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In this answer Yves de Cornulier mentioned a talk about the possible uses of persistent homology in geometric topology and group theory. Persistent homology is a tool from the area of topological data analysis, specifically designed to extract information from empirical data and used for various applications, ranging from changes in brain function under drugs to the study of fluid flows to the merging of maps of distinct scales, along with many others. So this definitely belongs to the realm of applied mathematics, and it being used in pure mathematics is very interesting.

It goes without saying that many applications inspired a lot of research in pure mathematics, both in order to establish the foundations for the tools used in applies mathematics and just for the sake of studying interesting objects that appear in such interactions. I am talking specifically about the applied tools themselves being used in research in pure mathematics.

As an example, interval arithmetics was used in the solution of Smale's 14th problem and in the proof of Kepler conjecture (the latter also used a lot of linear programming).

Going back in time, we find that a lot of methods that were initially developed mainly for some specific application, such as celestial mechanics, the stereometry of wine barrels or heat transfer, became the standard tools in pure mathematics. Now it seems that the flow of methods and techniques is mostly one-way, from pure mathematics to the applied. But not completely one-way, hence the question:

What are the recent uses of the tools from applied mathematics to the problems in pure mathematics?

If one requires a more specific indication what does "recent" mean, let's say last 30 years (but I would be delighted to hear about older examples as well).

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    $\begingroup$ Is Theoretical or Mathematical physics counted as "applied mathematics"? Enormous part of "pure mathematics" can be traced back to physics in its origin. $\endgroup$ – Alexandre Eremenko Aug 9 at 12:30
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    $\begingroup$ Wine barrels,not vine barrels :-) $\endgroup$ – Harry Wilson Aug 9 at 12:47
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    $\begingroup$ If theoretical computer science or information theory counts as "applied mathematics" then there are too many examples to list. $\endgroup$ – Terry Tao Aug 9 at 18:07
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    $\begingroup$ For certain values of "recent", the problem of finding the antiderivative of the secant function came from cartography. $\qquad$ $\endgroup$ – Michael Hardy Aug 9 at 18:11
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    $\begingroup$ The question seems too broad to me. "Applied math" and "pure math" need to be defined more narrowly or else a large fraction of all mathematics could qualify. $\endgroup$ – Timothy Chow Aug 9 at 21:49
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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:

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    $\begingroup$ Applications of physics to pure math seem too numerous to list. Mirror symmetry, supersymmetric proofs of index theorems, applications of statistical mechanics to enumerative combinatorics...where does one stop? $\endgroup$ – Timothy Chow Aug 9 at 21:39

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