Colloquium

In dimensions 2 and 3 (and in higher dimensions with some restrictions), there are complete theories of weak solutions that provide canonical evolutions of the Ricci Flow initial value problem. We will introduce these and discuss how, in dimensions 5 and above, uniqueness cannot be expected to hold. Specifically, we will show that in these dimensions, there exist smooth complete initial metrics whose forward evolutions under Ricci Flow are not unique after a first singularity forms, and whose topology may change at the singularity for some solutions but not for others.