Physicists' mathematics is often differently focused, depending on the field. So for example, Lie groups, aspects of topology and geometry, and partial differential equations are more present in the fields I know. Stochastic differential equations appear much more often in finance theory, but of course they also explain diffusive processes in gases, and the behavior of plasmas (as in fusion, or in the universe). There is a wonderful book by Cedric Villani about his proving a result in this field--you don't need to know math, just human nature.
A doctoral student in some fields in engineering is much more acquainted with all of this.
A macro guru such as Larry Summers is of an earlier generation. Also, at the levels of discourse he participates in, whatever you learn from the Lucas and other such books, and their research, has been made into a way of thinking-- BUT I suspect that it does not even have much of an effect there. Janet Yellen's staff does not show her a stochastic differential equation, or a Markov process, or a Martingale--but perhaps they have something useful to help her think about these things. There are wonderful ideas in dynamic programming (often, that a differential equation and an optimization problem are nicely related, and in effect the present is where all the action is). I do not expect that the head of the Fed in 25 years, having been trained in all the math stuff will ever talk about it, either. It will be interesting what the insights will be.