The Swampland conjectures are conjectures in quantum gravity that provide constraints on when a given theory of quantum gravity is consistent. The thesis of this talk is that these conjectures, and especially the cobordism conjecture of McNamara-Vafa, lead one to ask algebro-topological questions whose answers provide insights in string theory. I will introduce the cobordism conjecture and discuss how it leads to both theoretical and computational mathematical questions; then, I will go over work joint with M. Dierigl, J.J. Heckman, and M. Montero applying this to type IIB string theory.