Descripción del proyecto
Una innovación sin límites da lugar a nuevos enfoques de las integrales y el análisis numérico
Las ecuaciones diferenciales y las integrales se emplean con frecuencia para describir todo tipo de fenómenos en diversos ámbitos, incluida la biología, la química, la física, las matemáticas, las ciencias medioambientales y la ingeniería. Algunos casos especiales, como las integrales singulares en que el integrando alcanza un valor infinito, pueden presentar desafíos especiales en cuanto a su utilización práctica. El objetivo del proyecto AnalysisAtInfinity, financiado con fondos europeos, es superar este obstáculo. Los científicos desarrollarán nuevos enfoques para transferir la teoría del operador límite existente a los operadores integrales y aplicarlos a problemas de matemáticas e ingeniería, incluidos problemas aún por resolver.
Objetivo
The main objective of this project is to investigate fundamental properties of singular integral operators and apply our findings to concrete problems in mathematical physics and engineering. Our approach is to combine newly developed limit operator methods with Riemann-Hilbert analysis. Our plan is divided into three parts. In the first part we develop the limit operator fundamentals. We use the existing limit operator theory and transfer the methods to integral operators. In the second part we combine limit operator theory with Riemann-Hilbert analysis to obtain fundamental properties of Toeplitz operators like boundedness and Fredholmness. We will also use this combination to find double-scaling limits of Toeplitz determinants, which are used, for instance, to understand spontaneous magnetisation in the 2D Ising model. In the third part we will apply our results to concrete integral equations, e.g. the double layer potential. Our ultimate goal will be to resolve a long-standing spectral radius problem. The project combines the expertise of the Applicant (limit operator theory) very well with the expertise of the Supervisor (Riemann-Hilbert analysis) and the Host's analysis group (integral equations, mathematical physics). By combining these fields in a novel approach, this project opens up new research possibilities and greatly contributes to European research excellence in analysis and its applications. The results will be published in high-level journals and presented at international seminars and conferences. A workshop on the proposed topics will be organised at the Host university and a blog will keep everyone updated on the progress. The scientific research is accompanied by teaching, supervising students and workshops on complementary skills. This ensures that the Applicant will become a versatile and mature mathematician by the end of the project, who is capable of leading an international research group and acquiring a permanent position in academia.
Ámbito científico
Programa(s)
Régimen de financiación
MSCA-IF-EF-ST - Standard EFCoordinador
RG6 6AH Reading
Reino Unido