Mathematical Sciences: Topology and Dynamics in Dimension Two

Handel将研究拓扑学和动力学两个主题。一是延续前人关于Nielsen理论在无限欧拉特征曲面上应用的研究，重点研究了同胚的同位素分类，这些同胚在曲面两端的作用是（同伦）周期的。该项目的第二部分涉及表面微分同构的不变火车轨道（在Thurston的意义上）。在这个领域，我们所知甚少，亨德尔将从标准马蹄形图中产生的微分同态开始，通过放大一组有限的周期轨道。表面变换的动力学显然是一个物理概念，从纯粹的拓扑考虑中可以获得多少信息是令人惊讶的。

Handel will work on two topics in topology and dynamics. One is a continuation of previous research concerning applications of Nielsen theory to surfaces of infinite Euler characteristic, with emphasis on the isotopy classification of homeomorphisms whose action on the ends of the surface is (homotopically) periodic. The second part of the project concerns invariant train tracks (in the sense of Thurston) for surface diffeomorphisms. Very little is known in this area, and Handel will begin with a detailed study of those diffeomorphisms that arise from the standard horseshoe map by blowing up a finite set of periodic orbits. The dynamics of surface transformations is clearly a physical concept, and it is surprising how much information can be obtained from purely topological considerations.