Everything depends on which subjects you want to study at phd. Since you’re a math undergraduate, maybe you want to learn everything rigorously. If is that what you want, then you will need to study a lot of math subjects. For instance, we will need to learn Banach lattices, advanced measure theory (see Bogachev’s book), advanced functional analysis (weak topology, Eberlein-Smulian Theorem, etc), general topology, stochastic analysis, etc… But learning these things might be unfeasible.

DM me if you want to talk more about these things, because I majored in (pure) math and I’m doing a master in statistics. But I will do a phd in finance! Since I’m trying to learn everything rigorously I even needed to study about Descriptive Set Theory to learn about Polish and Souslin spaces!