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、化学和物理在内的所有科学领域中都很丰富。它们在我们的日常生活中也随处可见，从描述疾病流行可能发展的方式到预测天气。考虑到系统的初始状态，人们想知道系统的未来状态以及系统的长期行为。描述这种真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。对于严格的研究来说，它们通常太难了，科学家必须经常使用数值模拟来分析它们。然而，潜在的动力学现象通常可以通过研究更简单的系统来理解，这些系统的状态可以用一个或两个变量来描述。这个项目将支持对由两个复变量中的迭代有理映射组成的动力系统的研究，在这种情况下，可以使用复数分析和代数几何的强大工具。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产生令人惊讶的新结果，(2)从另一个领域获得额外背景的动力系统可以被更深入地研究，以及(3)对具体例子的研究通常会产生更一般的理论。除其他外，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从事这些研究课题，从而培训他们的动力系统。这笔赠款的更广泛影响将通过首席研究员对印第安纳波利斯地区有才华的高中生的指导，以及举办IUPUI高中数学竞赛进一步实现，该竞赛每年有大约60到100名印第安纳州的高中生参加。这个研究项目关注更高维度的复杂动态。主要目的是研究二维或更大维复流形的全纯(或有理)自映射的迭代，更一般地，研究有限生成的双全纯(或双调)自映射群的作用。所要研究的主题与其他数学领域相联系，包括：(1)来自Painleve 6微分方程的单行的复曲面上的全纯群作用，(2)与层次格子上的统计物理现象相关的Migdal-Kadanoff重整化映射，以及(3)具有超越一阶动力的有理映射的遍历理论。了解基本系统将导致在全纯动力学方面的有价值的理论结果。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.