Phil Isett
General Exam : 20 Oct 2009, 1:30 PM
Subjects: Partial Differential Equations, Analytic Number Theory
Committee: Klainerman (chair), Sarnak, Sorenson
When I entered Klainerman's office, the committee was discussing Obama's recent Nobel Prize
award. Sarnak informed me that my exam would be in politics, but it later turned out that
this comment was not sincere and that I would instead be subjected to a mathematics exam.
They then asked which subject I would like to do first, and I chose
---- Algebra ----
[Sor] Let's look at SL_2(F_3). How many elements are in that group?
- I tried something dumb, but got a hint. There are 24... (3² - 1)(3² -3)/2 was not
satisfactory.
[Sor] What group is it?
- This is a hard question when the answer is not "SL_2(F_3)". Someone mentioned that 24 =
4!. The committee guided me to consider a permutation representation of SL_2 on P^1 before I
could do anything stupid. But the action is not faithful.
[Sor] Perhaps you should consider the center of the group?
- So SL_2 is not S_4, but PSL_2 is A_4.
[Sor] Define A_4.
- I talked about permuting the edges of an oriented parallelogram in an exterior power, but
Sarnak didn't seem to like this definition much and wanted me to talk about cycle
decompositions.
[Sa] The second you see a three in there, you know it's a hard problem. Here's something
easier; take a prime p and look at F_p*. What kind of group is that?
- I offered a proof it was cyclic by counting elements of a given order, but could not
complete it fast enough and was led to use the classification of finite abelian groups, and
then to talk about finitely generated abelian groups.
[Sa] How would you prove that classification? What can you say about the uniqueness?
How would / when can you solve e^A = B ?
[Kl] For a linear ODE with constant coefficients, how would you solve it?
- I talked about Taylor expanding log(x) and e^{tx} about the eigenvalues in both these
applications (more generally you reduce an analytic function modulo the minimal polynomial).
I managed to avoid talking about Jordan normal form.
Sarnak suggested Sorenson ask me some representation theory before he had to go teach, but we
had already spent a lot of time on algebra, so instead we moved on to
--- Analytic Number Theory ---
[Sor] Do you know what zeta(0) is?
- No, so I spent some time figuring it out modulo a factor of 2.
[Sa] Can you say anything about zeta(2k) ?
How do you define a Dirichlet character?
How would you show that [ \sum \chi(n)/n ] is not zero ? How do you use positive coefficients?
Write down the class number formula ? I want to see if you know what all the terms mean.
How would you count the ways to write N as the sum of 10 cubes? (What is a crude way to
predict the order of magnitude?)
Why is the generating function big on the major arcs? What is the key ingredient in
estimating the minor arcs? What does Weyl's inequality say in this case?
What do you do for 3 primes? What goes wrong for two? Give a lower bound for the L^1 norm of
a generating function (with the hint to use the L^4 norm).
--- Partial Differential Equations ---
[Kl] Tell me about characteristics. What are they useful for? (Solving first order PDE,
propagation of singularities and characteristic IVP)
[Kl] For that equation you just wrote down [first order semilinear], what are the
characteristics? Derive from first principles the characteristics of Box = m^{ab}D_{ab} .
What is the name of that equation? (the Eikenol equation) How would you solve it? What is
the geometric meaning of the characteristics? (Null geodesics)
What is your favorite way to solve the Laplace equation? (Hilbert space methods)
Then after you have the solution, what do you do? (Regularity) How?
What about the boundary?
What is the most general theorem you know concerning existence for symmetric hyperbolic
systems? How does the proof go? (Do the energy estimate)
[Sa] What is a wavefront set?
They had me compute a few simple examples
[Kl] What is the fundamental solution to the wave equation?
[Sa] What is the difference between even and odd dimensions?
- (I noted that dimension 1+1 was an exception)
[Kl] Let's say we take the fundamental solution, and then we look at t = 1 and use the
fundamental solution as data, then evolve. Why do you get the fundamental solution, rather
than null geodesics emanating from the surface of intersection?
- I said something about the data involving the normal derivative, but he was looking for an
explanation involving the wavefront set.
[Kl] What is your favorite Strichartz estimate?
- I wrote down the general estimate for the wave equation with a right hand side.
[Kl] Do you know the picture when n = 3? Do you know what happens at the endpoint?
Remarks: I knew how to answer some of the questions. The PDE section is actually a
compilation of a diagnostic exam and the official exam; doing a practice exam with [Kl] was
extremely helpful. They decided rather early on to skip complex and real -- I guess because
of the topics.
Reading: For basic PDE, at least know what is in Fritz John's book and things in Klainerman's
essay for the Princeton Companion. Selberg has some very good notes on nonlinear wave
equations. If you're going to read Gilbarg and Trudinger, it's a good idea to start with
quasilinear equations. More generally, Tao has many good lecture notes and Klainerman's
analysis notes are very useful.