If you include mathematical physics (string theory, QFT) to "applied mathematics", the breakthroughs are too numerous to list them here.

Here is a very recent example not related to QFT or string theory. An important <a href="https://arxiv.org/abs/1212.1700">Connes Embedding Conjecture</a> from the theory of von Neumann algebras, has been apparently disproved by
the arguments from Computer Science (quantum computing):

<a href="https://dl.acm.org/doi/10.1145/3485628">MIP*=RE</a>

(The complete paper is not published yet, given its <a href="https://arxiv.org/abs/2001.04383">volume,</a> it will take some time to verity).

More examples (just a selection of those which I encountered during my career).

In 1987, <a href="https://www.mathnet.ru/links/32b93f06564682e565b9a8bb0dec1f76/sm1767_eng.pdf">Zograf and Takhtadzhyan</a> 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, <a href="https://archive.ymsc.tsinghua.edu.cn/pacm_download/117/6501-11511_2006_Article_BF02392568.pdf">Kari Astala</a> proved the old conjecture of Gehring and Reich by using ``thermodynamic formalism''. The proof has been <a href="https://www.math.purdue.edu/~eremenko/dvi/hamilt.pdf">substantially simplified</a> since, but still uses the main idea which comes from statistical physics. 

In the early 2000s Oded Schramm created the
SLE theory (Stochastic Loewner Equation, aka <a href="https://en.wikipedia.org/wiki/Schramm%E2%80%93Loewner_evolution">Schramm-Loewner evolution</a>) on the boundary of probability and complex analysis. It was directly inspired by questions from physics (statistical mechanics, percolation theory, Cardy's formula).

In 2005, Mukhin, Tarasov and Varchenko proved the <a href="https://arxiv.org/abs/math/0512299">B. and M. Shapiro Conjecture</a> from real enumerative geometry by unexpected argument
from mathematical physics (Bethe-ansatz from statistical mechanics). In 2020 a <a href="https://arxiv.org/abs/2006.08574">a surprising connection</a> of this former Shapiro conjecture with SLE has been found.