Homological Methods in Equivariant Fuller Index Theory

动力系统的研究是许多科学关注的基本问题。许多现有的理论在对某些物体的行为的纯粹的内在数学兴趣和在物理学或经济学中的大量应用之间提供了一座桥梁。额外对称性的存在使得用专门调整的方法对系统进行更有效的研究，因此往往大大降低了系统的复杂性。在这个研究项目中，我们想要研究具有额外对称性的动力系统。对这类系统的研究特别感兴趣的是它们的周期轨道，它们在某种意义上完全表征了这个系统。通过赋予一个有理数不变量来计算一个系统的周期轨道是可能的。这允许根据这个不变量的值对这些系统进行相当粗略但仍然重要的分类。直到最近，这个所谓的指数才被调整到具有对称性的系统。但仍有许多问题需要回答。例如，直接计算索引是极其困难的。在这个项目中，对称系统的指标应该被整合到一个更广泛的数学框架中。这将大大加深对这个不变量性质的理解，甚至可能最终导致更好的计算方法。特别令人感兴趣的是利用代数-拓扑方法对指标的几何解释，即所谓的等变同调理论，以提供几何和动力学之间的联系。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，这些理论具有重要的理论意义，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.