Analysis at Infinity: Integral Equations, Limit Operators and Beyond

ended

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.