CAREER: Dynamics in Several Complex Variables, in Context

随着时间而进化的系统似乎是几乎所有科学努力的核心，包括生物、化学、物理和社会科学。鉴于系统的当前状态，人们能预测未来的状态吗？系统状态的这种演变如何依赖于系统的参数？对于严格的研究来说，许多这样的动力系统太复杂了，所以人们求助于更简单的模型，这些模型有望表明人们应该在实验中预期的行为类型。这种更简单的模型的一个场所是全纯地图的迭代，这是本项目的主题。完成这些项目将更深入地了解动力系统在几个复杂变量中的基本性质，并将与其他数学领域建立联系。这个项目更广泛的影响主要是教育性的，重点是高中生和博士生。该项目的资金将有助于在当地和整个印第安纳州的高中生中培养对数学的兴趣和天赋。它将有助于在印第安纳大学-普渡大学培训至少一名受支持的博士生，并通过两个关于几个变量的全纯动力学的研讨会来培训来自美国各地的博士生。这个项目的主要研究目标是探索复流形的全纯自映射迭代与另一个数学或数学物理领域有关的几个例子。这种联系不仅允许其他领域的应用程序，而且来自该领域的上下文通常会给映射提供额外的结构，从而允许进行更深入的研究。提出了四个方案，将多复变量动力学与瑟斯顿理论、统计物理中的伊辛模型、由自相似群作用引起的某些算符的谱以及复时常微分方程组(Painleve方程)联系起来。建议的研究项目围绕三个原则设计：(1)探索两个或更多不同数学领域之间的联系可以产生令人惊讶的新结果，(2)具有来自另一个领域的额外背景的动力系统可以被更深入地研究，以及(3)对具体例子的研究通常导致对更一般定理的陈述和证明。与主要研究目标合作的是几个教育目标：(1)领导当地高中生进行特别定制的研究项目，(2)帮助举办和改进印第安纳大学-普渡大学高中数学竞赛，(3)领导博士生进行上述主要研究项目，以及(4)组织两个关于几个复杂变量的动力学的研讨会，这是博士生的培训部分。

Systems that evolve with time appear at the core of nearly all scientific endeavor, including biology, chemistry, physics, and the social sciences. Given the current state of the system, can one predict the future state? How does this evolution of the state of the system depend on the parameters of the system? Many such dynamical systems are far too complicated for a rigorous study, so one resorts to simpler models, which are hoped to indicate the types of behavior that one should expect experimentally. One venue for such simpler models is the iteration of holomorphic maps, the topic of this project. Completing these projects will provide a deeper understanding of fundamental properties of dynamical systems in several complex variables and it will build connections with other areas of mathematics. The broader impacts of this project are largely educational, focusing on both high school students and Ph.D. students. Funding for this project will help to develop an interest and talent in mathematics among high school students, both local and throughout Indiana. It will help to train at least one supported Ph.D. student at Indiana University-Purdue University and also to train Ph.D. students from all over the United States by means of two workshops on holomorphic dynamics in several variables.The main research goal of this project is to explore several examples where the iteration of a holomorphic self-map of a complex manifold has a connection with another area of mathematics or mathematical physics. Not only does the connection allow for applications in the other area, but the context coming from that area usually gives the mapping extra structure, allowing for a significantly deeper study. Four projects are proposed relating dynamics in several complex variables to Thurston theory, the Ising model from statistical physics, the spectra of certain operators arising from self-similar group actions, and complex time ordinary differential equations (the Painleve equations). The proposed research projects are designed around three principles: (1) exploring connections between two or more different areas of mathematics can lead to surprising new results, (2) dynamical systems having an additional context from another field can be studied significantly more deeply, and (3) a study of concrete examples often leads to statements and proofs of more general theorems. Partnered with the main research goals are several educational goals: (1) leading local high school students on specially tailored research projects, (2) helping to run and improve the Indiana University-Purdue University High School Math Contest, (3) leading Ph.D. students on the main research projects mentioned above, and (4) organizing two workshops on dynamics in several complex variables which have a training component for Ph.D. students.