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的交叉问题，涉及不动点理论、动力学和群论。在与加拿大拓扑学家a . Nicas的合作中，他发展了一种新的“单参数不动点理论”。由此得到了非奇异流的Fuller指数型不变量的迹式，以及悬浮流的k_1 -不变量和“非交换”zeta函数。将后者扩展到非悬浮流动是本项目的一个目标。他们的“一阶欧拉特征”是拓扑和群论中的一个新的不变量，具有类似于经典（“零阶”）欧拉特征的性质。他们将继续在具体情况下进行计算，并发展出一个完整的理论。该提案还涵盖了Geoghegan教授对几何群论的兴趣，特别是关于K（G,1）复合体的局部性质，这意味着G是Bieri和Eckmann意义上的对偶群。他寻找一个标准，它与对偶群的关系就像庞加莱对偶群的流形性质一样。这个项目是关于拓扑学（一种基本几何）和群论（关注对称的形式研究）之间的联系。外行人不习惯将物体所拥有的对称“群”与物体本身分开，但群论恰恰做到了这一点。它将对称性从它们最初产生的地方转移到新的数学对象上（在那里它们再次以对称的形式出现，但以不同的方式出现），从而揭示了群体微妙的新特征。这种一般哲学的一个方面，有时被称为几何群论，深入到代数拓扑的深处，用它丰富的方法和例子，为正在研究的群体找到新的和有启发性的设置。在这种情况下，时变不动点理论的方法是特别相关的。研究者和他的合作者开发了这部分拓扑的一个新方面，并使用它来检测某些类型群的新不变量。虽然数学以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体封闭轨道的行为，动力学现在被认为是非常适用的数学。此外，该研究人员还与一位应用数学家和一位电气工程师合作，将他的一些方法应用于电路理论。* * *

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. ***