Oral Transcript for Justin M. Curry
Major: Algebraic Topology
Minor: Partial Differential Equations
Committee Members: Robert Ghrist, Jonathan Block, Joachim Krieger
RG: "I feel as if we are the Council of Gods picking on a mortal,
toying with his or her fate until he or she does something vaguely
heroic."
JB: "Yes, but it is customary to let them choose which trial to first endure."
JC: "I brought a coin. I suppose I could let it be decided this way...
on second thought let's start with PDEs."
JK: "Suppose you have a bounded open subset U in R^n, and you have a
sequence of harmonic functions bounded above by some constant in the
L^{\infty} norm. What can you say?"
JC: "They converge locally uniformly to a harmonic function."
JK: "Right! How do you show that?"
I waffle for a little bit and it also becomes apparent I forgot to say
that a *subsequence* is actually want converges loc. uniformly.
Eventually, I realize the tool is Arzela-Ascoli. We figure out that
the appropriate boundeds on derivatives are needed. I provide the
inequality that uses the distance from the boundary, which is used to
prove harmonicity of harmonic functions. JK asks if I can give an idea
for the proof. I use mean value property. Everything I say, JK asks me
for the idea behind the proof. Green's representation comes up
naturally.
RG: "Ooo. I have a question about harmonic functions."
JK: "OK. Be my guest."
RG: "Suppose I am a freshman biology major. How would you explain
harmonic functions to me?"
JC: "Uhhhh... It solves laplace's equation"
RG: "Oh, Prof. Curry I don't understand all these multiple
derivatives. Can you *explain* it to me better?"
JC: "uhhhh...."
RG: "I'll give you a hint, you already have something on the board
that can help."
I pick out the mean-value property both in the integral around the
boundary and integral over the ball versions.
JC: "It's a function where the value at a point is encodes the
integral around the boundary."
JK: "That doesn't characterize it completely, but you're close."
RG: "Yeah I just learned integrals, can you make it simpler?"
JC: "In this case it's an average integral."
RG: "Oh good. I remember averages from high school."
Eventually JK and RG guide me to the following answer. I am told to
say it with confidence.
JC: (loud) "The value of a harmonic function on a set U is determined
by the average value of the function over any ball B centered at the
point where B is inside U."
RG: "So it's a nearest neighbor sort of function. There you go! Now
you can motivate your student's interest by talking about social
choice theory, voting models and all sorts of applications of possible
interest to them."
The committee talks about applications for a little while. We move
onto topology.
JB: "OK let's do Morse Theory. Suppose you have SU(3) and you take
some maximal torus. What can you tell me about the associated flag
variety?"
I was told in advance to calculate the homology of arbitrary flag
varieties. The basic idea is to let the Adjoint action act on a
specific element thereby embedding the variety into the lie algebra (a
vector space) picking a point off the variety and then using the
distance function as my Morse Function. I am told to summarize a lot
of results and give the big picture. JB starts grilling me on the
basics of Lie theory. I define roots and simple roots, but am told I
need something else. I am stuck for a long time. Eventually I realize
I need *positive* roots.
JB: "What about a different torus inside of U(3)? What is it's
centralizer? What is the associated flag manifold."
For the torus I write down the associated flag describes 2-spaces
inside 3-space, or CP^2.
RG: "I have a question about Morse Theory. Have we talked about Morse
Theory for degenerate critical points? Either way let's start with an
example. Are you familiar with a monkey saddle."
I draw the monkey saddle. JB asks why it is called that. RG replies
"It has two arms and a tail!" I discuss why it is necessarily
degenerate by looking at eigenspaces.
RG: "Ok so how might you define index for a critical point?"
JC: "The dimension of the unstable manifold."
RG: "Good! Most people make the mistake of always using the Morse
lemma, but this is a more robust notion. How might you assign an index
to the monkey saddle?"
We then proceed to recover some Conley Index Theory that I had learned
in passing over the summer -- not on my syllabus, but it's all in good
fun. The idea is to replace the numerical index with a homotopical or
homological concept, usually a wedge of spheres. The other committee
members are unfamiliar with the topic and start discussing among
themselves. JK is interested in the computational aspect of
determining the homological index.
Now I have a string of PDEs questions and I answer them:
JK: "How might you solve the equation \laplace u = u^2 +f?"
This is a Holder Theory question. You solve it my casting it in fixed
point form, viewing it as a contraction on the appropriate Banach
Space C^{k,a}.
JK: "How might you solve u_t + u^2u_x=0? subject to initial data u(x,0)=f(x)"
This is a first-order nonlinear PDE. We apply the method of
characteristics to foliate 1+1-d space. I am asked to describe the
characteristics explicitly. Explain why the value of u is constant
along characteristics. Describe shock formation.
JK: "State Sobolev Embedding for bounded sets and R^n."
I get stuck on the R^n case. JK tells me the answer and asks me why it
is true. He wants a one word answer. JK tries to offer lots of hints,
finally after a lot of pain and waiting, I realize scaling is the word
he is looking for. This all for PDEs. Now topology
JB: Classifying principle U(1) bundles over S^2. Classify the total
spaces up to homeomorphism."
There is an integers worth. I start tabulating the total spaces up to
homeomorphism via the Euler number. n=0 is the trivial bundle, n=1 is
the Hopf fibration and n>= 2 are given by lens spaces.
RG: "Ooo. What about the negative integers?"
JB: "Yeah, what about them?"
They wait for me to answer. I start saying something about reversing
orientation, but after a long while I figure out they must be
homeomorphic. JB says he is satisfied and the committee kicks me out.
I wait outside for a few minutes. The committee comes out and congratulates me.