What is Applied Math
It sounds funny, but I'm never sure about the meaning of the expression "applied math." To make it clear, consider numerical analysis. It's often considered to be a part of applied math, but is it? I'd say NO. I'd say that applications are an important for numerical analysts as a motivation. But it has no importance whatsoever in the analysis itself.
If you consider numerical analysis to be applied math, what about PDE? Dynamical systems? Functional analysis? Of course, the "link" goes weaker as the list goes, but where to stop? Anyway, I like the definition given by David Munford in an interview at SIAM News
Yes. Having worked in both pure and applied mathematics, I have been very conscious of the barriers between these two sides. Actually, I don't like to think of them as two sides. I like to think of pure math as being a core, and applied math as being a whole series of subjects arrayed around the core that bring these mathematical tools into all sorts of applications.
To lower the barriers, pure and applied mathematicians both have important jobs. Pure mathematicians should include more explanation and motivation in their papers and talks, and discuss the simple cases of their results. Simplify things---there is nothing wrong with that. The applied mathematician has the difficult job of looking at a problem in context with no explicit mathematics and trying to see what kind of mathematical ideas are under the surface that could clarify the situation. I think the most successful applied mathematicians are those who look in both directions, at the science and at the math.
You can't become too attached to one way of looking at things. Applied math has always rejuvenated pure, and theorems in pure math can unexpectedly lead to new tools with vast applications.