Homological Methods in Equivariant Fuller Index Theory

动力系统的研究是几门科学的基本问题。许多现有的理论提供了一个桥梁之间的纯粹内在数学的兴趣，在某些对象的行为和大量的应用程序在物理学或经济学。额外对称性的存在使系统的研究更有效，特别是调整的方法，因此往往显着降低其复杂性。在这个研究项目中，我们要研究具有额外对称性的动力系统。在研究这类系统时，特别令人感兴趣的是它们的周期轨道，它在某种意义上完全刻画了系统的特征。可以通过给它们分配一个有理不变量来计数系统的周期轨道。这使得根据这个不变量的值对这些系统进行一个相当粗略但却很重要的分类。只是最近，这个所谓的指数才被调整到具有对称性的系统。但是仍然有许多问题需要回答。例如，直接计算指数是非常困难的，在本项目中，对称系统的指数将被整合到一个更广泛的数学框架中。这将使一个显着的深化理解的性质，这个不变量，并可能在最后甚至导致更好的方法计算。特别感兴趣的是几何解释的指数使用代数拓扑方法，所谓的等变同调理论，提供了一个链接之间的几何和动力学。从数学的角度来看，这将是这些理论的一个非常有趣的应用，这些理论具有很大的理论意义，但由于它们的巨大复杂性，在应用中很难处理。

The study of dynamical systems is a basic concern of several sciences. Many existing theories provide a bridge between a purely innermathematical interest in the behaviour of certain objects and a plethora of applications in physics or economics. The presence of an additional symmetry enables a more efficient study of the system with specifically adjusted methods und therefore often a significant reduction of its complexity.In this research project, we want to investigate dynamical systems with additional symmetry. Of particular interest in the study of such systems are their periodic orbits, which in some sense completely characterize the system.It is possible to count the periodic orbits of a system by assigning them a rational invariant. This allows a rather rough but nevertheless important classification of these systems according to the value of this invariant.Only recently, this so called index was adjusted to systems with symmetry.But there are still many questions to answer. For example it is extremely hard to directly calculate the index.In this project, the index for systems with symmetry shall be integrated into a broader mathematical framework. This will allow a significant deepening of the understanding of the nature of this invariant and may in the end even lead to better methods of calculation.Of particular interest is the geometrical interpretation of the index by using algebraic-topological methods, so called equivariant homology theories, to provide a link between geometry and dynamics. From a mathematical viewpoint, this will be a very interesting application of these theories which have great theoretical meaning, but due to their enormous complexity are very hard to handle in applications.