Many theorems start by taking an existence theorem and asking "How
many?" or "How big?" or "How fast". The best-known example may be the
prime number theorem. Euclid proved that infinitely many primes exist,
and the prime number theorem describes how quickly they grow.
I'll discuss what happens when you apply the same idea to simple
connectivity. In a simply-connected space, any closed curve is the
boundary of some disc, but how big is that disc? And what can that tell
you about the geometry of the space?