Dr. Harron's currently studies two different subjects in number theory. The first, Iwasawa theory, traces its roots to the 19th century and began flourishing with the work of Kenkichi Iwasawa starting in the 1950s. Nowadays, it is a central subject in number theory, considered to be the most-promising approach to questions such as the Conjecture of Birch and Swinnerton-Dyer. The idea of the subject is to study the discrete objects of interest in number theory by placing them in continuously varying families of more general objects. Basically, while the objects appear to be discrete in the usual geometry of space (the real numbers), they lose that appearance from the perspective of so-called p-adic geometry where, for instance, two numbers are close if their difference is highly divisible by a fixed prime number p. His research in Iwasawa theory involves not only theoretical results, but also the development of software for computing examples of the families mentioned above.
Additionally, he also does research in the field of arithmetic statistics: taking collections of objects of number-theoretic interest and asking questions about their average behavior and other statistical measures. This field has really come to the forefront in the past decade with a lot of exciting developments. His work has also begun addressing the arithmetic statistics of Iwasawa theory, merging his two interests together.
Robert Harron recently presented his research at the 14th meeting of the Canadian Number Theory Association at the University of Calgary and at the 5th meeting of the Latin American Congress of Mathematicians at the Universidad del Norte in Barranquilla, Colombia. He obtained his PhD from Princeton in 2009 as a student of famous mathematician Andrew Wiles that solved Fermat's Last Theorem. He was then a postdoc from 2009 to 2011 at Boston University, and from 2011 to 2014 at the University of Wisconsin–Madison Before joining the department of Mathematics at UH Manoa in Aug. 2014.