Gergely Harcos
1:30 pm, May 18, 1999
Topics: analytic number theory, algebraic number theory
Committee: Sarnak (chair), Trotter, Luo
This is the true story of my general exam.
Sarnak asked me what I wanted to start with. I said complex analysis.
Trotter asked the following. If you have an analytic function in
C-{1,2i} how many power (Laurent) series does it have around zero and
how do I find them. Then Sarnak took over. Draw three concentric circles.
Suppose you have an analytic function in the annulus bounded by the
innermost and outermost circles, and you have an upper bound for its
absolute value on the innermost circle and another
bound on the outermost circle. How would you bound the function on the middle
circle. Let's say both bounds are 5. Then the bound is 5 by the maximum
principle, I said. What if one bound is 5 the other one is 7? I said, I
would apply the three circle theorem. What does it say? I told them.
Ok, let's see if you know what lies behind that theorem. Suppose you have
a domain bounded by a Jordan curve and you cut out two regions from that
domain bounded by Jordan curves inside the domain. We have an analytic
function in the remaining triply connected domain and have various
bounds - 3,5,7 - on the boundary components. How would you bound the
function in the domain? I said I would construct the harmonic measures
corresponding to those components and then the linear combination with
coefficients 3,5,7 will give a pointwise bound inside the domain. Let's
go back to the annulus and suppose that we have a bounded subharmonic
function inside it and we know bounds 5 and 7 on the boundary circles
with the exception of finitely many points. I said the same answer applies
by the generalized maximum principle. Sarnak then asked me to give the
definition of subharmonic functions and explain the proof of
the generalized maximum principle. So we discussed barriers a bit.
I was asked to construct the harmonic measures for a strip and also
to find explicitly the harmonic function inside a semidisc (bounded by
a line segment and a semicircle) which has boundary values 0 in the interior
of the line segment and 1 in the interior of the semicircle (just use the
Thales theorem from elementary geometry). Then he gave the following problem.
Let's assume we have a bounded subharmonic function u on the whole complex
plane. Can you conclude it is constant? I said yes. Why? My first idea to
write u as log|f| with f analytic was silly - as I realized quickly. Then
Sarnak told me that if I had not thought about this problem before I probably
shouldn't try to do it there. But I said I was sure I could do it quickly
using ideas as in the proof of the generalized maximum principle. I got the
right intuition that instead of u one should examine v=u+elog|1/z| where e is
any positive constant. But I got into trouble with the details, so Sarnak
helped me a bit. Here is the proof. Let M be the supremum of u and assume
that u is not constant. By upper semicontinuity, u attains its supremum on
the unit circle is, call it m. Then m1.
The same holds for |z|<1 as well. By letting e tend to 0 we get that u<=m
on the whole plane which is a contradiction to m