Much of modern complex analysis is concerned with the study of the tangential Cauchy-Riemann (CR) operator. This operator arises by restricting the classical Cauchy-Riemann operator in complex Euclidean space to a hypersurface or to an appropriate class of manifolds called CR manifolds. This operator is not invertible. Thus to understand it, we must understand a related singular integral operator, the Szego projection operator, which is the orthogonal projection of the space of square-integrable functions on a manifold onto the null space of the associated tangential CR operator. The Szego projection is relatively well-understood for boundaries of pseudoconvex domains of finite type. Comparatively little is known if one relaxes the finite-type hypothesis, and even less is known if one relaxes the pseudoconvexity hypothesis. Exploring these contexts systematically is the goal of the current project. More specifically, Halfpap and her collaborators will analyze explicit expressions for the integral kernel associated with the Szego projection for several classes of hypersurfaces and CR manifolds. The goals are 1) to determine the locations and sizes of the singularities of the Szego kernel, 2) to determine how these are connected to the geometry of the underlying manifold, and 3) to understand the mapping properties of the projection operator itself.
In general, the study of partial differential equations (PDEs) is important because PDEs give a language for describing relationships among changing quantities; some model heat flow, some wave propagation, and some are simply objects of study in their own right. For a general PDE, several fundamental questions arise: Can one solve for the unknown function in terms of the given data? Is the solution unique? If the data are known to have certain special properties (e.g., smoothness or square-integrability) what can be said about the solution? Often one seeks a so-called ``fundamental solution" -- a kernel function against which one may integrate the data to obtain a solution of the original PDE. Theorems relating properties of the solution to those of the data thus require detailed information about the associated integral kernel function. Frequently, the kernel function has singularities. In these cases one obtains a singular integral operator. This is the larger context in which the current project is situated, and thus the results of the current project have broader implications for other areas in which PDEs and singular integral operators arise. This mathematics research project also has the potential to increase diversity within mathematics; Halfpap is an active advisor of graduate students (as well as an effective teacher and mentor of undergraduates) at a university serving a largely-rural state.
