As mathematicians, we often wonder what happens when we take our favorite
mathematical objects to the extreme. Those interested in the geometry of
spaces, for instance, may talk about an ideal boundary of a space; the
points of the ideal boundary are actually limits of points traveling
infinitely far away from a fixed basepoint. Another geometric approach to
the infinite is to "zoom out" by re-scaling the space itself by
increasingly large scale factors and see what the geometry looks like in
the limit (the resulting limit is called an asymptotic cone). In this
talk, I will focus on CAT(0) spaces (metric spaces of nonpositive
curvature) and discuss a few things we can find out about our spaces by
looking at them from infinity. And I will relate the two approaches I
described earlier by showing how the ideal boundary of a CAT(0) space can
be thought of as "the slowest possible rate" of zooming out on the space
(even though this actually makes no literal sense). This is based on
joint work of mine with Curt Kent.