权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Complex Dynamics in Higher Dimensions

高维中的复杂动力学

基本信息

批准号：
1954335
负责人：
Jeffrey Diller
金额：
$ 28.65万
依托单位：
University of Notre Dame
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-05-01 至 2024-04-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1954335&HistoricalAwards=false
关键词：
Complex Dynamics Higher Dimensions

项目摘要

A dynamical system, at its most general, is any set of circumstances or physical object—e.g., the economy, a viral epidemic, or a solar system—that evolves from one moment to the next according to definite mathematical rules. It is often important but quite difficult to use such moment-to-moment rules to predict the state of the system in the relatively distant future. Will a change in regulation lead to a speculative bubble? Will present interventions suffice to stop an outbreak? Will the solar system fly apart? In a broad mathematical sense, all of these questions reduce to understanding whether some aspect of the dynamical system in question is 'stable' or 'unstable'. That is, does it vary slowly and predictably as the system evolves, or is it prone to change rapidly and chaotically with a small variation in the system. The research in this project aims at better understanding the mathematics of dynamical systems, and in particular at determining and describing those parts of a system which are most unstable. Funding for this project will, among other things, support the principal investigator's graduate student as well as several undergraduates who are helping to run math circles for local K-12 students. It's broader impact will be further felt through the principal investigator's involvement with the Riverbend Math Center, which is a local independent non-profit organization dedicated to promoting math education at all levels.This project concerns problems in the intersection between analysis, 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 prominently feature the particular case of rational maps preserving a meromorphic two form. Specific issues to be investigated include `algebraic stability' for degree growth of maps, the manufacture and intersection of dynamically natural closed currents to produce invariant measures, and combinatorial `train track' models for the dynamics of real automorphism on rational surfaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

从最普遍的意义上来说，动态系统是任何一组环境或物理对象，例如经济、病毒流行病或太阳系，它们根据确定的数学规则从一个时刻到下一个时刻演变。使用这种即时规则来预测系统在相对遥远的未来的状态通常很重要，但相当困难。监管的变化会导致投机泡沫吗？目前的干预措施足以阻止疫情爆发吗？太阳系会飞散吗？从广泛的数学意义上来说，所有这些问题都归结为理解所讨论的动力系统的某些方面是“稳定”还是“不稳定”。也就是说，它随着系统的发展而缓慢且可预测地变化，还是随着系统的微小变化而容易快速且混乱地变化。该项目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。除其他外，该项目的资金将用于支持首席研究员的研究生以及帮助当地 K-12 学生开展数学圈的几名本科生。通过首席研究员对 Riverbend 数学中心的参与，将进一步感受到其更广泛的影响，该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。该项目关注分析、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。该作品将突出展示保留亚纯两种形式的有理图的特殊情况。要研究的具体问题包括地图度数增长的“代数稳定性”、动态自然闭合电流的制造和交集以产生不变测量，以及有理表面上真实自同构动力学的组合“火车轨道”模型。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查进行评估，被认为值得支持标准。