In part the key word here is "explicitly." I have a PhD in Combinatorics and Graph Theory, and I have lectured in Calculus, Group Theory, Functional Analysis a

In part the key word here is "explicitly." I have a PhD in Combinatorics and Graph Theory, and I have lectured in Calculus, Group Theory, Functional Analysis and Topology. In my daily work for the past 18 years I have explicitly used that background exactly once. But the ongoing influence, the style of thought, the ability to visualise and the ability to abstract away from the details - these things I use all the time, every day. They have been enhanced and honed by all that math. I use my math background implicitly all the time, and I don't know how I'd do what I do without it. And this is perhaps the most important point about studying math. Often the greatest take-away is the abstract problem solving capability, not the material itself. I've never had to analyse the genus of a manifold in real life, but I have thought about objects moving in an 11 dimensional space with holes, because while everyone else was stuch in the detail, I was seeing things differently. It turned out that the combination of styles was critical to solving the problem.