Counting Solutions of Diophantine Equations

Oscar Marmon received his Ph.D. in Mathematics at Chalmers University of Technology in 2010. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position in Associate Professor Morten S. Risager’s group at the University of Copenhagen, Denmark.

Oscar Marmon’s project consists in research of Diophantine equations, which is study of whole numbers solutions of polynomial equations with integer coefficients. Study of such equations is one of the oldest endeavors in mathematics. The name is derived from Greek mathematician Diophantus of Alexandria who lived in the third century A.D.

Many mathematicians have studied Diophantine equations over centuries, including Pierre de Fermat, Carl Friedrich Gauss, and Leonhard Euler. Such research led to the development of some of the most important branches of mathematics. Yet, many problems remain unsolved.

Russian mathematician Yuri Matiyasevich proved in 1970 that there is no general algorithm determining if a given Diophantine equation has a solution. However, it is possible to describe solutions for certain classes of such equations. Further study of such classes of equations, as well as studying the solutions in some special cases is the goal of the project.

Diophantine equations occur naturally in a broad spectrum of pure and applied mathematics. The project will involve applications in several branches of mathematics, including harmonic analysis and number theory.