When can you infer the state of a system from measurements of a signal? In a surprisingly diverse set of situations, rather precise bounds can be obtained on the number of measurements needed to constrain a system from the Whitney embedding theorem. This result of differential topology is easy to state, easy to use, and intuitively satisfying. It is also remarkably easy to demonstrate with simple consumer-grade equipment, as I´ll explain.

When the state space is small, then the system has a canonical representation. The general theory and canonical representation leads to robust, practical nonlinear "topological filters", which generalize the linear filters already used extensively. Because they are built topologically, the local structure of these filters can be tailored easily and provides a solid theoretical grounding for nonlinear matched filters. I will advocate for the wider application of topological methods within engineering, and show how topological filters can improve the quality of maritime radar images.