As someone who did an applied math PhD before drifting towards ML, it's worth pointing out that these applied math groups typically talk about applications, but the real question is whether they are actually used for the stated application in practice due to outperforming methods that use less pretty math. Typically (in every case i have seen) the answer is "no", and the mathematicians don't even really care about solving the applied problems nor fully understand what it would mean to do so. It's just a source of grant-justifiable abstract problems.

I would love to be proven wrong though!