Eric Naslund
May 2014
Committee: Peter Sarnak, Charles Fefferman, Sucharit Sarkar
Special Topics: Analytic Number Theory and Ergodic Theory

COMPLEX ANALYSIS

Fefferman: What can you say about an analytic function on 0<Re(f)<pi such that |f|\leq 1 on the boundary?

We begin our discussion by mapping this region to the unit disk, and how the map I constructed (with several hints) misses two points on the boundary (namely 1 and -1). This leads to a discussion of Hadamard's three-lines theorem, Hadamard three circles theorem and subharmonic functions.

REAL ANALYSIS
Fefferman starts by asking me to define the Fourier transform, and what this transform is good for. I explain that the Poisson summation formula may be used to derive the functional equation for the zeta function. Luckily for me, rather than skipping this proof (which I know fairly well) it is decided that we'll get back to a full proof of that later. In the mean time the Sarnak asked what can I say about the Fourier transform of an L1 function. I prove that it is continuous, and to show that it vanishes at infinity I use the fact that C_0^infty is dense in L1 so that we can move to a compactly supported infinitely differentiable function. From here, I sketch how the continuity of this nice function combined with the extremely fast oscillations when summed will yield a very large amount of cancellation. 

Sarnak: While it is nice to give the intuition, can you tell us a less complicated proof? 

I stared at the board blankly for about a minute. I had never actually seen a different proof of the vanishing at infinity, so I wasn't sure how to approach it. I then realized that integration by parts works immediately, and gave that answer.

Sarnak: Why did you take so long?

Me: I had never seen that proof before.

Sarnak, Fefferman and Sarkar are all quite amused by this, and until the end of the exam they continue to ask me whether I had really never seen that proof before. (I hadn't)

From this, we conclude that for a C_0^infty function, the fourier transform decays faster than c_kt^{-k} for any k.

Sarnak: Does it decay exponentially fast? 

(I was unsure)

Sarnak: What would happen to the inverse Fourier transform?

It follows that the inverse would have to be an analytic function, and Sarnak asks ìAre there compactly supported C^infty functions that are not analytic?î Fefferman jokingly interjects ìAre there _any_ compactly supported analytic functions?î (The answer is of course no)

ALGEBRA

Sarnak: Can you solve e^A=B for matrices A,B?

Me: Solve for which one, A or B?

The committee is quite amused by this, and Sarnak comments that I would make a good lawyer. 

Sarnak: Solve for A. 

I then explain what happens to Jordan blocks, etc.

ANALYTIC NUMBER THEORY & ERGODIC THEORY

Unfortunately, I didn't write this part down immediately after, so I forgot what happened.