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，从描述疾病流行可能发展的方式到预测天气。 给定系统的初始状态，人们希望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。 它们通常太难进行严格的研究，科学家必须经常使用数值模拟来分析它们。 然而，基本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。 该项目将支持由两个复变量的迭代有理映射组成的动力系统的研究，这是一个可以使用复分析和代数几何的强大工具的环境。 该研究围绕三个原则设计：（1）探索两个或更多不同数学领域之间的联系可能会导致令人惊讶的新结果，（2）具有另一个领域的额外背景的动力系统可以更深入地研究，（3）对具体例子的研究通常会导致更一般的理论。 除此之外，该项目将支持博士学位。学生从印第安纳州普渡大学印第安纳波利斯从事这些研究课题，从而训练他们在动力系统。 这项补助金的更广泛的影响将进一步通过首席研究员的指导从印第安纳波利斯地区的高度有才华的高中生，并运行IUPUI高中数学竞赛，每年约有60至100名高中生从印第安纳州。这个研究项目是在更高的维度复杂的动态关注实现。 主要目标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，研究双全纯（或双有理）自映射的生成群的作用。 研究的主题与其他数学领域有联系，包括：（1）复杂曲面上的全纯群作用，来自Painleve 6微分方程的monodromy，（2）与统计物理现象相关的Migdal-Kadanoff重整化映射，以及（3）具有超越第一动力学度的有理映射的遍历理论。理解潜在的系统将导致在全纯动力学有价值的理论成果。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.