Random Holomorphic Sections and Complex Geometry

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Abstract:Bernard Shiffman will continue his research on the statistics of randompolynomials in several complex variables and more generally of randomsections of powers of positive line bundles on compact complex manifolds andon almost complex symplectic manifolds. He will investigate the "scalinglimit" statistics as the degree of the polynomial or the power of the linebundle goes to infinity when distances are rescaled so that densities arenormalized. He will study spacing of zeros, hole probabilities, paircorrelations for local maxima, and other topics. He will also studycorrelations of critical points of spherical harmonics. In anotherdirection, he will study the compact singularities of equidimensionalmeromorphic mappings into compact complex manifolds. He will also look fornew examples of Kobayashi hyperbolic hypersurfaces in complex projective3-space. Kobayashi hyperbolic spaces do not carry any entire holomorphiccurves; simple examples are the Cartesian squares of curves of genus greaterthan 1 and symmetric squares of generic curves of genus greater than 2. Hewill look for low-degree hyperbolic birational images of these surfaces incomplex projective 3-space.This research project is motivated by a need to understand complex quantummechanical systems. Quantum mechanics is the fundamental theory thatdescribes the behavior of atoms and molecules and their componentparticles--protons, neutrons, and electrons. These particles are describedby wave functions, which are solutions of Schrodinger's equation. The zerosand local maxima of wave functions give important information on states ofatoms and molecules; the zeros are known in quantum chemistry and physics asnodal lines. The behavior of random polynomials provide an elementary modelsimilar to complex quantum systems. Polynomials in several variablescorrespond to systems with several degrees of freedom, and those polynomialsof high degree correspond to wave functions for highly excited states. Theproject includes statistics on symplectic manifolds, which serve as themathematical models for the states of quantum systems. Another component ofthe research involves understanding the geometry of complex algebraicmanifolds, which play an important role in quantum field theory and providemodels for diverse physical phenomena.