To what extent is "advanced mathematics" needed/useful in A.I. research?
I do not want to sound condescending, but the math you are studying at the undergraduate and even graduate level courses is not advanced. It is the basics. The title of your question should be: Is "basic" math needed/useful in AI research? So, gobble up as much as you can, I have never met a computer scientist who complained about knowing too much math, although I met many who complained about not knowing enough of it. I remember helping a fellow graduate student in AI understand a page-rank-style algorithm. It was just some fairly easy linear algebra to me, but he suffered because he had no feeling for what eigenvalues and eigenvectors were about. Imagine the things AI people could do if they actually knew a lot of math!

I teach at a math department and I regularly get requests from my CS colleagues to recommend math majors for CS PhD's becase they prefer math students. You see, math is really, really hard to learn on your own, but most aspects of computer science are not. I know, I was a math major who got into a CS graduate school. Sure, I was "behind" on operating systems knowledge (despite having decent knowledge of Unix and VMS), but I was way, way ahead on "theory". It is not a symmetric situation.