If you have watched Going Deep shows of the Channel9 lately, one very frequently mentioned topic is mathematical duality in programming. TomasP has a good blog post about duality in object oriented programming.
This has been since Microsoft Research found that the observer design pattern is actually a mathematical dual of the iterator pattern. Since then they have used the duality concept in various ways.
My question is:
What mathematical dualities are there in programming?
Object oriented programming is a good start. The major GoF design patterns are: Decorator, State, Iterator, Facade, Strategy, Proxy, Factory Method, Adapter, Observer, Template Method, Composite, Singleton, Abstract Factory and Command. Here is a good object-graph-poster.