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.