Complex Analysis and Geometry

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

Abstract Award: DMS-1105586 Principal Investigator: Daniel M. Burns Some of the themes emphasized in these research projects are modern aspects of the Bohr-Sommerfeld theory from the early days of quantum mechanics in the 1920s, value distribution theory and Ahlfors currents, and Grauert tubes. In physics the Bohr-Sommerfeld theory was created as a method for quantizing an integrable classical mechanical system. The principal investigator interprets the Bohr-Sommerfeld conditions for quantization as a geometric property, name a triviality condition on the holonomy of a certain flat bundle. Ongoing work seeks to understand the connections between singularities of Hamiltonians and singularities of underlying complex spaces, and the geometry of the Bohr-Sommerfeld construction seems to be a central focus of the story. Value distribution theory is a framework for attempting to count solutions to an equation in complex variables. The Fundamental Theorem of Algebra tells us that a polynomial equation of degree n in one variable has exactly n solutions over the complex numbers, if you count repeated roots with care. More complicated equations in a single complex variable can have infinitely many solutions, but these are spread around the complex plane and the number of solutions contained in a ball of radius R about the origin can only grow at a limited rate as R becomes large -- a situation greatly clarified by ideas introduced by the Finnish mathematician Rolf Nevanlinna in the 1920s, at about the same time that the Bohr-Sommerfeld theory was developed in physics. One of the projects supported by this award is a project that applies modern geometric tools to study the distribution of solutions to equations in several variables.