Complex Analysis and Geometry

摘要奖：DMS-1105586首席研究员：丹尼尔·M·伯恩斯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，值分布理论和阿弗斯流，以及格劳特管。在物理学中，玻尔-索末菲理论是作为量子化可积经典力学系统的一种方法而创立的。PrintalInvestigator将量子化的玻尔-索末菲条件解释为一个几何性质，命名为某个平坦丛的完整的平凡条件。正在进行的工作试图了解哈密顿奇点和基本复数空间奇点之间的联系，而玻尔-索末菲结构的几何似乎是故事的中心焦点。值分布理论是试图计算复变量方程解的框架。代数基本定理告诉我们，如果你仔细计算重复的根，一元去绿的多项式方程在复数上恰好有n个解。单个复变量中的更复杂的方程可能有无限多的解，但这些解分布在复平面上，半径为R的球体中包含的解的数量只能随着R变大而以有限的速度增长--这一情况由芬兰数学家罗尔夫·内万林纳在20世纪20年代引入的想法大大阐明了这一点，大约与玻尔-索末菲理论在物理学中发展的同一时间。该奖项支持的项目之一是应用现代几何工具研究多变量方程的解的分布。

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.