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.

## What should I be (mathematically) interested in?

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:

• Geometric group theory. Groups as metric objects; quasi-isometry type of Cayley graph, topology and analysis on boundaries of groups. Cubulated groups. Surface subgroup conjecture.
• Low-dimensional dynamics. Usually nonabelian groups acting by transformations on low-dimensional manifolds (the circle, the plane, closed surfaces).
• Hyperbolic geometry. The theory of hyperbolic structures on 3-manifolds, its relation to number theory, topology, group theory, and so on.
• Structures on manifolds. Studying the existence and uniqueness of various kinds of structures on (usually low-dimensional) topological manifolds. For instance, smooth structures, symplectic structures, foliations, quasiconformal structures, etc.
• Extremal problems in topology. Stable commutator length, Gromov norm, 11/8 Conjecture, adjunction inequalities, the role of positivity in complex/symplectic geometry
• Riemann surfaces. Complex and quasiconformal analysis, Riemann surfaces and their generalizations (eg. laminations), rational maps etc.

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.

## How do I do research?

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.