Q. Are you taking students next year?
A. I am always interested in the possibility of taking on talented students wishing to study for the D. Phil degree at Oxford (called a PhD in almost all other universities) under my supervision. When funding allows, I aim to take on an average of one student per year. For each of the last four years I have been able to take on precisely one student. At the moment I have a grant from the Simons Foundation which I would be happy to use in funding an excellent student.
Q. How do I apply?
A. All students must apply through the formal process, full details of which may be found here. I would recommend that you aim to meet one of the earlier deadlines. In addition to the formal application, I very much welcome informal approaches from prospective students, perhaps setting out their particular interests and background. I am happy to recommend other people - both at this university and elsewhere - for the prospective student to consider working with if this seems appropriate given their interests. Shortlisted candidates are interviewed either in person or by skype.
Q. What projects do you have?
A. I maintain an open mind regarding what project a student might work on. In the past, I have had students arriving thinking that their interest is in combinatorics, but ending up writing papers in analytic number theory. I have supervised theses across a broad range of topics from quite classical analytic number theory, through additive combinatorics, to more algebraic topics bordering on group theory, as well as topics related to ergodic theory. The papers of my current and former students may be accessed from this page.
Q. How do you work with students?
A. This depends on the student. The aim is to get students to become fairly independent by the end of their degree. Some work independently almost from the beginning. I have a booklet (not publicly available) of potential questions to get students thinking, but always encourage them to follow their ideas and interests. Usually students have a few (two or three) different projects on the go simultaneously: this reduces the potential damage from spending ages going down a dead end, or of someone elsewhere in the world usurping the student's work. Generally we (me, my current students, some postdocs and occasionally interested others) have a weekly informal seminar in my office in which people take it in turns to present and discuss recent (or not so recent) papers. This provides further ideas for potential research. Usually, students aim to write papers by themselves. Occasionally students of mine have collaborated with one another, and now and then I have written joint papers with students. For more information you could contact one of my current students.
Q. What are you working on at the moment?
A. Most of my research is in the mathematical area known as additive combinatorics. You can read about what this is in this book review. I work on a variety of topics in number theory (additive and analytic), combinatorics, and related areas such as harmonic analysis. I have also written papers on topics close to group theory and some topics with a geometric flavour. Potential students interested in knowing more could read the abstracts of some of my recent papers. Those with more time could look at my ICM proceedings articles from 2006 and 2014.