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This seminar will look at tilings. We start with planar tilings, their properties, their generation e.g. by crystallographic groups, as well as at attempts of classification. (This quickly leads us to unsolved problems. For example, which pentagons tile the plane by congruent copies?) Then we look at 3-dimensional tilings and their properties. New questions arise here: Which (combinatorial types of) polyhedra can be used to tile space? How many faces can a polyhedron have whose congruent copies tile space? -- this seminar will mostly take place in English --
Extensions of polytopes The extension complexity of a polytope P is the minimal number of facets of a polytope Q that linearly projects onto P. This rather simple definition has interesting consequences and relations to areas such as discrete geometry, combinatorial optimization, information theory, and linear algebra. Determining the extension complexity of a polytope is extremely hard (even for polygons!) and obtaining exact values or even just bounds for special polytopes is an active area of research. The goal of the seminar is to develop a good understanding of extension complexity and the notions related to it. Topics might include
The seminar is aimed at students with an interest in discrete and convex geometry, discrete mathematics / combinatorial optimization, and linear algebra. The prerequisites for most topics is a basic understanding of polytopes (such as Discrete Geometry I).
The first meeting of the seminar will take place during the first week of the semester.
Extensions of polytopes
Federico Ardila und Richard P. Stanley: Pflasterungen, Math. Semesterberichte 53 (2006), 17-43. John H. Conway and Jeffrey C. Lagarias: Tiling with polyominoes and combinatorial group theory, J. Combinat. Theory, Ser. A, 53 (1990), 183-208. David Eppstein, John M. Sullivan and Alper Ungor: Tiling space and slabs with acute tetrahedra, Comput. Geometry: Theory & Applications 27 (2004), 237-255. Branko Grünbaum and Geoffrey C. Shephard: Tilings with congruent tiles, Bull. Amer. Math. Soc. 3 (1980), 951-973. Branko Grünbaum and Geoffrey C. Shephard: Tilings and Patterns, Freeman 1987. Egon Schulte: Tilings, in: Handbook of Convex Geometry (P. Gruber and J. Wills, eds.), North-Holland, Amsterdam 1993, 899-932.