Complex Analysis and Geometry

摘要奖：DMS-1105586主要研究者：丹尼尔M.伯恩斯在这些研究项目中强调的一些主题是20世纪20年代量子力学早期的玻尔-索末菲理论的现代方面，价值分布理论和阿尔福斯电流，以及格劳尔特管。在物理学中，玻尔-索末菲理论是作为一种量子化可积经典力学系统的方法而创立的。主要研究者将量子化的Bohr-Sommerfeld条件解释为一种几何性质，称之为关于某个平坦丛的完整性的平凡性条件。正在进行的工作旨在了解奇异性的哈密顿和奇异性的基础复杂的空间之间的联系，和几何的玻尔-索末菲建设似乎是一个中心focusoftheory.Value分布理论是一个框架，试图countsolutions的一个方程的复变。代数基本定理告诉我们，如果你仔细计算重根，一个单变量的多项式方程在复数上正好有n个解。更复杂的方程在一个单一的复变量可以有无限多的解决方案，但这些都是分散在复平面和解决方案的数量包含在一个球的半径R的原点只能增长在一个有限的速度作为R变大-一个情况下大大澄清了思想介绍的芬兰数学家罗尔夫Nevanlinna在20世纪20年代，大约在同一时间，玻尔-索末菲理论在物理学中得到了发展。该奖项支持的项目之一是一个应用现代几何工具研究多变量方程解的分布的项目。

AbstractAward: DMS-1105586Principal Investigator: Daniel M. BurnsSome of the themes emphasized in these research projects aremodern aspects of the Bohr-Sommerfeld theory from the early daysof quantum mechanics in the 1920s, value distribution theory andAhlfors currents, and Grauert tubes. In physics theBohr-Sommerfeld theory was created as a method for quantizing anintegrable classical mechanical system. The principalinvestigator interprets the Bohr-Sommerfeld conditions forquantization as a geometric property, name a triviality conditionon the holonomy of a certain flat bundle. Ongoing work seeks tounderstand the connections between singularities of Hamiltoniansand singularities of underlying complex spaces, and the geometryof the Bohr-Sommerfeld construction seems to be a central focusof the story.Value distribution theory is a framework for attempting to countsolutions to an equation in complex variables. The FundamentalTheorem of Algebra tells us that a polynomial equation of degreen in one variable has exactly n solutions over the complexnumbers, if you count repeated roots with care. More complicatedequations in a single complex variable can have infinitely manysolutions, but these are spread around the complex plane and thenumber of solutions contained in a ball of radius R about theorigin can only grow at a limited rate as R becomes large -- asituation greatly clarified by ideas introduced by the Finnishmathematician Rolf Nevanlinna in the 1920s, at about the sametime that the Bohr-Sommerfeld theory was developed in physics.One of the projects supported by this award is a project thatapplies modern geometric tools to study the distribution ofsolutions to equations in several variables.