The Late Benoît Mandelbrot
The late Benoît Mandelbrot helped heal the split between pure and applied mathematics. Before Mandelbrot, we had pure mathematicians working on things such as Cantor sets, non-differentiable curves, and point sets consisting entirely of branch points (which were obviously far more complex than anything in applied mathematics) and, on the other hand, we had applied mathematicians working on things such as line noise, financial markets, and the Eiffel Tower (which were obviously far more complex than anything in pure mathematics). Mandelbrot pointed out that, to a good approximation, the two lines of research were about the same thing.
Mandelbrot's research has another implication. I'm sure it's occurred to most people who have studied the hard sciences that they're far more useful than that fuzzy stuff. This is important because being useful is one of the best ways to ensure that you aren't fooling yourself. On the other hand, most pure mathematicians are bound to wonder if their apparently-useless specialties are just more of those fuzzy subjects. When Mandelbrot found a use for some of the most apparently-ridiculous parts of mathematics, he gave the rest of us an excuse to study the not-yet-useful stuff. Maybe someday we'll even find a use for the Banach–Tarski paradox.