Asymptotics of Toeplitz determinants, soft Riemann-Hilbert problems and generalised Hilbert matrices (HilbertToeplitz)

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My research proposal is concerned with topics in operator theory and complex analysis with applications to random matrix theory and mathematical physics. More precisely, I aim to focus on the following three areas: (A) double-scaling limits of Toeplitz determinants with Fisher-Hartwig (F-H) singularities and Riemann-Hilbert problems; (B) soft Riemann-Hilbert problems, which builds on a recent breakthrough of Hedenmalm and Wenmann; and (C) spectral properties of generalised Hilbert matrices. In theme A, I aim to compute the double-scaling limits of Toeplitz determinants in the presence of finitely many F-H singularities when at least two of them merge into one as the size of the determinants tends to infinity and an external parameter tends to a critical value simultaneously. I utilise Riemann-Hilbert problem (RHP) method and operator-theoretic techniques to study this problem. I also consider the applications of double-scaling limits of Toeplitz determinants in random matrix theory. In theme B, I investigate Soft RHPs that arise in two-dimensional determinantal point process models, such as the Random Normal Matrix, where the eigenvalues are complex, and tend to fill a two-dimensional set of positive area. Compared with the classical RHPs, the study of soft RHPs is at its infancy and my goal is to develop their theory further and apply it to problems in integrable nonlinear PDEs. In theme C, my main goal is to study boundedness and spectral properties of generalised Hilbert matrix operators on analytic function spaces. In particular, I aim to complete the spectral picture of these operators by describing it in the Hardy spaces, sequence spaces and Korenblum spaces. A goal is also to develop techniques to be used to characterise the spectra of a large class of Hankel operators acting on Banach spaces of analytic functions and methods for these spaces in general.