Complex Geometry via Analytic Techniques

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

Complex Geometry via Analytic Techniques Abstract 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.