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

Terry Tao's generals My examiners were Stein, Kleinermann and Rudnick. (Kleinermann came in about five minutes late). The first questions 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, essentially. Give examples of singular integrals (Calderon-Zygmund). What theorems do you know about them, how to prove boundnedness. (I stated a T(b) theorem, which nobody seemed very acquainted with, and said with some brief explanation that knowing boundedness on L^2 - L^2 was sufficient to prove L^p-L^p for 1

<\infty).

Since my T(b) theorem was martingale based, Stein asked me to give a definition of Martingales. I gave the sigma algebra, purely measure theoretic version, but stumbled a bit on the definition of conditional expectation. I think Stein was satisfied by my explanation of conditional expectation at the end, though.

Stein then asked me what other boundedness theorems I knew. I couldn't think of any concrete ones off hand, 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.

Some questions about the boundedness of the Hilbert operator, an easy way of proof. I said the Fourier transform was bounded, and that should do it for L^2 - L^2. They seemed to be hinting at generalizations, and eventually I clicked on to Riesz potentials, but again I didn't know enough results, only that they were bounded. (For example: if \Delta u = f and f is in L^2, what do you know about u? I guessed it was in some sort of Sobolev space). I did state though that all second derivatives of u would have L^p norms bounded by that of the Laplacian. Kleinermann asked me for an elementary explanation of this, but I couldn't think of one.

A couple definitions of H^1. I got the atomic definition right, and the Riesz potentials, but got the limit of harmonic-functions definition wrong. And I had no idea how to prove any of them were equivalent.. so they abandoned this after a while, but not until they made me realize that the latter definition I had gave was actually that of L^1, not H^1.

Then they asked me about fundamental solutions. I sluffed 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.

Kleinermann asked me about the connections between PDE's and singular integrals, but I didn't know any more than vague generalities.

Then they switched to Analytic number theory. Surprisingly, I fared much better on this topic!

Rudnick (mainly) asked: give several forms of the Zeta function, give one that allows analytic continuation (I used theta functions), prove the modularity of the theta function, and thus derive (give or take a number of pi's and Gammas) the functional equation. What do you know about the poles, zeroes. (I gave the De Vallee-Pouisson (sic?) zero-free region). State a form of the prime number theorem. Why li(x) instead of x/logx? Give an explicit formula for \psi(x). 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 an elementary proof of the infinitude of primes of the form 4n - 3. 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.") I gave the basic run-through of how it is sufficient to prove L(1,\chi) is non-zero. (With Rudnick's help, this was very smooth.) 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.) 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 seemed satisfied.

They asked me what else I had prepared, and I said some stuff on the circle method and the large sieve. So Rudnick asked me a few desultory questions on the sieve, and I stated one formulation, but didn't know of any useful applications.

Then Rudnick asked a few algebra questions only and it was over. Talk about Galois theory.. construct a field extension of order S_n. (polynomials over symmetric polynomials). Construct one of order A_n. I got the index and the order mixed up, but Rudnick corrected me, and so eventually it came down to finding a non-symmetric polynomial whose square was symmetric. I couldn't guess it, so Rudnick said "product of differences", and I said, "Oh, the Pfaffian" and they were satisfied.

"What else do you know about A_n?" (I said it was simple for n>=5). 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 professors talked among themselves whether the group of order 168 (which I mentioned) was sporadic.

After this, they decided to 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 2 hours.