Continued fractions are very classically studied objects in number theory. It is a way to represent a real number as a sequence of integers sort of like the decimal expansion. We'll see how to represent a geodesic of the modular surface SL(2,Z)\H by continued fractions. Then, we'll relate the geodesic flow with the shift map on sequences of integers N^Z. This construction can be generalized to all hyperbolic surfaces. Time permitting, some applications of this result will be discussed. Speaker(s): Chris Zhang (UM)