Mathematical Sciences: Applications of Algebraic Topology to Fixed Point Theory, Dynamics, and Cohomology of Groups

9401073 Geoghegan教授将研究代数和拓扑界面上的问题，涉及不动点理论、动力学和群论。他与加拿大拓扑学家A·尼卡斯合作，提出了一个新的“单参数不动点理论”。由此得到了非奇异流的Fuller指数型不变量的迹公式，以及悬浮流的新的K_1不变量和“非对易”Zeta函数。将后者扩大到非暂停流动是本项目的一个目标。它们的“一阶欧拉特征”是拓扑学和群论中的一种新的不变量，具有类似于经典(“零阶”)欧拉特征的性质。他们将继续在特定情况下计算这一点，并开发出完整的理论。该建议还包括Geoghegan教授在几何群论方面的兴趣，特别是K(G，1)复形的局部性质，这意味着G在Bieri和Eck mann意义下是对偶群。他寻找一个与对偶群具有相同关系的准则，就像流形性质与Poincare对偶群具有相同的关系一样。这个项目是关于拓扑学和群论之间的联系的，拓扑学是一种基本几何，群论是关于对称性的形式研究。外行人不习惯把一个物体所具有的对称性的“群”与这个物体本身分开，但群论做到了这一点。它将对称性从它们第一次出现的地方转移到新的数学对象(在那里它们再次以对称的形式出现，但以不同的方式出现)，从而揭示了这个群体的微妙的新特征。这一一般哲学的一个方面，有时被称为几何群论，深入到代数拓扑学的深处，以其巨大的丰富的方法和例子，为正在研究的群体找到新的和具有启发性的背景。在本实例中，含时不动点理论的方法是特别相关的。这位研究人员和他的合作者开发了拓扑学这一部分的一个新方面，并正在使用它来检测某些类型的群的新不变量。虽然数学之外的应用不是驱动力，但他们已经成功地将他们的工作应用于动力学，特别是在某些情况下流动的封闭轨道的行为，动力学现在被认为是认真适用的数学。此外，这位研究人员还参与了与一位应用数学家和一位电气工程师的合作，使用他在电路理论中的一些方法。***

9401073 Geoghegan Professor Geoghegan will investigate problems at the interface of algebra and topology, involving fixed point theory, dynamics and group theory. In collaboration with the Canadian topologist A. Nicas, he has developed a new "one-parameter fixed point theory." From this come trace formulae for Fuller index-type invariants of non-singular flows, and new K_1-invariants and "non-commutative" zeta functions for suspension flows. The extension of the latter to non-suspension flows is a goal of the present project. Their "first order Euler characteristic" is a new invariant in topology and group theory having properties analogous to the classical ("zeroth order") Euler characteristic. They will continue to compute this in specific cases and develop a full theory. The proposal also covers Professor Geoghegan's interests in geometric group theory, with particular reference to local properties of K(G,1) complexes which imply that G is a duality group, in the sense of Bieri and Eckmann. He seeks a criterion having the same relationship to duality groups as the manifold property has to Poincare duality groups. This project is about the connections between topology, which is a kind of fundamental geometry, and group theory, which concerns the formal study of symmetries. The layperson is not accustomed to separating the "group" of symmetries possessed by an object from the object itself, but group theory does just that. It transfers the symmetries from where they first arose to new mathematical objects (where they again appear as symmetries but in a different way), thus revealing subtle new features of the group. One aspect of this general philosophy, sometimes called Geometric Group Theory, reaches into the depths of Algebraic Topology, with its enormous wealth of methods and examples, to find new and revealing settings for the group being studied. In the present instance, the methods of time-dependent fixed point theory are particular ly relevant. The investigator and his collaborator have developed a new aspect of this part of topology and are using it to detect new invariants of certain kinds of groups. While application outside mathematics is not the driving force, they have had success in applying their work to Dynamics, specifically to the behavior of closed orbits of flows in certain situations, dynamics now being considered seriously applicable mathematics. Furthermore, the investigator has participated in a collaboration with an applied mathematician and an electrical engineer to use some of his methods in the theory of electrical circuits. ***