Complex Analysis and Geometry

复杂分析和几何

• 批准号: 1105586
• 负责人: Daniel Burns
• 金额: $ 30.83万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105586&HistoricalAwards=false
• 关键词: Complex; Analysis; Geometry

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.

Abstractaward：DMS-11105586原理研究者：这些研究项目中强调的主题的Daniel M. Burnsome是1920年代早期量子力学的Bohr-sommerfeld理论的Aromodern方面，价值分布理论和Ahlfors Courner和Grauert Tibes。在物理学中，创建了一种用于量化仿生经典机械系统的方法。主评论家将Bohr-Sommerfeld条件解释为Quantrization作为几何特性，将琐碎的条件命名为某个扁平捆绑包的载体。正在进行的工作寻求tounderstand的联系之间的联系，而hamiltonians和基础复杂空间的奇异性之间的联系，而Bohr-sommerfeld结构的几何形状似乎是故事的核心焦点。值分布理论是试图在复杂变量中计数方程式计算方程的框架。代数的基本理论告诉我们，如果您谨慎地计算重复的根，一个变量中的degreen的多项式方程在复合人的溶液上恰好具有N溶液。 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物理学。该奖项支持的项目之一是一个项目，该项目旨在研究现代几何工具，以研究几个变量中方程的分布。