Complex Dynamics in Higher Dimensions

最一般而言，动力系统是指根据明确的数学规则从一个时刻演化到下一个时刻的任何一组环境或物理对象--例如，经济、病毒流行病或太阳系。使用这种瞬息万变的规则来预测系统在相对遥远的未来的状态往往是重要的，但也是相当困难的。监管的改变会导致投机泡沫吗？目前的干预措施是否足以阻止疫情爆发？太阳系会分裂吗？在广义的数学意义上，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“稳定的”还是“不稳定的”。也就是说，它是随着系统的发展而缓慢而可预测地变化，还是容易随着系统中的微小变化而迅速而混乱地变化。这个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，该项目的资金将支持首席调查员的研究生以及几名本科生，他们正在帮助当地K-12学生管理数学圈。通过首席研究员与河湾数学中心的参与，它将进一步感受到更广泛的影响。河湾数学中心是一个当地独立的非营利性组织，致力于促进各级数学教育。这个项目涉及分析、复杂代数几何和动力系统之间的交叉问题。这些问题源于构造和分析射影空间有理自映射的极大熵测度的一个非常通用的程序。这项工作将突出地突出保持亚纯二型的有理映射的特殊情况。需要研究的具体问题包括地图度数增长的“代数稳定性”、产生不变度量的动态自然闭合电流的产生和相交，以及有理曲面上真实自同构动力学的组合“轨道”模型。这一奖项反映了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标准进行评估，被认为值得支持。

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.