The applied analysis group studies mathematical problems arising from physical, chemical, geophysical, biophysical, and materials sciences. These problems are often described by time-dependent partial, ordinary, or integral differential equations, together with sophisticated boundary conditions, interface conditions, and external forcing. Nonlinear dynamical systems offer a geometrical and topological framework for detecting, understanding, and quantifying complex phenomena of these time-dependent differential equations. Partial differential equation theory allows us to correctly formulate well-posed problems and to examine behaviors of solutions, and thus also allows us to set the stage for efficient numerical simulations. Nonlocal equations arise from macroscopic modeling of stochastic dynamical systems with Lévy noise and from modeling long-range interactions, and consequently give an understanding of anomalous diffusions.