Achieving mastery in your field of choice allows you to see and understand connections from outside your own area of expertise. I can’t help thinking of Richard Feynmann’s excursions outside the world of physics as an example. The recent Science Daily story Mathematicians Unlock Major Number Theory Puzzle provides another (emphasis mine):
It was during a flight to New Hampshire that Ono realized the full depth and meaning of Zwegers’ work. Skimming a journal to pass the time, Ono happened upon an old article by George Andrews on mock theta functions. Suddenly, he noticed that some of the mathematics in the paper seemed to resonate with parts of the Harmonic Maass theory, which he and Bringmann just happened to be developing at the time, for other reasons.
Or, as Louis Pasteur said way back in 1854, chance (and aha!) favors the prepared mind.
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no straight lines 