In the period from January 16 to January 19 of 2018, our guest is dr Axel Jänig from the Institute of Mathematics of the University of Rostock in Germany. During his visit, on Thursday of the 18th of January, at the Seminar of the Department of Nonlinear Analysis and Statistics, he will give a talk on Nonautonomous Conley index theory and asymptotically autonomous differential equations.
Abstract: The talk is divided into two parts. First of all, a nonautonomous flavour of Conley index theory is introduced. Just like in the classical, autonomous case, the index is defined by using so-called index pairs. Invariant sets are obtained from a skew-product formulation of the nonautonomous problem, so index pairs and invariant sets are located in different spaces. In addition to its basic construction, we will discuss important properties of the index. In the second part, either ordinary or semilinear parabolic asymptotically autonomous differential equations are considered. A generic subclass of these equations admits Morse-decompositions resembling Morse-Smale dynamical systems. We can relate the index of an isolated invariant set to the existence of generalised equilibria. In particular, a non-trivial index implies the existence of hyperbolic solutions of the asymptotically autonomous problem.