Accommodation (focusing the eyes to clear blurred near images) and vergence (turning the eyes inwards to look at near objects) usually occur together automatically. If accommodation and vergence are disordered, blurred vision, double vision, reading difficulties and eye strain occur. In children, serious consequences such as squint, long or short sight can permanently damage eyesight if not treated correctly. These are common causes of referral to hospital and community eyecare services, but treatments are frequently poorly targeted, especially in community services. Dr Anna Horwood will study 1) Premature infant development and extend studies into adolescents and typical ageing adults; 2) How much traditional treatment methods rely on true treatment, in comparison to practice / placebo effects; 3) How the visual system adapts to sudden or gradual changes in demand such as occur with new spectacles or presbyopia (the natural decline in accommodation that occurs with age). By understanding how commonly used, simple treatments act we will be able to target them better and so save time and resources. We will also collaborate with a commercial manufacturer of child vision screening equipment to develop simple and effective methods of measuring eye position at the same time as refraction (focus).

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.