We are surrounded by patterns that change their shape and structure with time -- particular examples include: chemical concentrations in materials, velocity profiles in fluids, population densities in ecosystems, etc. With modern information technologies it is relatively easy to collect enormous data on these patterns either through experimentation or numerical simulation. One goal of this project is to develop and employ computational topological techniques to use the observed patterns to identify, quantify and classify the time-dependent properties of the underlying systems.
Mathematical models for spatially dependent systems that evolve with time are typically extremely difficult to analyze using traditional mathematical techniques, and thus much of what we know about the detailed evolution of these systems comes from numerical simulations. However, the process of performing these numerical simulations introduces errors, which can potentially grow and propagate. In principle, algebraic topological properties remain invariant under small perturbations. With this in mind, another goal of this project is to combine the newly developed computational topological tools with standard numerical methods to verify that the solutions obtained through numerical simulation are indeed valid results for the systems being studied.
