University of Hawaiʻi at Mānoa Assistant Professors **Sarah Post** and **Rob Harron** were each awarded a collaboration grant for mathematicians from the Simons Foundation. These $35,000 five-year grants began September 1, 2016, and will end August 31, 2021 with the primary goal to stimulate collaboration in the field of mathematics.

The Simons Foundation’s mission is to advance the frontiers of research in mathematics and the basic sciences, and the funds of these awards will be used by the recipients to travel and to invite researchers to the department to collaborate with faculty, undergraduates and graduate students.

## Sarah Post: Quadratic Algebras and Orthogonal Polynomials

Post will use her grant to support her research on the study of orthogonal polynomials and how symmetries can be used in their classification and applications. Many of the most well studied physical systems, for example the hydrogen atom or the gravitational pull between two celestial bodies, have many symmetries that can be used to facilitate their analysis. Often, the solutions of the equations of motion can be expressed in terms of orthogonal polynomials. Orthogonal polynomials are also used in approximation theory and for numerical or asymptotic solutions to differential equations.

Post’s research has been to study the interplay between the symmetries of the physical systems, which gives rise to a particular quadratic algebra structure, and the systems of orthogonal polynomials.

Post recently presented her research at the Australia New Zealand Association of Mathematical Physics annual conference where she gave an invited plenary address. She will travel this winter to the Joint Math Meetings in Atlanta where she is co-organizing a special session titled Symmetries, Integrability and Beyond. She works with both undergraduate and graduate students including **Makana Silva**, who will be traveling to the annual Society for the Advancement of Chicanos/Hispanics and Native Americans in Science (SACNAS) conference. He will present results from their collaboration, which began in summer 2015 as a result of a research grant from the Undergraduate Research Opportunities Program at Mānoa.

Post received her PhD from the University of Minnesota in 2009, and was a postdoc at the Centre de Recherches Mathematiques at the University of Montreal from 2009 until 2012. She joined the UH Mānoa mathematics department in fall 2012.

## Robert Harron: Iwasawa theory and arithmetic statistics

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. 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. 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 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.

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 who 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 UH Mānoa mathematics department in August 2014.