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:

**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.

**Asymptotic invariants of infinite groups**M. Gromov, LMS (1993)**Metric structures for Riemannian and non-Riemannian manifolds**M. Gromov, Birkhauser (2000)**Volume and bounded cohomology**M. Gromov, Inst. Hautes Études Sci. Publ. Math. No. 56 (1982), 5-99 (1983)**Laminations par surfaces de Riemann**É. Ghys, Panor. Syntheses 8, 49-95**The Virtual Haken Conjecture**I. Agol, arXiv preprint**Three-dimensional geometry and topology**W. Thurston, Princeton Mathematical Studies vol. 35 PUP (1997)**Three-dimensional manifolds, Kleinian groups and hyperbolic geometry**W. Thurston, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357-381- My books,
**Foliations and the geometry of 3-manifolds**, and**scl**.

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.

- Thomas Mack, 2006. Quasiconvex subgroups and nets in hyperbolic groups
- Roberto Pelayo, 2007. Diameter bounds on the complex of minimal genus Seifert surfaces for hyperbolic knots
- Rupert Venzke (aka John Aaber), 2008. Braid forcing, hyperbolic geometry, and pseudo-Anosov sequences of low entropy
- Dongping Zhuang, 2009. A geometric study of commutator subgroups
- Joel Louwsma, 2011. Extremality of the rotation quasimorphism on the modular group
- Alden Walker, 2012. Surface maps into free groups
- Steven Frankel, 2013. Quasigeodesic flows from infinity