Riemann surfaces, Teichmueller theory and Kleinian groups, visual realization of geometric structures and applications.
Early in my career, I was an engineer. As students, a couple of us developed a control system to assist people with disabilities. It got lots of publicity. If you'd like to see what engineering was like in the age of Neanderthals, click here. At that time, older people were astounded that young people could do anything valuable; we were featured in the press, magazines and even the Voice of America.
Now young people don't have to carve out their
place in research or creative activity --- it is a natural part of the
university experience. I think that's great.
Here are some pictures of the Barycentric (Douady-Earle) extension of some mappings of the circle. Recently Cliff Earle (Cornell) and I have extended the methods used to produce the images to higher dimensional real and complex hyperbolic spaces.
Bill Harvey (King's College, London) and I have a history; I should refer to him as my dissertation supervisor but --- for now --- only in spirit. We've worked together, almost forever, on geometric structures. We use analysis, algebra and topology as tools to gain insight into geometry.
I like to think that hyperbolic manifolds approximate the universe in which we live. I'm still trying to find a physicist who will abandon the notion of a simple space with complicated properties in favor of a complicated space with simple properties. I also like to visualize those spaces. If that interests you, again, we should talk.
Ultimately, I'm a geometric function theorist. I study the interaction between a space (that's the geometry) and its phenomenology (that's its functions or what can happen in that space). If you find this interesting, contact me.