Science and mathematics depend to a very high degree on being able to think in images, to see analogies, to allow things to remain uncertain until an intuition comes to you that gives what’s called a gestalt, an overall form that makes sense of things.
— Iain McGilchrist
Certainly the best times were when I was alone with mathematics, free of ambition and pretense, and indifferent to the world.
— Robert Langlands
Zeta values are infinitely attractive objects.
— Kazuya Kato
Research Interests
For more than two decades I have been deeply and passionately interested in the theory of L-functions. Currently, one of my key problems is the search for possible mechanisms underlying the principle that two L-functions that look the same for essentially the first log(N) coefficients should in fact be identical. Here N refers to the conductor of an L-function (or a common maximal bound of the two conductors. In other words, log(N) should be viewed as the complexity of the L-function, but we do not know how or precisely why this is the case. The analogous problem for zeta functions over a finite field is resolved by the first Weil conjecture (Rationality), and in the case of curves, this complexity measure is simply the genus of the curve.
From a broader perspective, I am even more deeply interested in possible frameworks that could explain and unify a large number of different aspects of L-function theory.
Although I have not published as much as perhaps I should have, some of these research directions will appear as part of the RH Saga exposition at PeakMath.
In addition to these purely mathematical questions, I have also focussed on the philosophy and cognition of mathematical creativity and meaning-making, and the nature and structure of mathematical discovery.