Problems in three directions of geometric complex analysis are considered. The first collection of problems comprises basic research in sampling and interpolation, the theory that underlies the current technology of storing and retrieving information digitally. The second collection of problems studies a certain class of metrics for holomorphic vector bundles on compact complex manifolds, and is connected to a complex analog of Hilbert's 17th problem. It also ties in with current work by algebraic and analytic geometers on multiplier ideal sheaves. The third collection of problems concerns the so-called Density Property for complex manifolds, introduced by the PI in his PhD thesis and developed significantly since then. The three directions are intimately related, and cannot be completely separated.
The main goal of the work on sampling and interpolation is to use the geometry of the space under consideration to determine the density of points needed to sample functions sufficiently so that they are faithfully stored in some digital incarnation. Conversely, the reconstruction of the functions from their digital realization requires the use of sufficiently few points. The geometric information that comes into consideration is the curvature of the metrics on the space. This idea was already considered by Einstein in his work on the General Theory of Relativity. The need to understand metrics rather well brings us in contact with the second circle of problems; here we consider certain metrics that arise naturally in problems of complex and CR geometry. Finally, the ideas considered have a completely different nature when discussed on spaces with the density property. These spaces, which are highly symmetric, have incredible flexibility properties, and enable one model to rather diverse dynamical phenomena, while still being able to use the complex structure to impose certain constraints on the results of the dynamics.
