Alex Peng
January 14, 1999, 2pm
Committee: Sarnak (c), Sinai, Ellenberg
Special topics: Theory of Probability, Analytic Number Theory
Sarnak began by asking me which of the three general subjects that
I'd like to begin with. I said complex analysis. Sarnak then
gestured for Sinai to pose the first question. Sinai asked me what
an entire function was, which I defined following on the
definition of a holomorphic function. Sarnak interjected to ask if
I knew of any regularity condition on the function. To which I
responded with the standard proof of existence of power series
representation. Sinai was satisfied at this point, but Sarnak
pressed on with another question in the same vein--if we know f to
be entire, and that |f(z)|<|z^100| what can I say about this
function? I blanked out for a while before their persistent hints
prodded me to the idea of dividing and then applying Liouville's
theorem. Sarnak then remarked that since we were in this entire
function mood, we might as well do another one--f entire, nowhere
vanishing, and |f(z)|