What you need to know depends largely what you are working on - and that also holds for a successful Ph.D. candidate. Of course, it is good advice to be open-minded beyond one's narrow field of inquiry, as the OP suggests, but a long list of maths topics may not be helpful a lot to beginners. The Ph.D. period is the time when you have some time available to acquire additional skills, and my advice is: try to strength…
> Read the main textbooks of your field and read and re-read ALL relevant papers for your actual Ph.D. topic, once you've been able to identify I'm gonna be real as someone in grad school. Basically no PhD, grad students, or professors I know read full text books. I hear these ideas a lot, and they often sound like one of those Instagram influencer diets, that's completely unreasonable if you have any constraints in…
(Hands-on machine learning, by Geron, back when I made the jump math->ML.)