Mathematical Sciences: Complex Dynamical Systems and Mobius Groups

小行星9400999 该奖项支持对复杂动力系统、莫比乌斯群及其拓扑对应物研究中出现的问题进行数学研究。 这项工作的重点是复变函数的复变量和域上，这些功能的定义。 第一组问题涉及有理函数集和它们通过合成确定的半群。 这些半群是由单个函数的迭代得到的自然扩张动力系统。 该理论导致了大量的动力学，组合和代数问题，并表现出丰富的结构，包括属性和动力学行为的正规集和Julia集。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。 第二条线的调查涉及的问题的特点收敛群在圆方面的不动点，以及与问题的确定在什么情况下收敛群在平面上是拓扑共轭的莫比乌斯组。 在二维和更高维中，有相当数量的收敛群，并非所有的收敛群都与莫比乌斯群共轭。 重要的是，要努力确定必要的约束，以保证一个群具有相同的拓扑性质作为莫比乌斯群。 经典函数论引入了几何技术，这些技术已被证明具有比最初想象的更广泛的适用性。 该科目利用强大的分析技术与几何解释之间的相互作用，产生一些分析领域最完整的数学结果。在这个项目中，变分技术和几何推理预计将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. ***