Final Report Summary - CIRCLE METHOD (The circle method, character sums, and quadratic forms) The Marie Curie fellowship entitled 'The circle method, character sums, and quadratic forms', funded the research of Dr Lillian Pierce at the University of Oxford for two years, with the mentorship of Prof. Roger Heath-Brown, FRS. During the fellowship period, Dr Pierce engaged in research in pure mathematics on problems in analytic number theory and harmonic analysis. The project aimed to investigate a series of open problems in analytic number theory, all linked by the themes of the circle method, exponential and character sums, and quadratic forms. We describe here the main results produced by the collaboration of Dr Pierce and Prof. Heath-Brown. First, the researchers developed a new version of the power sieve, which enabled them to prove new bounds for the number of perfect power values of certain types of polynomial. As a result, they were able to prove, and in fact improve one, a well-known conjecture of Serre concerning the number of rational points on smooth cyclic covers. This work appeared in the peer-reviewed paper 'Counting rational points on smooth cyclic covers', Journal of Number Theory, 132 (2012), pp. 1741-1757, co-authored by Prof. D. R. Heath-Brown and Dr. L. B. Pierce; a preprint is available on the open access repository ArXiv. Furthermore, the researchers developed a new technical innovation on the circle method of Hardy and Littlewood, one of the most important techniques in modern analytic number theory. The novel two-dimensional Kloosterman refinement produced by this work enabled Dr Pierce and Prof. Heath-Brown to prove new results about pairs of integers simultaneously represented by two quadratic forms with integer coefficients. This work will appear in the paper 'Simultaneous prime values of pairs of quadratic forms', co-authored by Prof. D. R. Heath-Brown and Dr L. B. Pierce. This manuscript is in the final stages of being prepared for submission to a peer-reviewed journal. One of the key goals of this project was to solidify a long-term collaboration between Prof. Heath-Brown and Dr Pierce, with the intention that work on certain aspects of the project would continue past the conclusion of the funding period. This has been accomplished, and the collaboration continues actively on several problems, in particular on several questions on bounding short character sums via Burgess-type methods. It is anticipated that this research will lead to several more collaborative publications. Another key goal of the project was to connect Dr Pierce with the international research scene and to facilitate her establishment as a researcher and collaborator. This has also been accomplished; we describe briefly below two further ongoing research interests that she has initiated with mathematicians in the international community. Inspired by her work on the circle method and quadratic forms, Dr Pierce developed an interest in Carleson operators of Radon type. These operators are motivated by questions from classical analysis, but can be redefined in a discrete setting, in which case considerable number theoretic understanding and innovative applications of the circle method are required. Dr Pierce initiated a collaboration with Dr P. L. Yung, and they successfully proved a significant theorem on such Carleson operators. This will appear in the paper 'Polynomial Carleson operators of radon type', co-authored by Dr L. B. Pierce and Dr P. L. Yung. This manuscript is in the final stages of being prepared for submission to a peer-reviewed journal. A second phase of this project will address the discrete analogue, and will require the application of circle method techniques, which Dr Pierce has learned during the project. Moreover, Dr Pierce is currently engaged in a collaboration with Dr Frank Thorne on the 3-part of class numbers of quadratic fields. This work requires the application of exponential sums, sieves, and the circle method, and is closely tied to several of the original objectives of the project. It is anticipated that this ongoing research will result in several future publications. The work of this long-term research programme on the circle method, character sums, and quadratic forms, has the potential to influence current mathematical research ranging from analytic number theory to harmonic analysis. The work on power sieves has developed a new tool with wide applicability to problems in number theory, and has pushed forward current understanding of a conjecture of Serre. The work on the circle method has broken a long-standing barrier; this has also opened the door to new and advantageous applications of the circle method to higher dimensional problems. The work on Carleson operators of Radon type has introduced an entirely new element to the setting of Carleson operators, and it is expected that this will generate wide-ranging interest in such problems. Pierce is now strongly placed to maintain productive collaborations with researchers in both Europe and the United States. At the completion of the Marie Curie funding period, Pierce is now a research fellow at the Mathematical Institute at Oxford, funded by the National Science Foundation of the United States. In 2013-2014, Pierce will be a Bonn junior fellow (w-2 professor) at the Hausdorff Centre for Mathematics in Bonn, Germany. In 2014, she will join the faculty at Duke University in the United States. The opportunity for her to receive advanced training at Oxford and to initiate collaborations with mathematicians in Europe has now established Pierce's strong long-term links between research institutions in Europe and the United States. 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