More information can generally be found in my publications, but some general headings are:
AviaNZ: BioacousticsI'm interested in every part of the problem of estimating animal populations from data. My main focus is the mathematical analysis of soudn recordings, with the aim of recognising bird species calling, and turning the call rates into abundance estimates. Our opensource software, and much more information, is available on our wepage. 

Euler Equations on Diffeo GroupsI have a longstanding interest in the geometric, analytic, and numerical aspects of generalised Euler equations on diffeomorphism groups. The focus is a better understanding of the equations and their behaviour. This is part of the field of shape analysis, which was motivated by the work of D'Arcy Wentworth Thompson in `On Growth and Form'. (Picture is my brain, sometimes it's nice to know you have one!) 

Information Geometry and Machine LearningUnderstanding why machine learning algorithms work is very challenging. One approach is to consider either the manifolds on which data live, or the manifolds where the parameteric distributions underlying them live, and to write down mathematical formulations of the algorithms. I'm also interesting in using the algorithms in conservation technology. (Picture from http://www.nada.kth.se/~asa/Game/BigIdeas/ai.html) 