Euclidean Traveller in Hyperbolic World

This lecture reports on a journey of an imaginary traveller, who travels along a Euclidean line. She will be travelling to many different geometric worlds such as the Euclidean n-torus guided by Kronecker in 1884, closed hyperbolic surfaces guided by Hedlund in 1936 and closed hyperbolic n-manifolds guided by Ratner in 1991. She also adventures into hyperbolic manifolds of infinite volume in the 21st century. We will describe what she sees (the closure of the line she travels on) and explain the mathematics involved in understanding the various closures with a number of illustrations.