Homological quantities provide robust computable invariants of dynamical systems well-adapted to numerical methods. As a consequence, several groups have actively implemented algebraic topological invariants to characterize the qualitative behavior of dynamical systems. Examples include tracking patterns of nodal domains, proving the existence of invariant sets in infinite-dimensional systems, and generating forcing theories for invariant sets based on topological indices. Extensions of Morse-theoretic ideas due to Conley (in the finite-dimensional setting) and Floer (in the infinite-dimensional cases) are especially relevant and are active foci of current techniques in applied dynamical systems. This workshop will bring together researchers in topological methods with those working in differential equations and the physical sciences.