Saturday Evening Plenary Lecture:

"Symmetries, Ping-Pong and Outer Space"

Abstract:
A group is a very basic algebraic structure in mathematics. One can often find an alternate description of this algebraic object as symmetries of a geometric object. In fact it turns out that there is a natural way to think of any algebraically-described group as symmetries of a geometric object, and the geometric properties of this object can reveal features of the group which were not obvious from the original algebraic description. One of the simplest of all types of groups are the so-called free groups. I will explain the geometry of free groups, and use it to show how to recognize when a group which looks like it might be very complicated is in fact just a free group. The method is to use some geometric object as a ping-pong table. The reason for this terminology will be clear when the method is described. The symmetries of free groups themselves form a group; the ping-pong table for this group is known as Outer space. I will describe this space and show how to play ping-pong in Outer space.