CIRCLE METHOD

"The Circle Method, Character Sums, and Quadratic Forms"

Coordinatore	THE CHANCELLOR, MASTERS AND SCHOLARS OF THE UNIVERSITY OF OXFORD    more  Organization address address: University Offices, Wellington Square city: OXFORD postcode: OX1 2JD contact info Titolo: Ms. Nome: Linda Cognome: Polik Email: send email Telefono: +44 1865 289800 Fax: +44 1865 289801
Nazionalità Coordinatore	United Kingdom [UK]
Totale costo	173˙185 €
EC contributo	173˙185 €
Programma	FP7-PEOPLE Specific programme "People" implementing the Seventh Framework Programme of the European Community for research, technological development and demonstration activities (2007 to 2013)
Code Call	FP7-PEOPLE-IIF-2008
Funding Scheme	MC-IIF
Anno di inizio	2010
Periodo (anno-mese-giorno)	2010-08-01   -   2013-01-31

 Partecipanti

#	participant	country	role	EC contrib. [€]
1	THE CHANCELLOR, MASTERS AND SCHOLARS OF THE UNIVERSITY OF OXFORD  Organization address address: University Offices, Wellington Square city: OXFORD postcode: OX1 2JD contact info Titolo: Ms. Nome: Linda Cognome: Polik Email: send email Telefono: +44 1865 289800 Fax: +44 1865 289801	UK (OXFORD)	coordinator	173˙185.81

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 Obiettivo del progetto (Objective)

'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.'