Different areas of mathematics have differing amounts of background required to get started on a research project.
Because my work is at the intersection of a few fields, there is a significant amount of background to cover, and it is important
that you start early. If you are interested in being my student, you should aim to approach me by the end of your first year, and expect to read with me over the summer.
That means that I'll suggest material for you to read, and we'll meet about once a week for some portion of the summer to discuss it.
When you approach me, you should have a background in algebraic topology, differential topology (manifolds), real analysis (measure theory), and some algebra/group theory. Of the graduate core courses, I recommend you take at minimum real analysis, algebra 1, and both topology courses. Having a solid working knowledge of these fundamentals and having done a lot of problem sets will serve you better in research than having sat in on lectures in advanced topics courses. I highly recommend that all graduate students take 3 core courses per semester during their first year.
In second year you'll take more graduate courses to finish out the requirements, and continue reading in preparation for your A exam. In most cases, I expect students to take the A exam by the end of their second year. This is more of a presentation with question period than an oral exam. The intent is for you to demonstrate proficiency in some specialized knowledge (from the reading we have done) that means you're ready to take on a research problem. You will write a short proposal with a description of what material you plan to cover, and circulate it to your committee. The exam itself is typically a ~30 minute presentation followed by questions where we will ask you to go into more technical detail about some of the material you presented.
Here is a big reading list with some possibilities of areas to work in.
Typically we will meet for 1 hour every week, this will move to every other week as you become more independent and/or start to work with others.
When we start working together, the way I run meetings is a kind of practice for your A exam. I expect you to speak at the board, and explain what you have been working on without reading from a book/article. (You may have your own notes and read from them, but it is painful for both of us if either of us spends time looking through pages of a text trying to find the right statement of a lemma, etc., or copying words out of a book verbatim!) Give precise statements and write clearly. For a long proof, it is better to give the "two-minute summary" first, then we can talk about a few specific details in it.
Later on in your studies this format may shift to more discussion based or to group meetings with other students.
My goal is to help you become an independent researcher. To this end, I will suggest what I think are fruitful directions, themes or families of questions, recommend reading, and offer guidance on coming up with your own problems. I prefer to lead students to explore the unknown rather than give them an assigned question that I already know the answer to. This is, in my opinion, the essence of research! Learning how to find and pose good problems will set you up well for the future (even if you don't spend your life in academia!) Other advisors have different styles, you should seek an advisor who's advising style matches your learning style. What seems inspiring and helpful for one student may seem boring or too structured (or too unstructured!) for another.
Ordered roughly by overlap with my own interests, we have
Jason Manning (geometric topology and geometric group theory)
Ben Dozier (Riemann surfaces, billiards, moduli spaces)
John Hubbard (complex dynamics, Teichmuller theory, geometric topology)
Justin Moore (set theory but also group theory and dynamics)
Tim Riley (classic geometric group theory)
Tara Holm (symplectic geometry)
Martin Kassabov (further towards group theory/algebra)
and a bit further afield are the algebraic topologists...
The mathematics students resource website has a collection of links to helpful advice for graduate students
It is the companion website to the topology students workshop a bi-annual conference and professional development program for grad students in topology.