Akshay Venkatesh's generals. Committee: Sarnak (chair), Fefferman,
Katz. 10am, May 5, 1999
They asked me what topic I wanted first; I said algebraic
geometry, to which Sarnak replied that we should start with the
basic topics. So we started with complex analysis.
Complex analysis.
Suppose that I have a complex function whose growth is bounded by
sqrt(|z|) or |z|^20 - what can I say about it?
State uniformization for Riemann surfaces, proof for domains in
the plane (i.e., Riemann mapping theorem). How did Riemann do
this? Do you know what a Green's function is? How could you do it
with Green's functions?
Infinite product for sin(z); relation to Gamma function. Talk
about Hadamard factorisation, order, genus and so forth. What is
the order of the Riemann zeta-function?
Algebra. Finitely generated abelian groups. Are the summands
uniquely determined? How would you determine the invariants?
Relation to Jordan canonical form. Katz stopped me very soon.
What's the structure of a finite subgroup of the multiplicative
group of a field?
(K) Give me a thirty second sketch of Galois theory. Is normality
transitive?
Do you know what a local field is? Define inertia group,
decomposition group...what is the decomposition group at a
unramified prime? What is the structure of the inertia group?
(This I couldn't quite remember.)
Then there was some discussion of Dirichlet's theorem in primes in
arithmetic progression, key steps in proof, and somewhere along
the line I was asked to talk more about cyclotomic fields.
Talk about ideal classes (proof of finiteness) and Dirichlet's
unit theorem.
State Wedderburn's theorem. I stated the one about division rings.
They then asked me to state the `other' one.
Then we talked about group algebras...complete reducibility,
structure of the group algebra over C and what this tells you
about representations; there was some question about S_n that I
couldn't answer.
After that, we discussed compact groups and the Peter-Weyl
theorem. While talking about Peter-Weyl, I mentioned compact
operators, and so we moved into what I suppose was real analysis:
Real analysis. What is a compact operator? Give some examples.
Relation to finite rank operators.
Fourier transform; why does it map Schwarz space to itself? Is
there some L^p on which it behaves particularly nicely?
How about the behaviour of Integral(f(x) exp(i lambda g(x)))? Katz
asked what the name is for such results. I didn't know; he said it
starts with "stationary," so I of course said stationary phase...
What's the Poisson sum formula? I was then asked about the
behaviour of Sum(Exp(-n^2 z)), for z small; we talked about theta
functions, their transformation properties and the relation to the
functional equation of the zeta function.
Talk about modular forms, why is the space of modular forms of a
given weight finite-dimensional?
How can I use theta-functions to work out the number of ways of
representing an integer as a sum of squares? (I didn't know,
although of course I remarked it was a coefficient in a power of
the theta series.)
Why is every Riemann surface an algebraic curve?
Can I prove the existence of meromorphic functions on a Riemann
surface? (No!)
What is a measurable set? Can you exhibit a Lebesgue set that is
not Borel? I said I couldn't, but by counting arguments you can
show they exist. We talked about this for a while, and then took a
break.
Algebraic Geometry (all qns from Katz unless noted)
(S) What is the Nullstellensatz? What are the maximal ideals? What
about over a nonalgebraically closed field? Can you prove the
Nullstellensatz? (No! - I felt quite bad about this, but they
didn't mind.)
What is the Jacobian of a curve? I defined it over C and
mentioned Abel-Jacobi. (There ensued some discussion of how to
pronounce "Jacobi.")
What is an abelian variety? Is the Jacobian an abelian variety?
(Yes.) I mentioned that any abelian variety over C is a complex
torus and was asked: Are complex torii abelian varieties? (I said
yes and committee disagreed.) I remembered that they're usually
not algebraizable, and may not admit any meromorphic functions.
Why is the Jacobian algebraizable for an algebraic curve?
Tell me about elliptic curves in thirty seconds. Do you know the
bound for the number of points over a finite field? (Not
really...I made a guess which wasn't quite right, but Katz didn't
mind and moved on.)
What is the genus of a curve? What is the arithmetic genus? Why
are they the same? State Serre duality. Can you sketch a proof?
(No, but...)
Can you prove it for projective space? (I talked about Cech
cohomology of the O(n)). What's the canonical bundle on P^n?
Prove it.
Question about the vector bundles on P^1 and P^n.
Say I have an elliptic curve over a not-necessarily algebraically
closed field. Does it necessarily have rational points? How about
over a finite field? Construct a plane curve over a finite field
with no rational points. (These questions I answered with guesses
or with help.)
You talked before about class groups and Jacobians. What is a
context in which these are related?
What is the Hilbert polynomial? How do you use it to compute the
genus of a curve? Compute the Hilbert polynomial of the Fermat
curve.
What do you know about surfaces? (I mentioned intersection
theory.) Katz asked me some questions about that.
Can you tell me about the relation between algebraic curves and
coding theory? I said no, and we then moved on to the next topic.
Representation theory (Always Sarnak unless noted.)
Compact Lie groups, maximal torii. What do you know about maximal
torii? (Their union is the group and any two of them are
conjugate.) Prove that any two of them are conjugate. Weyl
character formula, Weyl dimension formula. Apply to SU(2).
(K) What is the relation between SO(3) and SU(2)? What is the
relation between the irreducible representations?
(K) How do I get the n-dimensional representation of SU(2) from
the 2-dimensional one? (Symmetric power.)
(K) Some question about spherical harmonics and representations of
SO(n); I said I had no idea.
Define induced representation; state Frobenius reciprocity.
Then we moved on to SL_2(R):
What are finite dimensional representations? Prove they are never
unitary. What is Weyl's unitary trick? Are the representations
self-dual?
How do you construct other representations? (Induce from
parabolics.) What are these called? (principal series) Talk about
unitarizability. What's the norm for complementary series
representations?
What does the Plancherel formula say for a compact group? Which
representations occur in the Plancherel formula for SL_2(R)? He
then wanted to know why, which I couldn't really answer. I said
the tempered ones should occur, and he noted that this is true for
a semisimple Lie group but not for R! Then we talked awhile about
matrix coefficients and their growth.
Which Lie groups have discrete series? Does SL_3(R) have discrete
series?
(K) Work out the Haar measure on SL_2(R) - explicitly, in terms of
the matrix coordinates. (Previously I had done it in the Iwasawa
decomposition.) I worked out the measure for GL_n and then had no
idea what to do for SL_2(R). Then Katz gave me the following hint:
(K) What are the holomorphic differentials on the Fermat curve.
What does this have to do with previous question? [!] (I had no
idea...with much help from Katz and Fefferman I worked out the
answer, I was really quite confused at this point and was having
difficulty working out things like dx ^ dx... )
Comments:
I found myself absolutely unable to think through things at the
board, so my answers were a combination of knowledge and
guesswork, but they were very nice and didn't mind. The exam
lasted about 2 1/2 hours.