If mathematical developments in physics count as "applied mathematics" there are many examples, here is a recent one and an older one:
- Gauge theory spawned fundamental resuls in differential geometry, see for example A proof via the Seiberg-Witten moduli space of Donaldson's theorem on smooth 4-manifolds with definite intersection forms
- Dirac delta functions were introduced to simplify the normalization of quantum mechanical wave functions, and were formalized by Laurent Schwartz in the theory of distributions.