Before and after the cobordism hypothesis

In 1995, James Dolan and I formulated a series of “hypotheses” about topology and higher categories, before the theory of higher categories was sufficiently developed to make these into precisely stated conjectures. In 2009 Jacob Lurie reformulated one of these using infinity-categories, called it the “cobordism hypothesis”, and gave a 111-page outline of a proof. Simply put, the cobordism hypothesis gives a purely algebraic description of smooth manifolds, helpful for constructing topological quantum field theories. I’ll say a bit about what led Dolan and me to this hypothesis, what it says, why it’s useful, and why I quit working on it.