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

复杂动力学：群作用、Migdal-Kadanoff 重整化和遍历理论

• 批准号: 2154414
• 负责人: Roland Roeder
• 金额: $ 28.76万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154414&HistoricalAwards=false
• 关键词: Complex; dynamics; group; actions; Migdal

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.

动力系统是研究系统状态如何随时间变化的数学领域。 这种系统在科学的所有领域都很丰富，包括生物学，化学和物理学。 它们在我们的日常生活中也很容易看到，从描述疾病流行的方式到预测天气的方式。 鉴于系统的初始状态，人们想知道系统的未来状态以及系统的长期行为。 描述这种现实现象的方程式非常复杂，通常是针对手头系统定制的。 对于严格的研究而言，它们通常太困难了，科学家必须经常使用数值模拟来分析它们。 但是，通常可以通过研究更简单的系统来理解潜在的动态现象，这些系统可以用一个或两个变量来描述。 该项目将支持对两个复杂变量中迭代理性映射组成的动力系统的研究，该设置可以使用复杂分析和代数几何形状的强大工具。 该研究围绕三个原则进行设计：（1）探索两个或更多不同的数学领域之间的联系可能会导致令人惊讶的新结果，（2）可以更深入地研究具有额外背景的动态系统，并且（3）对具体示例的研究通常会导致更普遍的理论。 除其他外，该项目将支持博士学位。印第安纳大学普渡大学印第安纳波利斯的学生参与这些研究主题，从而在动态系统中培训。 这笔赠款的更广泛影响将通过主要研究人员指导来自印第安纳波利斯地区的高级高中生，并举办IUPUI高中数学竞赛，该竞赛每年与印第安纳州的大约60至100名高中生参与。这项研究与更高层中的复杂动力学有关。 主要目的是研究两次或更大维度的复杂歧管的全体形态（或理性）自变态的迭代，并且更普遍地研究有限生成的生物形态（或生育）自我塑料的行为。 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.理解基本系统将导致全体形态动态的有价值的理论结果。该奖项反映了NSF的法定使命，并使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来获得支持。