Aravind Asok
May 2000
Committee: Katz (Chair), Aizenman, Wee Teck Gan
Length: 4 Hours (Is that the longest yet?)

The exam began by Katz, Wee Teck and I clearing enough space for
three people to sit down and for me to be able to reach the board (which
was guarded by a boxes of books through the course of the exam).  Then,
Katz turned over a recycling bin to act as a coffee table.  Aizenman
walked in and the coffee table was promptly rattled and they spent several
minutes cleaning up the mess.

After the escapade at the beginning, the questions began.  We started with
complex analysis. (more or less)

A: Suppose I have a function f which is analytic on the upper half plane
and which is continuous along the boundary.  Suppose that f(x) is bounded
on the real axis between -1 and 1 by 5 and for |x|>1 is bounded above by
1.  What can I say about the f.

"Well, map it to the disc using a conformal transformation."

K: Since you brought it up, construct an explicit conformal
map between the upper half plane and the disc.

A: What is Liouville's thm.  Pf?

K: When can a function on the interior of the unit disc be represented by
a Poisson integral?

A: Suppose that g is some analytic function which is not necessarily
continuous on the boundary of the above region, but which actually
satisfies g = 5 on the real line between 1 and -1 and g = 1 on the
exterior of the region.  What can I say about the relationship between g
and the f above.  Again I know that g is analytic on the upper half plane.

A: What's your favorite thm in complex analysis?
"I don't know, I like the gamma function"

A: Ok, well, let's do this.
A: Suppose a is a real constant bigger than 1.  What can I say about
solutions of the equation e^z = z^n e^a.  Prove Rouche's theorem.

"What does that have to do with the gamma function?"
A: Nothing.

(I'm not sure if I remember this question properly, i.e. it could be
false as stated, I haven't checked)

A: Suppose f_n is a sequence of analytic functions which are convergent
pointwise on a neighborhood of the boundary of the unit disc.  Show f_n
are an equicontinuous family, that there exists a function f which is analytic
to which the f_n converge.

K: OK, Let's do some representation theory.

A: Aizenman began to ask some strange question about hearing the shape of
a circular drum.  Then doing some symmetry breaking.

K: Start by talking about the representation theory of D_8 (= Dihedral
group with 8 elements).  What are the irreducible representations?

K: Ok, let's talk about representation theory of O(2).  How do I get
representations?
(Induce from representations of SO(2) which is conveniently abelian)

K: How do I tell whether these representations are irreducible?
I mentioned something about Mackey's irreducibility theorem and he said
that's not what he wanted.
K: What about in the last century?
I guessed that he was driving at Frobenius Reciprocity.
K: Why don't you tell me what that is?
(Then we talked about the characters of induced representations, inner
product on characters, irreducibility of induced representations etc.  All
fairly computational)

A: How do the eigenspaces of the Laplacian (which are a priori
eigenspaces for the group 0(2)) degenerate when we break this symmetry?

K: I'm going to shoot your question with my revolver.  So you're asking
how each 0(2) representation decomposes as an irreducible representation
of D_8?

A: Yes.

K: Ok, go ahead.

K: Ok, talk about the Weyl Integration Formula.  What is the Haar measure
on SU(2).

K: Describe the representation theory of SU(2).

K: Ok, why don't you write down a formula for the character of a
representation of SU(2).

K: Characters are supposed to be orthogonal with respect to an inner
product on SU(2).  How can you get the Haar measure from this information?

K: What is the relationship between SU(2) and SO(3).  How can I use that
to obtain the Haar Measure on SO(3)?  What is it explicitly?

W: What is the Frobenius-Schur indicator.  (How can I tell if a
representation is real?)

(I guess that was the end of complex analysis/representation theory)

At 1:30 we took a break.. As we walked to the common room, Aizenman
mentioned something about the "Lemma of the Rising Sun".  He didn't get to
finish on the elevator ride down, but he finished his statement after the
break.

A: As it turns out, the level sets of a function with the properties
described in the first question can be described very explicitly.  They
turn out to be circle which pass through the two points where the bounds
change. The level sets at the integers look like a rising sun. What can
you say about the image of these circles when I map them to the unit disc?

K: Define the genus of a curve in every way that you know how.
(Geometric Genus, Arithmetic genus using Hilbert polynomial and Euler
characteristic, 1/2 dim of first homology group over the complex numbers)
"Is the curve complete non-singular etc?"

K: Sure.  Why are these equal?  State Serre Duality.  Define the Hilbert
Polynomial.

K: Compute the genus of the fermat curve in every way you can.  (Degree
genus formula, Riemann-Hurwitz, explicitly writing down a basis for the
space of holomorphic differentials using adjunction or "Manjul's Trick")

K: What does adjunction say in the case of the fermat curve?  How about a
degree d hypersurface.

We talked about adjunction for degree 4 hypersurfaces in P^3 and
then:

K: What are such hypersurfaces called (K3 surfaces)?

K: What happens if I know the number of points of the variety over a
finite field with p^r elements for all r?  What can I say?

K: Suppose I know that the variety has p^n+1 points over field of
characteristic p^n what can I say?
(The variety is P^n (i.e projective 1-space over a finite field with p^n
elements))  Do you know how to prove that?  (I didn't)  Do you know what
the Weil Conjectures are?

K: (Some comment about wild ramification).  Suppose that my equations are
defined over Z.  What can I say when I base change to various fields?
(For characteristic zero, I mentioned something about cohomology commutes
with flat base change and he was happy.)  How about other characteristics.
I really didn't have any idea.

K: State Riemann-Roch
"For curves or what?"

K: You just opened yourself up to a whole new line of questioning.

K: What is Riemann Roch good for?
"Proving the Group Law for an elliptic curve"

I made some comment about Weil Divisors and Cartier divisors being equal
and then.

K: What's the relationship between divisors and sections of invertible
sheaves?

K: What would Weierstrass say about elliptic curves?
"I guess that's a pretty obvious question."

K:  Good, talk about the p-function.  Define it, differential equation.
Poles etc.

A: How would you know that there should exist such a differential
equation.

K: What was the only thing Eisenstein knew about?  (In relation to the
differential equation for the P-function).

K: Well you said that the P-function has a pole of order 2 at the lattice
points.  What about its derivative.  How do you get a relation?

K: Give me a 30 second sketch of everything you know about Riemann-Roch in
higher dimensions.  (I gave a really quick definition of the intersection
pairing for divisors on a surface and then defined Riemann-Roch in terms
of the Euler characteristic of a divisor.)  What is this useful for?

K: I guess we should ask some algebra questions.

K: What is Dirichlet's theorem about primes in arithmetic progression?
What can you say about the density of such primes?

K: Calculate the Galois group of X^5 -2

(By this point, I was absolutely stupid)

K: (To Aizenman) Ok, ask some questions about real analysis and then we'll
be done.

A: Do you know what a point of density is?  (He then proceeded to define
it)  If I have a set of positive measure on the real line, show that it
has a point of density.

A: State the fundamental theorem of calculus.  What's the relation to the
above?

That was it.  Katz sent me out of the office.  I walked down the Hall to
talk to Jordan.  Jordan asked if I was done and how I did.  I told him
that I knew I had failed.  He said, "that's alright, sometimes it's
difficult to work under pressure."  Then, I heard the door to Katz's
office open and Katz peeped his head out.  He had a broad smile.  I
started walking towards him expecting to hear "You Failed!"  Instead he
said "Congratulations."  I said "What?" He said "You passed."  My response
was: "How!?"  He said "Your not the only one who forgets how to think
when they're under pressure."

All in all, it was a grueling experience.