Random Holomorphic Sections and Complex Geometry

Johns Hopkins University, Baltimore MD

Investigators

Abstract

Abstract: Bernard Shiffman will continue his research on the statistics of random polynomials in several complex variables and more generally of random sections of powers of positive line bundles on compact complex manifolds and on almost complex symplectic manifolds. He will investigate the "scaling limit" statistics as the degree of the polynomial or the power of the line bundle goes to infinity when distances are rescaled so that densities are normalized. He will study spacing of zeros, hole probabilities, pair correlations for local maxima, and other topics. He will also study correlations of critical points of spherical harmonics. In another direction, he will study the compact singularities of equidimensional meromorphic mappings into compact complex manifolds. He will also look for new examples of Kobayashi hyperbolic hypersurfaces in complex projective 3-space. Kobayashi hyperbolic spaces do not carry any entire holomorphic curves; simple examples are the Cartesian squares of curves of genus greater than 1 and symmetric squares of generic curves of genus greater than 2. He will look for low-degree hyperbolic birational images of these surfaces in complex projective 3-space. This research project is motivated by a need to understand complex quantum mechanical systems. Quantum mechanics is the fundamental theory that describes the behavior of atoms and molecules and their component particles--protons, neutrons, and electrons. These particles are described 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. The behavior of random polynomials provide an elementary model similar to complex quantum systems. 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. The project includes statistics on symplectic manifolds, which serve as the mathematical models for the states of quantum systems. Another component of the research involves understanding the geometry of complex algebraic manifolds, which play an important role in quantum field theory and provide models for diverse physical phenomena.