Multivariable complex dynamics

这一建议涉及多个复变量、复代数几何和动力系统之间的交集问题。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。这项工作将从保持亚纯二型的有理映射的特殊情况开始并由其指导。要研究的具体问题包括映射度增长的“代数稳定性”，余维不变流大于1的层状性，以及如何与动态自然闭合流相交以产生不变度量。动力系统，在其最一般的情况下，是任何“事物”--例如。天气、菌落、经济或太阳系--它们是根据明确的数学规则在时间上进化的。鉴于目前的系统状态，人们经常想知道一些关于其未来状态的信息。地球在下个世纪会显著变暖吗？太阳系会分裂吗？一项特定的减税或政府干预的长期财务影响是什么？在广义的数学意义上，所有这些问题都归结为理解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。这项研究的目的是为了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。这项提议的资金也将支持国际和平协会目前建议的两名博士生。它的更广泛的影响将通过PI参与圣母大学为高中数学教师举办的暑期项目以及致力于促进各级数学教育的当地独立非营利性组织河湾数学中心而感受到。

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.