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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 (1995)
• 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.

If mathematical developments in physics count as "applied mathematics" there are many examples --- as requested by the OP here is a recent one (< 30 years old) 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 (1995)
• 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.

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.

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 (1995)
• 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.