Danny as an advisor

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:

• Metric geometry. Coarse or asymptotic geometry in the sense of Gromov. For example, studying groups via the quasi-isometry type of their Cayley graphs.
• 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.
• Hyperbolic geometry. The theory of hyperbolic structures on 3-manifolds, and its relation to the geometrization conjecture. Also, the topological consequences for the manifold of admitting a hyperbolic structure.
• Riemann surface laminations. General questions about complex analysis in the setting of abstract Riemann surface laminations; also, detailed understanding of specific examples which arise naturally. For example, solenoidal Riemann surfaces arising in the theory of rational maps.

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.

What should I read?

Here are some excellent books/papers relevant to the material above.

• Asymptotic invariants of infinite groups M. Gromov, LMS (1993)
• Metric structures for Riemannian and non-Riemannian manifolds M. Gromov, Birkhauser (2000)
• Bounds, quadratic differentials and renormalization conjectures D. Sullivan, AMS Centennial Publications vol. II (1992)
• Laminations par surfaces de Riemann E. Ghys, Panor. Syntheses 8, 49-95
• Three-manifolds, foliations and circles, I W. Thurston, eprint math.GT/9712268
• Three dimensional geometry and topology W. Thurston, Princeton Mathematical Studies vol. 35 PUP (1997)

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 as above. 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.

How structured would the advisor-advisee role be?

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.

Does Danny want students?

Currently, yes!

What are some sample thesis problems?

There are millions of problems out there, but for problems in low-dimensional topology, you could start by looking in Kirby's problem list. Also, you could look in my problem list in the theory of foliations of 3-manifolds.