Selected topics in discrete geometry

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Selected Topics in Discrete Geometry: Discrete geometry is a young and fast developing area on the border line of pure mathematics and computer science. The phrase discrete geometry, which at one time stood mainly for the areas of packing, covering and tiling, has gradually grown to include such areas as combinatorial geometry, convex polytopes, and arrangements of lines, circles, planes, spheres and other geometric objects in the plane and in higher dimensions. Also, discrete geometry has a number of applications in other fields such as in digital communications. Namely, one of the fundamental questions in communications theory is to determine the densest packing of equal balls in multidimensional space. Actually, the proposed research is a rather broad one investigating densest sphere packings in spherical space, curved analogues of convex polytopes called ball-polyhedra, (partial) coverings by cylinders and planks, illumination problems as well as (short) billiards of (convex) billiard tables. The proposed collaborators include my graduate students (resp., advanced undergraduate students) at the University of Calgary, the visitors of the Center for Computational and Discrete Geometry at the University of Calgary and leading researchers from institutions in Canada, USA and Europe.