If you are considering Danny as an advisor for your PhD, here are some typical questions and answers. These things are negotiable, and you should not consider them hard-and-fast rules.
Broadly speaking, you should be interested in geometry. This is a word which means different things to different people, so it is worth clarifying what is meant by this. Here are some areas of geometry which I find particularly interesting:
As a general rule, I'm not especially interested in Algebraic Topology. However, I am quite interested in the algebraic topology of groups of homeomorphisms of certain quality of simple manifolds, like n-dimensional space and the circle.
Here are some books/papers relevant to the material above.
It's up to you, but what I find helpful is to always have some hard problem in mind - usually some well-known conjecture - and when I learn new material, try to relate it to my problem. Another approach, apparently practiced by Shelah, is to always have three problems that you're working on - an easy one, usually an exercise to master a recently learned concept, a moderately difficult one, maybe a generalization of some recent result of someone else, and a very hard one, perhaps a well-known conjecture. When you're stuck on one problem, move to the next.
Also very important is to build up a library of examples of geometric phenomena which you understand very well, from many points of view. Whenever you learn a new abstract concept, try to see how your examples fit into this idea, and try to generate new examples which illustrate the main point.
Finally, I believe it is important to do experiments. These could be thought experiments, calculations to test ideas, or (more usually), computer experiments. Part of the point of this exercise is just to think explicitly about the problem of what it would take to translate a mathematical idea, concept or problem, into something that can be analyzed by computer. This act in itself leads to new perspectives, new mathematical ideas, and new mathematical questions which are interesting in their own right.
Initially, quite structured. I will expect you to meet with me every week at a regular time. I will expect you to read, or to attempt to read, papers that I suggest. I will want you to have prepared material to discuss in detail at these meetings. As you progress, however, I expect the relationship to become less structured, as you start to generate your own ideas and develop as an independent researcher.
This is a good question. My own research tends to suggest many problems, and I only have the time to follow up a small fraction of them. I am more likely to be an attentive and useful advisor to someone working on a problem close to my current strengths and interests. Sometimes I discuss open problems and other things I'm thinking about on my blog.
Here is a list of Danny's previous PhD students, with a link to their web page (if one exists) and thesis.