At the honours level I can supervise a thesis in algebraic geometry or in homological algebra. Further detail will be determined in consultation with the student, and will depend on his/her interests and background.
When it comes to PhD supervision it's important that the topic should lead to a thesis that the experts will find interesting - after all the graduating student will want a job, and his/her job prospects will inevitably hinge on what the experts will think of the PhD thesis. What's exciting and new changes with time - my recommendation to students interested in working with me is that they read my latest postings on the math archive, to be found at https://arxiv.org/archive/math.
At the time of this writing the postings that might be most informative are https://arxiv.org/abs/1806.06995, https://arxiv.org/abs/1806.03293 and https://arxiv.org/abs/1901.01453 each of these gives a survey of work I have done in the last few years.