Aim of the course:
To become familiar with algebraic topological tools for studying and understanding nonlinear (partial) differential equations and dynamical systems.
The evolution of a dynamical system is sensitive to initial conditions and model parameters. Physically it is impossible to measure initial conditions and model parameters with arbitrary precision. This is a sad state of affairs: the exact evolution of a dynamical system is not something that can be predicted and/or measured. It is therefore important to understand features of dynamical systems that do not change under perturbations of the system. Tools from (algebraic) topology are well-suited to this task as topological invariants are invariant under large classes of deformations.
In the first part of the course a variety of algebraic topological techniques are discussed that are important in the modern treatment of partial differential equations and dynamical systems. Among these are degree theory (finite and infinite dimensional), nonlinear Fredholm maps, variational techniques, Morse theory and Conley Index theory.
In the second part of the course Morse and Conley theory are applied to the geodesic flow. For example, we will show that the geodesic flow on any closed Riemannian manifold admits a non-trivial closed geodesic. The ideas discussed in this course are a double edged sword: Algebraic topology is not only a tool to understand the dynamics. In some cases the dynamical system is better understood than the ambient topology itself. In these cases it is possible to extract topological information from the dynamics. As an application of this idea we will compute the homology of loop spaces. If time permits we will discuss Bott periodicity.
- manifolds, differential forms, Riemannian metrics (J.M. Lee, “Introduction to Smooth manifolds”, 2nd Ed., Ch.’s 1-16)
- fundamental group, singular/simplicial homology* (A. Hatcher, “Algebraic Topology”, Ch.’s 0-2)
- basic functional analysis, definitions of Banach/Hilbert spaces and bounded operators (I. Gohberg and S. Goldberg, “Basic Operator Theory”, Ch.’s 1-2)
- ordinary differential equations, existence + uniqueness (L. Perko, “Differential Equations”, 3rd Ed., pp. 1-70)
* this is useful, basic homology theory will be reviewed during course
Dr. T.O. Rot (VU University)
Prof. dr. R.C.A.M. van der Vorst (VU University)