Colloquium: Kevin Powell (Snow College)


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Title: Long Division, Hyperbolic Geometry, and Bernoulli Numbers

Abstract: There are interesting connections between arithmetic and geometry. I address two of these. First, the Euclidean (division) algorithm is only a special case of shuffling through hemispheres in 3-dimensional hyperbolic geometry. Further, this generalized “pseudo” Euclidean algorithm allowed me to generalize a computational technique. The very results from using the technique reveal the second connection between arithmetic and geometry: an uncanny correlation between Bernoulli numbers, which encode arithmetic information, and geometric invariants of 3-dimensional hyperbolic spaces. (These spaces are analogous to the classic modular curve.)