Multivariable complex dynamics

多变量复杂动力学

• 批准号: 1066978
• 负责人: Jeffrey Diller
• 金额: $ 18.52万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-15 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1066978&HistoricalAwards=false
• 关键词: Multivariable; complex; dynamics

This proposal concerns problems in the intersection between several complex variables, complex algebraic geometry and dynamical systems. The problems stem from a very general program for constructing and analyzing measures of maximal entropy for rational self-maps of projective space. The work will begin with and be guided by the particular case of rational maps preserving a meromorphic two form. Specific issues to be investigated include `algebraic stability' for degree growth of maps, laminarity of invariant currents in codimension greater than one, and how to intersect dynamically natural closed currents to produce invariant measures. A dynamical system, at its most general, is any 'thing'-e.g. the weather, a bacteria colony, an economy, or a solar system-that evolves in time according to definite mathematical rules. Given the present state of the system, one often wants to know something about its future state. Will the earth warm significantly in the next century? Will the solar system fly apart? What are the long term financial effects of a given tax cut or government intervention? In a broad mathematical sense, all of these questions reduce to understanding whether some aspect of a dynamical system is 'stable' or 'unstable'. That is, does it vary slowly and predictably as the system evolves, and or is it prone to change rapidly and chaotically with a small variation in the system. The research proposed here aims at better understanding mathematics of dynamical systems, and in particular at determining and describing those parts of a system which are most unstable. Funding for this proposal will also support the two PhD students the PI is currently advising. It's broader impact will be felt through the PI's involvement with a Notre Dame summer program for high school math teachers and with the Riverbend Math Center, which is a local independent non-profit organization dedicated to promoting math education at all levels.

该提案涉及多个复杂变量、复杂代数几何和动力系统之间的交叉问题。这些问题源于一个非常通用的程序，用于构建和分析射影空间理性自映射的最大熵度量。这项工作将从保留亚纯两种形式的有理映射的特殊情况开始并以此为指导。 需要研究的具体问题包括地图度数增长的“代数稳定性”、大于一的余维不变流的层流性，以及如何动态地与自然闭合流相交以产生不变测度。从最普遍的意义上来说，动态系统可以是任何“事物”，例如：天气、细菌群落、经济体或太阳系——它们根据明确的数学规则随时间演化。考虑到系统的当前状态，人们常常想了解其未来的状态。 下个世纪地球会显着变暖吗？ 太阳系会飞散吗？ 减税或政府干预的长期财务影响是什么？ 从广义的数学意义上来说，所有这些问题都归结为理解动力系统的某些方面是“稳定”还是“不稳定”。 也就是说，随着系统的发展，它是否会缓慢且可预测地变化，或者是否容易随着系统的微小变化而快速而混乱地变化。 这里提出的研究旨在更好地理解动力系统的数学，特别是确定和描述系统中最不稳定的部分。 该提案的资金还将支持 PI 目前正在指导的两名博士生。 通过 PI 参与圣母大学高中数学教师暑期项目以及 Riverbend 数学中心，我们将感受到其更广泛的影响。Riverbend 数学中心是一个当地独立的非营利组织，致力于促进各级数学教育。