My research is in the branch of abstract discrete geometry called matroid theory. You can find an introduction to matroid theory in the survey paper by James Oxley entitled What is a matroid? I gave a non-technical introduction to geometry and matroid theory in a talk for the Royal Society of New Zealand. You can watch a video recording of the talk. For a more technical discussion, take a look at the Matroid Union blog, of which I am an editor.
I have been visiting Bryan Crump's programme, Nights, on Radio New Zealand. Here are links to our conversations.
For several years now I have been visiting high schools and talking to their students about mathematics. I was the 2013 Bevan Werry Speaker. If you would like to arrange a visit to your school, please feel free to contact me.