John Stogin's generals -- May 2013
Committee: Klainerman (chair), Warren, and Gunning
Topics: Hyperbolic PDE and Riemannian Geometry
Total time: 1hr 45min
Note: For the questions I didn't answer right away, I tried to include
the hints I received so that you can have a better idea of what is
actually expected on the exam.
Warren and I were waiting outside Klainerman's office when Gunning
showed up and muttered something like, "let's pass him quickly and get
out of here!". Soon after, Klainerman showed up and let us in to his
office.
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I chose to start with Complex Analysis.
Gunning: Classify singularities of a holomorphic function on the
punctured disk. Provide examples. How does the principal part relate
to these singularities? [I wasn't sure if having infinitely many terms
in the principal part meant there was an essential singularity, so
Klainerman suggested I redefine essential singularity to be precisely
that.] What properties determine these singularities? Prove
Casorati-Weierstrass.
Gunning: State precisely the Riemann mapping theorem. Why does it not
apply to the entire complex plane?
Gunning: What is your favorite special meromorphic function? [gamma
function] What are its properties? How do you extend it? Can you plot
it along the real line?
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Then Complex Analysis turned into (elliptic) PDE theory
Klainerman: What properties of holomorphic functions can you conclude
from PDE theory? [Cauchy integral formula, maximum modulus principle,
reflection principle, Morera's theorem] Continue discussing the
maximum modulus principle. Does this generalize to other operators?
What is the statement precisely? [I stated the strong maximum
principle for elliptic operators.] Can you prove it? [I did, but
forgot a detail or two which they provided.]
Warren: Suppose you have a positive harmonic function on the ball of
unit radius such that the function takes values less than or equal to
1 on the boundary. Can you bound the oscillation of this function on
the ball of radius 1/2? [It didn't automatically occur to me that he
was referring to Harnack's inequality, so I started proving some
pointwise bound for the magnitude of the derivative instead. Then he
reminded me the meaning of the word oscillation and I realized he was
talking about Harnack. They asked me to sketch a proof.]
Klainerman: Talk about estimates for solutions to Poisson's
problem. [I mentioned the trivial estimate for the homogeneous H^2
norm and also the Schauder estimates. I think he wanted me to state a
more general Sobolev inequality.] What if you have \triangle u = dfdg
(in 2 dimensions) and f,g are in H^1? Is u in L^\infty? [umm...] Try
working it out. [I bounded dfdg in L^1 without too much trouble. Then
I assumed I could bound u in W^{2,1} by \triangle u in L^1 (which
turns out to be impossible) and wondered whether W^{2,1} embeds in
L^\infty. Since I didn't know, I was asked a few simple additional
questions.] Does the inequality (embedding W^{2,1}\subset L^\infty)
scale appropriately? [yes] Can you prove this for 1 dimension?
[yes--fundamental theorem of calculus] Ok. So repeat the proof for 2
dimensions. [Ok. so the embedding is true.] Now, are you willing to
bet that you can control u in W^{2,1} by \triangle u in L^1? [I said
yes.] Actually you can't. [oops. ok]
Klainerman: Ok. Same problem as before except what if you have instead
*(df\wedge dg). Can you conclude something then? [I observed this was
Wente's inequality from his lecture notes (so yes)] Can you prove it?
[I said I wasn't familiar with the proof. A smile formed on
Klainerman's face...] Ah--you should have come to class yesterday!
[Everybody laughed. He didn't seem to really care that I couldn't
prove it.] Ok. Let's move on to Algebra.
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Algebra
Gunning: What is the structure theorem for finite abelian groups? How
about for finitely generated abelian groups?
Gunning: State the fundamental theorem of Galois theory. What if the
extension is not Galois? [I gave the standard example Q adjoined
cube-root 2 and said its automorphism group is trivial. That was
sufficient.]
Gunning: Groups of order 15? State the Sylow theorems.
Gunning: What special forms can be used to show similarity between
matrices? [I stated Jordan canonical form and then my mind completely
blanked on what the second was called. (I knew this would happen at
least once during the exam!) Gunning said the word "rational" quietly,
and then I mentioned the rational canonical form.] Define them.
Gunning: How can you quickly tell if a matrix is positive definite?
[He wanted Sylvester's theorem, but I didn't know what that was.]
Ok. I'm satisfied. Let's do some Geometry.
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Geometry
Warren: With a lower bound on Ricci curvature, bound the volume of a
ball by the volume of a ball with twice the radius. [I didn't know how
to do this, so Klainerman suggested we do something more closely
related to general relativity.]
Warren: If sec<0 and dim=3, then show a stable hypersurface has
negative curvature. [I wrote down Gauss's equation, and worked it out
after much mumbling and a few hints. After I solved it, they asked for
a simpler observation which had me stuck for quite a while despite
their very straightforward hints: "how does /that/ expression [the
determinant of sectional curvature] relate to /that/ expression [two
terms in Gauss's equation] for a 2D hypersurface in an orthonormal
frame?" (they're equal)]
Klainerman: How about some general relativity? Tell us about the
initial value problem for the vacuum. [I began by describing initial
data.] What are the constraints? [I wrote down the constraints and
then began talking about the need to choose a gauge. I asked if they
wanted proof of propagation of the gauge and Klainerman said yes. In
the proof, I mistakenly said div(ric)=0, so they asked me to compute
what it actually is. I then clarified my statement (it is true in
vacuum, but I couldn't assume that yet.) When I wrote down Bianchi's
2nd identity, Klainerman said that was enough.] Step outside and we'll
decide. Just don't jump out of any windows or anything.
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After a minute or two, they stepped outside and congratulated
me. Interestingly enough, there didn't seem to be any real analysis
topics, and the "hyperbolic PDE" topic seemed a lot more like elliptic
PDE.
Advice: It seems there are generally two types of questions that are
asked. Some are very straightforward (eg. define Jordan canonical
form) and you should be able to answer most of these (Klainerman told
me this a while ago). But obviously you don't have to know all of them
(eg. I didn't know Sylvester's theorem). Then there are those which
require you to do some more serious thought, but the committee will
guide you through the parts where you get stuck.
Good luck!