Topology, Algebraic Geometry, and Dynamics Seminar (TADS): Higher-dimensional continued fractions

Speaker:  Anton Lukyanenko, GMU

Title:  Higher-dimensional continued fractions

Abstract:  Continued fractions and their applications have been studied over the real numbers since at least Huygens (1682). Early generalizations include Hamilton's quaternionic CFs (1852) and Hurwitz's complex CFs (1887). We show that CFs can be defined in even more exotic spaces, such as the octonions (non-associative), the Heisenberg group (not even a division algebra!), R^3, and Minkowski space. Recovering standard properties such as convergence of the CF representation and ergodicity of the generalized Gauss map (which removes the first digit of the CF) leads to both interesting hands-on analysis of cylinder sets and to connections to geodesic flow on certain hyperbolic manifolds. In particular, it yields new results for both real and complex CFs. This is joint work with Vandehey and ongoing work with Burelle, Barreto-Rosa, and Hartt.

Time:  Friday, September 27, 1:30pm – 2:30pm

Place:  Exploratory Hall, room 4208