Spectral Stability, the Maslov Index, and Spatial Dynamics
Understanding the spectral stability of solutions to partial differential equations is an important step in predicting long-time dynamics. Recently, it has been shown that a topological invariant known as the Maslov Index can play an important role in determining spectral stability for systems that have a symplectic structure. In this talk, the notions of spectral stability and the Maslov Index will be introduced and an overview of recent results will be given. If time permits, the perspective of spatial dynamics will be discussed, as well has how these recent developments have led to a notion of spatial dynamics in multiple space dimensions.