My research has traditionally been in Singularity Theory, a long and venerable subject dating back to Newton. Recently I have been working in Discrete Differential Geometry, in particular recent ideas surrounding generalizations of the discrete Laplacian.
For example, any scanned shape will have noise due to scanner inaccuracy. Can we smooth away the noise? The picture on the right shows a scanned object that has been smoothed using a diffusion-type flow. The top two show the original scan and the bottom two show the smoothed. The colours represent the mean curvature. As you can see from the colours the smoothed version has lot, but not all, of the noise removed. The problem is to remove the other noise. First we need to decide what we mean by noise...
If you fancy doing a PhD in this area, then please contact me: k.houston(at)leeds.ac.uk
My main teaching interest is encouraging (read forcing) students to think, hence the title of my best-selling book, How To Think Like a Mathematician. You might be interested in this taster for it: a free booklet called 10 Ways To Think Like a Mathematician.