of luck, only one of the three examiners had an extensive background in
number theory, but had mistakenly thought that Terence's second special topic was algebraic
number theory instead of analytic number theory. And after realising his mistake,
he wasn't able to use any of the difficult algebraic number theory questions which
he had prepared, because that wasn't what Terence needed to be tested on. So instead,
Terence was given just the standard analytic number theory questions,
which weren't that difficult. Rudnick mainly asked give several forms of the zeta function,
give one that allows analytic continuation, prove the modularity of the theta function
and thus derive the functional equation. What do you know about the poles and zeros?
Analytic number theory is using the methods of analysis to study number theory. The Riemann zeta
function is perhaps the most famous example, where complex analysis is used to study the distribution
of prime numbers. State a form of the prime number theorem. Throughout I didn't have to
back up anything with messy calculations, Rudnick always stopped me when I tried.
Talk about primes in arithmetic progressions, give have an elementary proof of the infinitude
of primes of the form of four n minus three. These were more standard questions which Terence
had prepared for and so breezed through them. The prime number one is funny because Terence later
went on to prove quite a famous theorem about prime number progressions, which are a series
of numbers which are equally spaced. For example, three, seven and 11 constitute a progression of
prime numbers, where the gap between each number is four. Terence Tao and Ben Green proved that
it is always possible to find somewhere in the infinity of integers, a progression of prime
numbers of equal spacing and any length. What algebraic tools are used in Dirichlet's proof of
infinitude of primes in arithmetic progressions? It took a while before I understood, Oh, you want
me to talk about characters? And 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 realised 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 sign
log formulas. Then they made a comment that this would put 1000s of people out of work,
or something like that. Anyway, I stated what the class number was vaguely, but didn't give
out an explicit equation. Again, Rudnick waved me off when I tried this, and they seem satisfied.
It seems like here that Terence has accidentally insulted anybody who uses
the class number theorem. But when he said it was disgusting, it was because he was
thinking about writing out some other messy expression. A nice, awkward miscommunication,
but maybe they all laughed about it. Then Rudnick asked a few algebra questions only and it was
over. Talk about Galois theory. What do you know about A_n? What other simple groups do you know?
The monster but I backed down quickly, saying I knew not how to define it. Then the discussion
meandered a bit as the professor's talked among themselves about whether the group
of order 168 which I mentioned was sporadic. After this, they decided to barely pass me,
though they said that my harmonic analysis was far from satisfactory. They didn't ask
any real or complex analysis, but I guess from my handling of the special topics, they decided
that wasn't necessary. Besides, we were almost getting snowed in. The exam lasted two hours.
After the exam Stein, who was his thesis advisor, told Terence that he was disappointed,
that his letters of recommendation had been strong, and this wasn't what he was expecting.
Terence said that this moment left him in a state of shock because it was the first time he had done
poorly on an exam that he had been genuinely interested in doing well in. It's not the first
time he had done poorly on an exam. In fact, he had failed two of them back in undergrad. But one
was for a Fortran class where he had refused to learn it on the basis that he already knew how to
program in BASIC. And the other one was a quantum mechanics class, where Terence ignored the warning
that he would have to write an essay about the history of the subject during the final exam,
and he had to be escorted from that final exam in tears. He said that the Princeton general
exam was a wake up call and a turning point in his career, he began to take studying and
his classes more seriously, because he had finally seen consequences for not studying.
I listened more to my fellow students and other faculty,
and I cut back on my gaming, I worked particularly hard on all the problems
that my advisor gave me in the hopes of finally impressing him. In retrospect,
nearly failing the generals was probably the best thing that could have happened to me at the time.
Being told that his harmonic analysis was unsatisfactory is kind of funny now,
because Terence Tao was awarded the Fields Medal for his contributions
to partial differential equation,s combinatorics, harmonic analysis,
and additive number theory. Either he did become a lot better at harmonic analysis,
or these exams are not great at and measuring one's ability to get a Fields Medal. Reading
about Terence Tao struggles with this exam is i'm sure comforting to anyone who has struggled with a
test. It's a reminder that tests themselves don't say too much about your potential. But instead,
what's more important is what you do after getting a bad result. It also shows that really dedicated
and intelligent people can sometimes mess up opportunities by overestimating their own ability,
or assuming they can coast by purely on natural talent. Terence Tao is a top tier mathematician
because of his creativity and problem solving and in coming up with new mathematics. These aren't
skills that are being tested on a qualifying exam, but experience and being well read also
matter. Terence himself says that one does not need a genius gene to become a mathematician.
Talent is important of course, but how one develops and nurtures it is even more so.
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