Homological Methods in Equivariant Fuller Index Theory

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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.