For more information about this meeting, contact Svetlana Katok, Becky Halpenny, Flossie Dunlop.

Title: | Geometry and arithmetic of Apollonian gasket |

Seminar: | PMASS Colloquium |

Speaker: | Alexandre Kirillov, University of Pennsylvania |

Abstract Link: | http://www.math.psu.edu/katok_s/Gasket.pdf |

Abstract: |

One of oldest and beautiful examples of a fractal set is the Apollonian gasket, arising as a maximal tiling of 2-dimensional sphere by non-intersecting discs. There are (at least) two remarkable facts:
1. The Descartes theorem, which gives the algebraic relation between the curvatures of four pair-wise tangent circles on the plane. I shall give the proof based on the geometry of Minkowski space.
2. The existence of integral realisations of the Apollonian gasket, where all curvatures are integers. I show some arithmetic properties of these curvatures and formulate several open questions.
I attached a picture of one integral realisation. |

### Room Reservation Information

Room Number: | MB113 |

Date: | 02 / 28 / 2013 |

Time: | 02:30pm - 03:20pm |