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), Xingwang Xu (Singapore and
Nanjing) and I have extended the methods used to produce the
images to rank one non-compact symmetric spaces.
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).