If you include mathematical physics (string theory, QFT) to "applied mathematics", the breakthroughs are too numerous to list them here. Statistical mechanics is especially productive in its influence on pure mathematics.
Here is a very recent example not related to QFT or string theory. An important Connes' Embedding Conjecture from the theory of von Neumann algebras, has been apparently disproved by the arguments from Computer Science (quantum computing):
(The complete paper is not published yet, given its size, it will take some time to verity, but it has already 348 citations on Google Scholar at the time of this writing).
More examples (just a selection of those which I encountered during my career). All of these are examples of breakthroughs in pure mathematics inspired by physics.
In 1987, Zograf and Takhtadzhyan made a breakthrough in the classical accessory parameter problem of uniformization theory by proving a conjecture of physicists Belavin, Polyakov and Zamolodchikov inspired by string theory.
In 1993, Kari Astala proved the old conjecture of Gehring and Reich by using ``thermodynamic formalism''. The proof has been substantially simplified since, but still uses the main idea which comes from statistical physics. More generally "thermodynamic formalism" is a part of pure mathematics directly inspired by physics, see, for example this survey paper by David Ruelle on the influence of statistical mechanics on pure mathematics.
In the early 2000s Oded Schramm created the SLE theory (Stochastic Loewner Equation, aka Schramm-Loewner evolution) on the boundary of probability and complex analysis. It was directly inspired by questions from physics (statistical mechanics, percolation theory, Cardy formula).
In 2005, Mukhin, Tarasov and Varchenko proved the B. and M. Shapiro Conjecture from real enumerative geometry by unexpected argument from mathematical physics (Bethe-ansatz from statistical mechanics). In 2020 a surprising connection of this former Shapiro conjecture with SLE has been found (Peltola and Wang, https://arxiv.org/pdf/2006.08574 )
In cases 1,2,4, the initial problem had nothing to do with physics, and solving them by methods coming from physics was quite unexpected.
There are various opinions on this "unreasonable effectiveness of physics in pure mathematics", for example, Ruelle (loc. cit.) ascribes a deep philosophical meaning to it. Another opinion that I've heard is simply that "some physicists are very clever people, and when they start thinking of mathematics, they come with clever ideas" (this was proposed by V. G. Drinfeld when he was listening my talk on Astala's theorem).