The principal focus of the project is the interplay between complex geometry and probability. In particular, 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. 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 using curvature invariants to describe the off-diagonal asymptotics of this kernel for large powers of the line bundle. Shiffman will apply the off-diagonal Bergman kernel asymptotics to obtain optimal sup-norms for orthonormal bases of spaces of holomorphic sections of powers of ample line bundles. He will also continue his investigation of the distribution of random zeros of systems of polynomials, or more generally random sections, in order to obtain central limit theorems for the numbers of zeros in smooth domains as the degree increases. He will investigate the zeros of random polynomials of increasing degree containing a fixed number of monomial terms. Shiffman will also study point processes given by critical points of random holomorphic sections.
In the physical sciences it is often necessary to handle disorder, where a certain amount of randomness is inserted into a system. Random functions can be used to model 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 can be used to study systems with several degrees of freedom, and those polynomials of high degree correspond to wave functions for high energy states. The mathematics of point processes--the spatial and/or time distribution of random occurrences--has been used in many diverse fields such as signal and image processing, quantum mechanics, epidemiology, seismology, astronomy, and economics. This mathematics research project includes the development of geometric methods to study the statistics of point processes arising from mathematical equations with some random input.
