ya lol and Hamiltonians. sometimes people just reel off all the math they've heard of to sound impressive. next we'll have people talking about de rham cohomology because of TDA (or something like that)
Hamiltonian mechanics, along with many other seemingly out of place 'advanced' maths, show up in modern Bayesian statistics pretty frequently. Hamiltonian Monte Carlo/Riemannian Manifold Monte Carlo are pretty cutting edge (although are implemented in popular libraries like MC-Stan and Pymc3) and both require fairly advanced physics to really understand.
Additionally, we're seeing the introduction of even more sophisticated stochastic samplers (stochastic gradient hamiltonian monte-carlo, etc) that require even more esoteric branches of math and physics to really grok. I have a strong math background but frequently find myself struggling with a lack of knowledge in statistical mechanics when trying to read papers in these areas.
So yeah - there's plenty of bullshit and exaggeration. But there's also some wicked cool stuff happening which requires very sophisticated (and specialized) knowledge to understand.
:) Personally I've never used any serious complex analysis in my job (I'm very grateful too, because I always struggled a bit with it). The closest thing I've seen, which I did run into recently, is the use of complex numbers to compute very accurate finite differences. It's one of those delightful tricks that is both elegant and useful: https://blogs.mathworks.com/cleve/2013/10/14/complex-step-di...
I've been working in golang, which fortunately has built-in complex128 types, so it's proved very helpful in a project!
oh cool, I really like Cleve Moler's Matlab posts.
I took a complex analysis class and did OK, but I get the feeling that EEs are the ones who really benefit from it (at least in the applied world). They seem to have some very rich analyses of linear dynamical systems using frequency domain methods.
I have only seen it used to evaluate some integrals in a few papers via Residue Theorem, so maybe I should have said Residue theorem instead of contour integration. I am sure there were other methods that the authors could have used, but I was sure glad to know of contour integration then. I'd say some complex analysis still deserves to be on the list to have a base understanding of Fourier transforms. Of course, you can arrive there without complex analysis.