Random Holomorphic Sections and Complex Geometry

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Bernard Shiffman will continue his research on the statistics of random functions of several complex variables and more generally of random sections of powers 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. The research focuses on asymptotic results 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 find the variance of the number of simultaneous zeros of systems of random polynomials or entire functions in a domain. Another problem is to estimate "hole probabilities;" i.e., the probabilities that a random function has no solutions (or critical points) in fixed domains. 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. Shiffman will also study the distribution of zeros of real and complex "fewnomial" systems---polynomials of high degree with few terms---and he will investigate similar problems for spherical harmonics. The research program on critical points has two aspects: studying how the metric on a positive line bundle on a compact complex manifold affects the distribution of critical points of holomorphic sections, and secondly, how to count the distribution of vacua in string theory.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. One then looks for a way to statistically describe a random "landscape"---the hills and valleys of the graph of a function of several variables. One aspect of the geometry of the landscape that this project studies is the distribution of the local maxima (peaks), local minima, and saddle points (passes), the totality of which are called "critical points." 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. Another application of the research is to the "string theory landscape." String theory, which is a framework that unifies general relativity and quantum mechanics, postulates that space has 6 hidden dimensions rolled up into a small compact shape, called a Calabi-Yau manifold. This manifold is described by many parameters, whose possible values are given as critical points of a function. This project investigates the enumeration and distribution of the critical points corresponding to possible descriptions of our universe.