CAREER: Algebraic structures in complex dynamics

最简单但非平凡的复杂动力系统是代数的：在一个复变量中的不可逆多项式的迭代，更一般地，复代数簇的有理自映射。 与这些全纯系统相关的某些解析量与代数结构有着惊人的联系。 这个项目的例子说明了在复杂的动力学设置丰富的代数结构包括：（1）一维多项式的研究，通过其相关的树和连接上的领域的正规函数的赋值模空间;（2）分析正规自映射环面品种，这种映射的结果，并与多位理论的连接;（3）正则自映射的模空间的结构（例如射影空间）和它们作为代数簇的“动力学”紧化（描述这种系统如何退化）。这种动力学系统最熟悉的例子之一是牛顿方法，一种求多项式根的迭代算法，首先在微积分课程中介绍。 即使对于这个著名的、看似简单的例子，我们也只是在最近才开始利用现代数学技术来理解它的结构，以及它的一般故障。 用牛顿的方法（它的历史，它的应用，和最近的研究）作为指导，我将追求学生的教育项目的集合：（1）本科研究项目，重点是计算机探索的例子;（2）在动力学本科课程的发展，目前的理论和最近的应用;（3）两个研讨会的研究生;（4）定期举办学生研讨会。 上述研究旨在探索混沌动力系统（如牛顿方法及其推广）和任何潜在的代数结构之间的相互作用。 这样的结构为一个看起来难以处理的系统增加了额外的对称性或规则性。 在复杂动力系统的这些方面的专家集中在法国，英国，日本和美国（与智利，西班牙，波兰和德国，仅举几例较小的群体）;与这个项目，首席研究员将汇集这些专家与学生一起工作，并澄清代数和动力学之间的这些连接。

The simplest yet non-trivial complex dynamical systems are algebraic: iteration of noninvertible polynomials in one complex variable and, more generally, rational self-maps of a complex algebraic variety. Certain analytic quantities associated to these holomorphic systems have surprising connections to algebraic constructions. Examples from this project that illustrate the rich algebraic structures in a complex dynamical setting include: (1) the study of one-dimensional polynomials by their associated trees and the connection with valuations on the field of regular functions on the moduli space; (2) an analysis of regular self-maps on toric varieties, the resultant of such mappings, and connections with pluri-potential theory; (3) the structure of the moduli spaces of regular self-maps (e.g. of projective spaces) and their "dynamical" compactifications (describing how such systems degenerate) as algebraic varieties.One of the most familiar examples of such a dynamical system is Newton's method, an iterative algorithm for finding roots of polynomials, first introduced in calculus courses. Even for this famous and seemingly simple example, we have only recently begun to understand its structure, and its failure in general, using modern mathematical techniques. With Newton's method (its history, its applications, and recent studies) as a guide, I will pursue a collection of educational projects for students: (1) undergraduate research projects, with an emphasis on computer exploration of examples; (2) the development of an undergraduate course in dynamics, to present both the theory and recent applications; (3) two workshops for graduate students; and (4) regular student seminars. The research described above is designed to explore the interplay between chaotic dynamical systems (such as Newton's method and its generalizations) and any underlying algebraic structures. Such structures add an extra element of symmetry or regularity to a system which might otherwise seem intractable. The experts in these aspects of complex dynamical systems are concentrated in France, England, Japan, and the US (with smaller groups in Chile, Spain, Poland, and Germany, to name a few); with this project, the Principal Investigator will bring together some of these experts to work with students and clarify these connections between algebra and dynamics.