Intellectual Merit: The project concerns problems fundamental to the discipline of analytic methods in complex geometry, including problems of Hilbert space methods in complex and algebraic geometry, Hermitian algebraic functions, and interpolation and sampling of holomorphic functions and sections of holomorphic line bundles. The objects being studied are fundamental and central in complex geometry, and the problems address basic research of these key objects. Solutions of the problems will likely result in the creation and development of new techniques and theories, which will advance knowledge in various areas of mathematics related to complex geometry, as well as in adjacent areas of engineering, physics and other hard sciences.
Broader Impact: The problem of interpolation and sampling is closely related to problems in engineering, particularly signal analysis, and our methods and results will have numerous applications. Moreover, the interpolation and sampling problem is closely related to higher-dimensional algebraic geometry. In this setting, our analytic methods can be combined with globalizable metrics to construct precise weights and obtain sharp results about sections of line bundles on compact Kahler manifolds. Hermitian algebraic functions lie at the interface of analysis and linear algebra, where the algebra is no longer manageable via direct, brute-force calculation, and limit processes must be brought in to simplify arguments, even from the computational perspective. There are a number of directions created in this project that can be used to train graduate students wanting to do research related to or making use of mathematics addressed in this project. The PI has also gained enough background in the field that he has felt it important to produce some notes from which students of complex geometry and researchers in other related fields could gain the background necessary to understand the current state of the art.
The project has resulted in completion of a majority of the goals set out in the original description, as well as new developments that came about naturally in the course of pursuing the reseach and problems originally laid out. Great advances have been made in development and application of the twisted techniques that have become fundamental in complex analytic geometry, and progress continues. A number of articles have been published, in which the NSF was acknowledged for its support. A graduate text was published, as was a more advanced book chapter on fundamental analytic techniques in the IAS/PCMI series, edited by McNeal and Mustata. In short, the intellectual merit was very high, and has both solved fundamental problems and opened new avenues of research at the top level. The extension problems should have a huge impact in problems where one wants to sample and interpolate finite energy signals. Such work is not only of theoretical use, but has important and obvious practical applications. The broader impact, in the immediate sense, will be in training graduate students and researches in related branches of mathematics and other close-lying fields of science. There are even some interesting problems for advanced undergraduate students, which I hope to help train in the REU setting, as I have done in the past.
