"The numerical simulations that are used in science and industry require ever more sophisticated mathematics. For the partial differential equations that are used to model physical processes, qualitative properties such as conserved quantities and monotonicity are crucial for well-posedness. Mimicking them in the discretizations seems equally important to get reliable results.This project will contribute to the interplay of geometry and numerical analysis by bridging the gap between Lie group based techniques and finite elements. The role of Lie algebra valued differential forms will be highlighted. One aim is to develop construction techniques for complexes of finite element spaces incorporating special functions adapted to singular perturbations. Another is to marry finite elements with holonomy based discretizations used in mathematical physics, such as the Lattice Gauge Theory of particle physics and the Regge calculus of general relativity. Stability and convergence of algorithms will be related to differential geometric properties, and the interface between numerical analysis and quantum field theory will be explored. The techniques will be applied to the simulation of mechanics of complex materials and light-matter interactions."