Complex dynamics: group actions, Migdal-Kadanoff renormalization, and ergodic theory

动力系统是研究系统状态如何随时间变化的数学领域。这样的系统在包括生物学、化学和物理学在内的所有科学领域都很丰富。它们在我们的日常生活中也很容易看到，从描述疾病流行的可能发展方式到预测天气。给定系统的初始状态，人们想知道系统的未来状态是什么，以及系统的长期行为。描述这种现实世界现象的方程非常复杂，通常是根据手头的系统定制的。它们通常很难进行严格的研究，科学家必须经常使用数值模拟来分析它们。然而，潜在的动力学现象往往可以通过研究更简单的系统来理解，其状态可以用一个或两个变量来描述。该项目将支持由两个复杂变量的迭代有理映射组成的动力系统的研究，在此背景下，复杂分析和代数几何的强大工具是可用的。这项研究是围绕三个原则设计的：(1)探索两个或多个不同数学领域之间的联系可以带来令人惊讶的新结果；(2)从另一个领域获得额外背景的动力系统可以得到更深入的研究；(3)对具体例子的研究通常会产生更一般的理论。除此之外，该项目将支持印第安纳大学-普渡大学印第安纳波利斯分校的博士生参与这些研究课题，从而训练他们在动力系统方面的能力。通过首席研究员对来自印第安纳波利斯地区的高天赋高中生的指导，以及每年约有60至100名来自印第安纳州的高中生参加的IUPUI高中数学竞赛，这项资助的更广泛影响将进一步实现。本课题研究的是高维复杂动力学问题。主要目标是研究二维或二维以上的复流形的全纯（或有理）自映射的迭代，更一般地说，研究有限生成的生物全纯（或双全）自映射群的作用。本文将研究与其他数学领域相关的主题，包括：(1)基于painlevel 6微分方程单一性的复杂曲面上的全纯群作用，(2)与分层格上统计物理现象相关的Migdal-Kadanoff重整化映射，以及(3)具有超越一动力度的理性映射的遍历理论。理解底层系统将导致全纯动力学中有价值的理论结果。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Dynamical systems is the area of mathematics that studies how the state of a system changes with time. Such systems are abundant in all areas of science including biology, chemistry, and physics. They are also readily visible in our everyday lives, ranging from describing the ways in which a disease epidemic is likely to progress to predicting the weather. Given the initial state of the system, one would like to know what the future state of the system will be, as well as the long-term behavior of the system. The equations describing such real-world phenomena are very complicated and are usually custom tailored to the system at hand. They are typically far too difficult for rigorous study and scientists must often use numerical simulations to analyze them. However, the underlying dynamical phenomena can often be understood by studying simpler systems whose states can be described in terms of one or two variables. This project will support the study of dynamical systems consisting of iterating rational mappings in two complex variables, a setting where the powerful tools of complex analysis and algebraic geometry are available. The research is 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 more general theories. Among other things, this project will support Ph.D. students from Indiana University-Purdue University Indianapolis to engage in these research topics, thus training them in dynamical systems. The broader impacts of this grant will be further achieved through the principal investigator's mentoring of highly talented high-school students from the Indianapolis area, and running the IUPUI High School Math Contest which engages approximately 60 to 100 high-school students from Indiana each year.This research project is concerned with complex dynamics in higher dimensions. The main goal is to study the iterates of holomorphic (or rational) self-mappings of a complex manifold of dimension two or larger, and, more generally, to study the actions of finitely generated groups of biholomorphic (or birational) self-mappings. The topics to be investigated, which draw connections with other areas of mathematics, include: (1) holomorphic group actions on complex surfaces coming from the monodromy of the Painleve 6 differential equation, (2) Migdal-Kadanoff renormalization mappings associated to phenomena in statistical physics on hierarchical lattices, and (3) ergodic theory of rational maps with transcendental first dynamical degree. Understanding the underlying systems will lead to valuable theoretical results in holomorphic dynamics.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.