Cel The project consists of three distinct parts, on which the applicant will work in parallel.Part 1 : We intend to work on analogues of Hilbert's tenth problem for various function rings, such as the rings of analytic functions and the fields of meromorphic functions on subsets of both the p-adicand the complex plane (for example on the unit discs). At a first stage we intend to develop new techniques in order to create new tools for the study of this kind of problems. The techniques that we have in mind relate to characterizing maps from (sufficiently large subsets of) algebraic curves (over the complex numbers) to elliptic surfaces.Part 2: We will study the decidability problem for a field of power series in positive characteristic, in the language of addition, divisibility and the Frobenious map, and some variants. We expect to apply the results to problems rising in number theory.Part 3: We will study a problem which has standed for a long time: what the elementary classes of function fields are. The field of complex rational functions is one of them. Despite various recent progresses towards a general answer to this question, some cases remain with no complete answer. This part of the project deals with the case of some elliptic fields with complex multiplication. The underlying theme of all the parts of the project is the connection of model theoretic notions to algebraic properties of various domains of interest in algebra, algebraic geometry and number theory .In recent years mathematical logic has been used increasingly for applications in other domains of mathematics, such as algebraic geometry. Also, knowledge in many domains, such as algebra, algebraic geometry, analysis, p-adic analysis and number theory, is a prerequisite in order to solve problems of interest in model theory. For example, the use of the theory of algebraic surfaces in order to solve some analogue of Hilbert's tenth problem will increase the ability of the applicant to make both fields interact. It will be an asset for his further research opportunities. Also the study of p-adic cases will strengthen the interaction that already exists intrinsically between Hilbert's tenth problem and number theory. Due to the large range of its research subjects, Oxford university would be an ideal place to carry out such a multi-disciplinary project. The expected impact for the host is to have some (probably young) researchers become interested and work on various aspects of the problems of the project, and thus to promote the interaction of ideas through discussions and seminars. Dziedzina nauki natural sciencesmathematicspure mathematicsdiscrete mathematicsmathematical logicnatural sciencesmathematicspure mathematicsarithmeticsnatural sciencesmathematicspure mathematicsgeometrynatural sciencesmathematicspure mathematicsalgebraalgebraic geometry Program(-y) FP5-HUMAN POTENTIAL - Programme for research, technological development and demonstration on "Improving the human research potential and the socio-economic knowledge base" (1998-2002) Temat(-y) Data not available Zaproszenie do składania wniosków Data not available System finansowania RGI - Research grants (individual fellowships) Koordynator THE CHANCELLOR, MASTERS AND SCHOLARS OF THE UNIVERSITY OF OXFORD Wkład UE Brak danych Adres St Giles' 24-29 OX1 3LB OXFORD Zjednoczone Królestwo Zobacz na mapie Koszt całkowity Brak danych
