I will show how computational algebraic topology can be used to study high dimensional nonlinear dynamics. The talk is broken into three sections:
1. How topology combined with numerics can be used to rigorously studying chaotic dynamics in infinite dimensional systems
2. A brief discussion of homology and how it can be efficiently computed
3. How computational topology can be used to study high dimensional data sets. In particular, I will talk about how we are using it to study the dynamics of Rayleigh-Benard convection.