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Thursday, October 11, 2018

**Abstract:** Abstract: We will discuss results from a long-running project between myself and Anton Lukyanenko to understand the connections between continued fractions and hyperbolic geometry in higher dimensions. Our recent breakthrough has allowed us to consider the connections between complex continued fractions and three-dimensional real hyperbolic space, quaternionic continued fractions and five-dimensional real hyperbolic space, octonionic continued fractions and nine-dimensional real hyperbolic space, and also Heisenberg continued fractions and two dimensional complex hyperbolic space. We will discuss the implications of these connections on a variety of number theoretic problems, including rational approximation and the study of algebraic irrationals, and many new problems these results have led us to. This talk will have connections to dynamical systems and hyperbolic geometry.