My committee consisted of Prof. Fefferman, Kohn and Oh.
Topics: Elliptic PDE and Representation Theory
Since Prof. Kohn complained about just having returned from the dentist,
Prof. Fefferman started with questions about Cplx. Analysis:
-Tell me your favourite function from cplx. Analysis.
I mention the Gamma function, and proceed to define it by means of the
product development for its inverse.
-Why does this converge?
Weierstrass factors etc.,the usual rigmarole.
-Give properties of the Gamma function. In particular, prove the
basic functional equation directly from the product identity.
-Can you define the Gamma function via an integral formula? How does the
functional eqn. follow from this? Why is this the Gamma function?
-Prove the Stirling formula
After this barrage of questions on the Gamma function, I mention that I
actually know more about the Riemann Mapping thm.
-Why is the Riemann Mapping Thm. wrong for the whole plane?
-What happens, if instead of a simply connected region, we take a doubly
connected region?
-If you have a biholomorphic mapping from a doubly connected region to an
annulus, why does the bdry. of the region get mapped to the bdry. of the
annulus?
-When are 2 annuli conformally equivalent?
-How about C-(infty)-extendability to the bdry? (I first mention
the Schwarz reflection principle in case of real analytic bdries, then
regularity of solns. to elliptic eqns., provided the bdry. is smooth. One
can reduce the problem to 0-bdry values)
Next, Prof. Oh starts to ask about Algebra:
-State the structure thm. for modules over PIDs. What does this look like
for finitely generated Abelian groups?
-Describe the Jordan Normal Form of a matrix. Relation to previous question?
-What is the Galois Group of the polynomial x^n-1 over Q?
I mention that I think x^5-2 is more fun, so they ask me to do that, too.
-Do you know Witt's theorem on real quadratic forms?
After this, we move on to my main topics, it being understood that Real
Analysis is essentially a prerequisite for my first topic. Prof. Kohn starts:
-Define an elliptic equation in as general terms as you can.
-What can one conclude from the vanishing of a solution to an elliptic
eqn. in an open subset of a domain in R^n? What are the general conditions
under which such a conclusion is valid?
-We know that certain regularity assumptions on the coefficients of an
elliptic PDE ensure the existence of solutions with certain regularity.
Can you give an example with somewhat weaker regularity on the
coefficients where solvability of the eqn. breaks completely down?
-State and prove your favorite thm. on elliptic PDE from Gilbarg and
Trudinger.
I discuss the De Giorgi Nash result on Hoelder continuity of weak solns. to
2nd order PDE.
-Describe the Schauder Method.
Finally, we move on to Rep. Theory.
Prof.Oh asks me basic structure questions.
-Given a cpct. group G,and x(in)G, is there a representation of G which is
nontrivial at x, if x(neq)e ?
I wind up proving the existence of a faithful representation.
-Describe the way in which L^(2)(S^1) splits as an S^1-module.
-Discuss highest weights, roots, etc.
-Describe the Rep. Theory of SL(2,C), i.e. the infinite dimensional
irreducible unitary reps.
When I offered to prove the irreducibility of the discrete series Rep.
my committee decided that I didn't have to do that.
All in all, my Generals lasted one hour and 20 minutes.