My Generals, by Spyros Alexakis
May 8 2000, 10:30, Fine 1102
Commitee: Fefferman (chair), Schlag, Kollar
Duration: 1 hour
F: In what order would you like to take your topics?
Me: Real-Harmonic, PDE, Complex, Algebra (actually many of the questions 
they asked combined topics)
F: OK, what is a singular integral? What are their properties? Why do we 
care about them?
Me: (I gave the definition) We can use them in PDE, for instance obtain 
estimates for Laplacian(u)=f.
F: Illustrate?
Me: L^p bddness of D^2(u) by L^p norm of f for 1<p<infty
F: Does L^1 bddness hold? Counterexample?
Me: for f= delta(0) doesn't hold, so approximate delta(0) by approximate 
identities.
F What would u look like then? 
Me: Use Riesz potentials to get u. (He wanted me to be precise and helped 
me in drawing a graph of u. I was pretty pathetic)
K: Why do you need a kernel of a singular integral to satisfy the 
cancellation condition?
Me (I mumbled something about getting an operator which sends the 
characteristic function of an interval to infinity at every point in the 
interval if cancellation condition doesn't hold. I also mentioned you 
need cancellation to prove weak L^1 bddness)
S: Here's a question which combines harmonic, PDE, complex: Take a C^2 
harmonic function on a bdd domain with smooth bdry which is continuous on 
the bdry, 0 on a bdry segment and whose normal derivative on that segment is 
0. What can you say?
 Fefferman made a cool suggestion and I showed it is 0 using complex 
analysis. Schlag said he hadn't thought of that proof and hinted towards 
the solution he had in mind (it involved weak solutions and was valid 
for higher dimensions). I eventually got the solution he wanted.
 F&S to K: How about some Algebra? 
 K: Take the ring of locally holomorphic functions around a fixed point 
in C^n. What can you say?
 I admitted I was clueless about the case n>1 and so worked out the case 
for C. (It is a PID)
K: OK now do C^n. 
F&S helped me in proving that the ideals can be the functions that vanish not 
only at the point but also on a complex line, eg.
K: OK now do the ring of functions locally C^infty(R) around 0.
 After quite a bit of hinting I said the ideal of functions with a 0 
of infinite order at 0 is not principal. I eventually managed to prove 
it too.
F asked me to leave the room and after a few seconds all 3 came out 
and shook my hand, congratulating me. (for what I have no idea).

Comments: All three created a pleasant atmosphere, to the extent that 
my tenseness allowed it. They want to see you come up with ideas of your 
own. 
Especially for the algebra questions, it is hard to describe the 
enthusiasm that overcame Schlag and Fefferman when I mentioned anything 
that could be conducive to a solution.