I was curious if you had any discussions on what kind of math background new graduate students need to have? For instance, if the undergraduate institution did not have a good math program to support the CS curriculum, what specific topics should students self-study before going to graduate school?Most importantly you should have some familiarity with mathematical proofs. Mathematical maturity is more important than specific knowledge in any single topic.
Theoretical computer science is mostly discrete mathematics and other areas of discrete math play an important role: Discrete probability, combinatorics, algebra especially group theory, logic and number theory.
Depending on your interests analysis, measure theory, topology and algebraic geometry might be important. Almost every branch of mathematics has played some role in theoretical computer science.
I don't mean to scare you. As I said best to take any real math course (one with proofs, not just Plug-and-Chug Engineering math) and you can later pick up more specific math knowledge when you need it.