Random Holomorphic Sections and Complex Geometry

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Bernard Shiffman will continue his research on applications of pluripotential theory and the Bergman-Szego kernel to the statistics of random functions of several complex variables and more generally of random sections of positive line bundles on compact complex manifolds. The principal focus of the project is the interplay between geometry and probability. One of the goals of the research is to refine our understanding of the distributions of zeros and critical points of polynomials, holomorphic sections of ample line bundles, and entire functions. A fundamental ingredient in the study of random sections is the Bergman-Szego kernel, and this project involves an examination of the asymptotics of this kernel for large powers of the line bundle. Shiffman will investigate asymptotic statistics for polynomials of increasing degree, holomorphic sections of increasing powers of a line bundle, or in the case of entire functions, on domains of increasing size. One problem is to estimate "hole probabilities" and "overcrowding probabilities" (i.e., the probabilities that random systems of equations have no solutions or too many solutions in fixed domains). Shiffman will also study the distribution of zeros of real and complex "fewnomial" systems--polynomials of high degree with few terms. He will also investigate the asymptotic distribution of critical points for spherical harmonics as the dimension increases.The distribution of critical points and zeros of random functions is relevant to many areas in physics, signal and image processing, and other areas of engineering. In the physical sciences it is often necessary to handle disorder, where a certain amount of randomness is inserted into a system. Random polynomials provide an elementary model for many systems, such as systems of atoms and molecules and their component particles--protons, neutrons, and electrons. Quantum mechanics describes these particles by wave functions, which are solutions of Schrodinger's equation. The zeros and local maxima of wave functions give important information on states of atoms and molecules; the zeros are known in quantum chemistry and physics as nodal lines. Polynomials in several variables correspond to systems with several degrees of freedom, and those polynomials of high degree correspond to wave functions for highly excited states. While Shiffman's recent research was concerned primarily with demonstrating that average states of such large systems are typical, as in the law of large numbers, this project will also include the study of "rare events," which has applications to various areas of current interest, such as economics and the study of climatic extremes.