Discrete Differential Geometry and Cluster Algebras via TCD maps

In this PhD thesis we develop the frame work of triple crossing diagram maps (TCD maps), which describes constrained configurations of points in projective spaces and discrete dynamics on these configurations. We are able to capture the constraints and dynamics of a large list of examples that occur in discrete differential geometry (DDG), discrete integrable systems and exactly solvable models. We explain how to apply various geometric operations to TCD maps, including projections, intersections with hyperplanes and projective dualization. In fact, we show how many examples in the literature are related by the aforementioned operations. Moreover, we introduce a hierarchy of cluster structures on TCD maps, thus answering the open question how objects of DDG relate to cluster structures. At the same time, the general cluster structure reproduces cluster structures known for the pentagram map, T-graphs and t-embeddings. We also explain how the cluster structures behave under geometric operations. Via the cluster structures, the TCD maps are also related to the probabilistic dimer model. The spanning tree model and the Ising model can be obtained as special cases of the dimer model, and we investigate how these special cases relate to geometry. This also leads to two new incidence theorems in relation to quadrics and null-polarities in $\mathbb C \mathrm P^3$. Finally, we also show how TCD maps relate to the Fock-Goncharov moduli spaces of projective flag configurations.