"Discrete structures appear throughout mathematics not only as approximations to continuous objects, but also as mathematical objects of their own right. The ""right"" discrete models should have analogous theory to the continuous limit, but often more transparent, more interesting structure, it ""tells you more"". The proposed project has the agenda to connect, and make substantial progress in, a number of interesting, but rather diverse instances for this, including- Convex Polytopes as models for linear, semi-definite and non-linear optimization problems,- Polyhedral Surfaces as models for differential geometry, including questions of (discrete) integrability,- Structured meshes as the ""right"" discrete structures for solving systems of partial differential equations with quality guarantees,- Triangulation models as they appear as models for space in quantum gravity.In this simultaneous treatment of these topics we hope to capture connections and identify analogous and parallel structures in different parts of mathematics. This is a theory proposal, but a number of the core topics are suggested by applied research, as done e.g. in the framework of the Berlin DFG Research Center MATHEON in Berlin. It will connect to, and rely on, other major structured research groups in Berlin, such as the ""Polyhedral Surfaces"" DFG Research Group, and the Research Training Group led by the PI. In collaboration between individuals and groups with diverse mathematical expertise in Berlin, throughout Europe and beyond we are set to establish an additional ""theory backbone""; for applied research in Berlin."