Research Projects

Weighted kernel functions, such as the weighted Bergman kernel and the weighted Szegő kernel, play a central role in complex analysis. While the unweighted case is well studied, the weighted case poses intricate challenges and is of great current interest and importance. In this project, we aim to study various themes with respect to weight functions. The first focus is on the form of holomorphic isometries with respect to the weighted Bergman metric for weights of the form $K^{-d}$, where $K$ denotes the classical Bergman kernel and $d$ is a non-negative integer. The second focus is on the structure of the zeroes of the weighted Szegő kernel and the weighted Garabedian kernel. These kernels solve many important extremal problems; for example, the weighted Ahlfors map which generalizes the Riemann map, is given as the ratio of the weighted Szeg\H{o} kernel and the weighted Garabedian kernel. We further aim to develop numerical methods to compute the weighted Ahlfors map and the solution of the Dirichlet problem using Szegő kernel-based methods. In the context where the Cauchy kernel is used to approximate the weighted Szegő kernel, we will also study the spectrum of the weighted Kerzman–Stein operator, which relates these kernels. Some of the hypotheses that we are going to test are as follows. We have observed that holomorphic isometries as explained above from the unit disc into the bidisc or tridisc can be written in terms of the diagonal embedding and Mok's root embeddings. We aim to prove that this holds for holomorphic isometries into any polydisc. Next, the (classical) Szeg\H{o} kernel and Garabedian kernel do not vanish when one variable lies in the domain and the other lies on the boundary. We want to test whether this holds for the weighted setting as well. We also plan to extend recent work on classical quadrature domains to quadrature domains with respect to weight functions; an area of research that has the potential to be applicable in science and engineering. Next, the Kerzman-Stein operator is zero if and only if the domain is a unit disc. We aim to test whether some condition on the spectrum may serve to classify domains beyond the unit disc. The project broadly requires complex analytic methods, geometric function theoretic methods, and numerical implementation in Python. Some of the problems also require combinatorial methods and numerical analytic methods. Achieving the stated objectives would advance the fundamental understanding of kernel function theory with respect to weight functions. Moreover, the project bridges abstract theory with computational methods, aiming to make substantial contributions to both the analytical foundations and practical tools in complex analysis.