My interaction with Brendan Murphy began in the most humorous
possible way. I was the graduate director at the University of
Rochester when Brendan was a first year student. At the end of the
academic year, Brendan approached me for an advice on what he should
be doing during summertime. I strongly suggested that he stay in the
area and study for the preliminary exams, but Brendan decided to
take a bike trip across the USA! To put the icing on the proverbial
cake, Brendan sent me a postcard from Colorado during the trip.
Krystal Taylor still remembers the expression on my face when the
postcard arrived. But everything worked out in the end. Brendan came
back from the trip energized and passed all his preliminary exams.
I co-advised Brendan with Jonathan Pakianathan. During this time,
the three of us wrote a paper that continues to influence my work to
this day. We showed, roughly speaking that if one considers a
sequence of two-dimensional moduli over finite rings of odd size
tending to infinity, and if the Fourier transform of the hyperbola
over these moduli satisfies the optimal square root bound with
uniform constants, then these rings are eventually fields. The theme
that square root cancellation is impossible in the presence of zero
divisors and its connections with complexity theory is the subject
of some of my recent explorations.