Events
DMS Graduate Student Seminar |
| Time: Feb 06, 2019 (03:00 PM) |
| Location: Parker Hall 249 |
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Details: Speaker: Osman Yardimci Title: On the Number of Cylinders Touching a Sphere Abstract: The kissing number problem is a packing problem in geometry where one has to find the maximum number of congruent non-overlapping copies of a given body so that they can be arranged with each touching a common copy. The most studied version of this problem is about the kissing number of the unit ball. A similar question was proposed by Wlodzimierz Kuperberg in 1990. Kuperberg asked for the maximum number of non-overlapping, infinitely long unit cylinders touching a unit ball. He conjectured that not more than six disjoint, infinitely long unit cylinders could touch a unit sphere. W. Kuperberg’s so called six cylinder problem is a well known, 28 year old problem in discrete geometry, and it is still an open problem. In 2015, Moritz Firsching showed an arrangement of 6 disjoint cylinders with radii of 1.0496594, where each cylinder touched a given unit ball. We worked and solved several variants of W. Kuperberg’s problem. For example, new bounds were proven concerning the number of tangent cylinders with various radii. We improved some already known bounds by elaborating on the method introduced by Brass and Wenk. An application of a deep theorem on circle packing by Musin also provided some non-trivial bounds. By a joint work with Andras Bezdek, we proved that seven infinitely long cylinders of radii 1.04965 (Firsching’s radius) cannot touch a unit sphere. In view of Firsching’s construction, this settles the Kuperberg question for radius 1.04965. |