Furthering research in mathematics in Europe
Mathematical analysis and number theory have helped further our understanding of the world in unprecedented ways. Analytic number theory uses mathematical analysis to solve problems about integers. 'The circle method, character sums, and quadratic forms' (CIRCLE METHOD) is an EU-funded project for investigating problems in analytic number theory linked to the circle method, character sums and quadratic forms. Project members worked on refining the circle method based on the work of a Dutch mathematician, Hendrik Douwe Kloosterman. The Kloosterman refinements were meant to answer questions on whether two quadratic forms must simultaneously attain prime values. Works and theories of other mathematicians were also introduced into the project team's analysis. They focused on research in pure mathematics related to problems in analytic number theory and harmonic analysis. Using circle method, new approaches and technical innovations developed demonstrated that pairs of integers can be simultaneously represented by two quadratic forms with integer coefficients. CIRCLE METHOD team involved partners from the United States and the United Kingdom. They established long-term collaboration to continue working on the project even after its official end. Many of the achievements and conclusions that emerged during the project and even after it ended have appeared in peer-reviewed journals. The ongoing research and collaboration are expected to lead to more publications. Research on the circle method, character sums and quadratic forms could significantly influence mathematical research in fields ranging from analytic number theory to harmonic analysis. Project outcomes have helped solve problems in number theory and paved the way for tackling higher dimensional problems. This project has attracted outstanding postdoctoral researchers and will strengthen collaborative ties between Europe and United States to further research.
