This is a question that I've been thinking about for a while, as my background is mainly on pure mathematics.
From a few OR papers that I've looked at, I could identify some areas of pure mathematics that can be applied to OR:
(Analytic) Number Theory: deriving asymptotics such as polynomial time
Metric Spaces: use of the Euclidean metric in TSP
Differential Geometry: optimal transportation problems1 in this case, the Monge and Kantorovich formulations.
Are there other branches of pure mathematics that are currently applied in OR? For each branch that you specify, at least one reference to literature would be much appreciated.
 Loeper, G. (2009). On the regularity of solutions of optimal transportation problems. Acta Math. 202:241-283. doi: 10.1007/s11511-009-0037-8.