Committee: Peter Sarnak (Chair), Elon Lindenstrauss, Alireza Salehi Golsefidy

Real Analysis:

What's Lebegue density theorem (it's just Lebegue differentiation theorem on 
characteristic functions). How would you prove the Lebegue differention 
theorem? (I was stopped after I mentioned the Vitali covering lemma and the
Hardy Littlewood maximal function, though they pointed out that it's not
necessary to use the Hardy Littlewood function).

Prove that if E is a set in R of positive measure, E+E contains an interval.
(It follows from the fact that E has density points. They also made me give
a proof by convolving the caracteristic function of E with itself).

If f(x+y)=f(x)+f(y), what can you say about f? What if f is measurable? (I
knew that the answer was that f is linear but I suffered while trying to
prove it. I was finally pushed to the answer with a huge number of hints.)

Where does the Fourier series converge to the function? (I said at twice
differentiable points). Take a charecteristic function of an interval. Does
the Fourier series converge absolutely? (No) In what sense do they
converge?

Complex Analysis:

Take a simply connected region and remove some region from it's interior.
What can you say about it? (It can be mapped conformally onto an annuli
but I had no idea how to prove it). When are two annuli conformally
equivalent? Prove it? (My proof involved showing that a holomorphic function
on C\{0} bounded about 0 can be extended to 0. Sarnak then showed me how to
prove it using L2 norms.)


Algebra:
State the main theorem of Galois theory.

If A is a non-singular matrix over C, prove that there exists B such that
exp(B)=A.
What is Jordan canonical form?
What is rational canonical form? (I didn't recall the exact form so I derived
it with the usual modules over a PID theory).
If the traces of all powers of A are 0, what can you say about A? (Nilpotent).

What is the Jacobson radical? If R is a finitely generated algebra over a 
field what can you say about it? (I had no idea. In fact I don't even 
remember if this was the exact question. But the answer was that it's equal
to the nil radical).
Give an example of an Artinian ring. (I gave C[G] hoping to steer the
questioning towards rep theory which I knew better than non commutative
algebra but they didn't take the hint).
State the structure theorem for semisimple Artinian rings. (Artin-Wedderburn)

Number Theory:
Take Q(root(5)). What are the ring of integers? What is the class number?
What is the group of units? What is a fundemental unit? How do primes behave?
What is the Minkowski bound? Where do the n! and the n^n come from?

If K over L is Galois, are there primes in K that split completely in L?
(Chebotarev). Prove Chebotarev. (I sketched a proof using Artin L functions).
How would you prove the Meromorphicity of L functions (Brauer's theorem).
Chebotarev is unnecessary for the question, give a one line proof that
infinitely many primes split. (Because the zeta function of L has a pole at
1). Use that line of arguement to bound the number of primes of bounded
norm that don't split.
(We then had a long discussion about grossencharacters which I have managed
blocked out of my memory)

Representation Theory:
Given a compact Lie group, how would you construct a non trivial
representation? (Peter Weyl theorem. I indicated how the proof went and was
quickly stopped.)
How do you get all the irreducible representations of a compact Lie group?
(The fact that all of them are finite dimensional come from the Peter Weyl
thm. The finite dimensional ones correspond to points in the weight lattice,
which follows from the proof of the Weyl charecter formula.)
Representations of SO(3). Representations of sl3 (here I was slightly
embarrassed by my inability to draw an acceptable hexagon.)
Prove the conjugacy of maximal tori.