The theory of simplicial and polyhedral surfaces has combinatorial notions of paths and geodesics, quite distinct from the case of smooth surfsces. Thus it requires a separate theory of "how to move information along paths on the surface", which also has useful and interesting extensions beyond the two-dimensional case of surfaces. The project is centered around the interplay of combinatorics and topology with a focus on graph colorings and related subjects. This field is both classical and very active.

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This is a continuation of the previous project "Branched coverings and combinatorial holonomy". The main goal of this project is to exhibit and to analyze high genus surfaces that appear embedded (or immersed) in higher-dimensional cubical manifolds. For this we build on techniques such as combinatorial holonomy concepts and branched coverings that were developed in the first funding period. Additionally, our methods will use discrete concepts of combinatorial curvature in the sense of Alexandrov and Gromov in an essential way. In order to tackle known open problems about cubical (and other polyhedral) surfaces we want to access a wider class of interesting candidates (in particular, high curvature/high genus surfaces, and surfaces with extremal f-vector).

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