Mathematical Sciences: Complex Dynamical Systems and Mobius Groups

该奖项支持在复杂动力系统、莫比乌斯群及其拓扑对应物的研究中出现的问题的数学研究。工作重点是复变量的复函数和定义这些函数的域。第一组问题涉及有理函数集和它们通过复合确定的半群。这些半群是由单个函数的迭代得到的自然扩展动力系统。该理论引出了大量的动力学、组合和代数问题，并展示了包含正态集和Julia集的性质和动力学行为的丰富结构。研究这些半群的另一个原因是某些多项式半群与离散Mobius群的模空间紧密相连。动力系统的观点为研究和理解这种模空间提供了一种新的工具。第二个研究涉及到用不动点来描述圆上的收敛群的问题，以及确定在什么情况下平面上的收敛群与莫比乌斯群拓扑共轭的问题。在二维及高维中，有相当数量的收敛群，但并非所有的收敛群都共轭于莫比乌斯群。重要的是要努力确定必要的约束，以保证一个群具有与Mobius群相同的拓扑性质。经典函数理论引入了几何技术，这些技术已被证明具有比最初想象的更广泛的适用性。该学科利用几何解释与强大的分析技术之间的相互作用，在分析领域产生一些最完整的数学结果。在这个项目中，变分技术和几何推理有望导致有价值的物理意义的结果。＊＊＊

9400999 Hinkkanen This award supports mathematical research on problems arising in the study of complex dynamical systems, Mobius groups and their topological counterparts. The work focuses on complex functions of a complex variables and the domains on which these functions are defined. The first set of problems concerns sets of rational functions and the semigroups they determine via composition. These semigroups are natural extensions dynamical systems derived from iterations of a single function. The theory leads to a great many dynamical, combinatorial and algebraic questions and exhibits a rich structure involving properties and dynamical behavior of the set of normality and the Julia set. another reason for studying these semigroups is that certain polynomial semigroups are closely connected to moduli spaces of discrete Mobius groups. The dynamical systems point of view offers a new tool for studying and understanding such moduli spaces. A second line of investigation involves the question of characterizing a convergence group on the circle in terms of fixed points, as well as with the problem of determining under what circumstances a convergence group in the plane is topologically conjugate to a Mobius group. In dimensions two and higher, there are a considerable number of convergence groups, not all of which are conjugate to Mobius groups. It is important that efforts be made to determine the constraints necessary to guarantee that a group have the same topological properties as a Mobius group. Classical function theory has introduced geometric techniques which have proved to have far wider applicability than first imagined. The subject exploits the interplay between geometric interpretation with powerful analytic techniques to yield some of the most complete mathematical results in the field of analysis. In this project, variational techniques and geometric reasoning are expected to lead to results of valuable physical significance. ***