Eduardo Duenez' Generals
Committee: P.Sarnak (Chair), D.Christodoulou, T.Hewett.
Date: 10-21-97
Time: 9:15 (actually 9:30)
Duration: Slightly over 2 hours.
Special topics: Analytic Number Theory and Riemannian Geometry.
Following Dan Grossman's advice, I arrive with this enormous plastic cup
(the "Cool Stuff" one) filled with water, and when Sarnak sees it the
first thing he says is "You must have a big bladder!".
The exam was scheduled to be held in Fine 314 but (to Steve Miller's great
disappointment!) Sarnak suggests that we move to someone's office,
so we go to Christodoulou's.
COMPLEX ANALYSIS (Sarnak and Christodoulou)
S: Draw one doubly connected region in the plane. Now another. Can they
be mapped conformally to one another? Prove that such a region can be
mapped conformally to an annulus (if no component of the complement in the
sphere is a single point...). Now what about triply connected regions?
Can they be canonically mapped to a circle with circular holes punched?
Cultural parenthesis in which Sarnak says that the Riemann Mapping
Theorem isn't actually Riemann's. He asks what the key step in the proof
is, so I say Montel's Theorem and he says Riemann actually used the
existence of a function attaining the supremum without proving it.
S: What is a Green function? Can you have a harmonic function on the
sphere with just a logarithmic singularity at infinity?
C: Say you have a Riemannian surface (i.e., Riemannian manifold of dimension
2). How can you connect it to complex analysis? (conformal coordinates
give it a complex structure, so it becomes a Riemann surface.)
S: State the theorem of uniformization for Riemann Surfaces.
C: What would uniformization say in connection with the Riemannian
surface above? (I thought this was very vague.)
S: More specifically, say you have a Riemannian surface, diffeomorphic to
the sphere, with a metric g. What does uniformization let you conclude?
(we can map it conformally to the Riemann Sphere and pull back the
sphere's metric to find a metric g', conformal to g, such that the
curvature of g' is identically 1.)
At this point our host (Christodoulou) leaves the office and doesn't
return until I'm almost done with algebra, which takes maybe half an hour.
ALGEBRA (Hewett... also Sarnak)
H: Do you know the statement Weddenburn's theorem? (any finite
division ring is a field.) Prove it.
H: Now give an example of a division ring which isn't a field
(Hamilton's quaternions). Define the quaternions. Why are there inverses?
S: What is the group of unit quaternions topologically? What does it
have to do with SO(3)? Prove a finite subgroup of the multiplicative
group of units of a field is cyclic. Can a polynomial over a division
ring have more roots than its degree? (there are eight fourth roots of
unity in the quaternions.)
H: Classify all groups of order eight. Now find the Galois group of x^4-2
over Q.
REAL ANALYSIS (Sarnak and Christodoulou)
C: Define the total variation of a real-valued function on a closed
interval. What can be said about differentiability of such a function?
S: Say that F defined on [a,b] satisfies F(x)-F(a) = int_a^x f(t)dt for
an L^1 function f. Is F of bounded variation? What is its total
variation? What is the relation between functions of bounded variation
and monotone functions?
C: If a function is of bounded variation on [a,b] is it necessarily equal
to the integral of its derivative? (assuming the latter exists a.e.!) (I
say Cantor's function is a counterexample, and they make me construct it).
S: What is special about functions that DO equal the integral of their
derivative? (absolutely continuous). State the Radon-Lesbesgue-Nykodym
theorem (for complex measures on R, with respect to Lesbesgue measure).
What does mutual singularity mean? Give an example of a measure that's
mutually singular to Lebesgue measure (the one associated to the Cantor
function). How about an easier example? (a point mass).
He then asks if I have ever thought about the Fourier transform
of the Cantor function. I say no and he says it is unfair to ask me that
question if I have never thought about the matter, but that the answer is
neat. He also asks me what the Hausdorff dimension of the Cantor set is.
S: What kind of set is the following?
{x in R | For infinitely many positive integers q there exists a
rational number a/q such that abs(x-a/q) < exp(-q)}
Sarnak starts to seem hurried, because he has a seminar at noon, so he
asks Christodoulou to do geometry.
RIEMANNIAN GEOMETRY (Christodoulou... and Sarnak)
C: What is the exponential map? How do you know geodesics exist? Why is
the exponential map defined on a neighborhood of the origin of Tp(M)?
What are normal coordinates in a neighborhood of a point?
What does the Taylor expansion of the (components of the) metric
around the point p look like? (with respect to normal coordinates.) What
does this say that happens to the lengths of vectors in T(TpM) upon
exponentiation?
S: State the Gauss-Bonnet theorem (first for two-dim Riemannian
manifolds, the he asks if I know a generalization and I state it for
arbitrary even-dimensional Riemannian manifolds). What is that form that
you are integrating? (the Gauss-Bonnet form). Describe it explicitly (I
write it down in terms of Cartan's curvature two-forms). What is the
Euler Characteristic of a compact manifold? How can you express it in
terms of Betti numbers?
At this point Sarnak is more than eager to finish as quickly as possible,
so we turn to
ANALYTIC NUMBER THEORY (Sarnak, Sarnak, Sarnak!)
Talk about the proof of the Three Prime Theorem. Where is N odd
used? What is the key step in proving that the minor arcs contribute a
smaller amount? (Vinogradov's estimates for the generating function
S(N)). What is the really powerful idea behind Vinogradov's method? What
happens if you try to adapt the proof to prove Goldbach's conjecture?
Here Sarnak decides to stop and they ask me to go out, only to be
summoned again shortly thereafter and have Sarnak shake my hand
enthusiastically after telling me I pass. I pick up the laaarge cup whose
contents I never touched, and leave after thanking them.