"This proposal investigates a series of problems in analytic number theory linked by the themes of the circle method, character sums, and quadratic forms. The first set of objectives, for the circle method, proposes innovative extensions of Kloosterman refinements. In particular we propose a novel two-dimensional Kloosterman refinement, the first of its kind, with impact on questions such as whether two quadratic forms must simultaneously attain prime values. Additionally, we propose to develop further the so-called double Kloosterman refinement in order to obtain asymptotics for solutions of non-singular homogeneous cubic equations in seven or more variables. The second set of objectives, for character sums, intersects the circle method with a fresh idea for an improvement of Hua’s inequality via estimates for exponential sums; this has immediate implications for Waring’s problem, as well as for discrete fractional integral operators in harmonic analysis. For more general character sums, we propose two novel approaches to bounding character sums by Burgess’ method, in one case by a highly original method using the modularity of elliptic curves. The third set of objectives, for quadratic forms, linked by the first problem to the circle method, additionally proposes to investigate two new approaches to counting values of quadratic forms, with applications in both number theory and analysis. This proposal presents innovative methods at the cutting edge of number theory for reaching each of these objectives. This project would bring an outstanding mathematics post-doctoral researcher to Europe, introducing her exceptional research expertise into the European mathematical community, providing her with critical training for the foundation of an independent research career, and establishing a long-term collaboration between mathematicians in Europe and the US, upon her return to her home country."