Scheme: Royal Society Research Professorship
Organisation: University of Cambridge
Dates: Oct 2014-Sep 2019
Summary: My mathematical research area is known as additive combinatorics. This could be described as the study of finite subsets of groups, and how properties that such subsets might have are related to one another. Results in this area are proved using a very interesting mix of techniques from areas such as Fourier analysis, probabilistic and extremal combinatorics, dynamical systems, number theory, and even algebraic geometry. Moreover, they often feed back into those areas.
I also have a strong interest in theoretical computer science, which has (initially quite surprising) connections with additive combinatorics.
Finally, I am also extremely interested in artificial intelligence, and in particular in the task of trying to program computers to prove mathematical theorems. Ultimate success in this area would be if I could put myself out of a job, but I think that I will have long since retired anyway before this goal will be anywhere near being achieved!
Dates: Oct 2009-Sep 2014
Summary: I am currently working in two areas, one mathematical and one "metamathematical" (meaning that it is about the processes of doing mathematics). The mathematical area is known as combinatorics. It studies discrete (as opposed to continuous) structures, such as networks or sets of numbers. Combinatorial questions arise in many contexts, both in the real world and within mathematics. For example, networks arise in contexts as diverse as the internet, design of electronic circuits, traffic flow, timetabling, and assigning radio frequencies. Combinatorics also has intimate connections with theoretical computer science. One of my favourite problems is the famous P versus NP problem, which asks, roughly speaking, whether finding answers to mathematical questions is as easy as checking whether answers are correct once they have been found. (The latter sounds much easier, and indeed most experts believe that it is easier. But it turns out to be extraordinarily difficult to prove this.)
The metamathematical side of my research belongs at the interface between mathematics and artificial intelligence. I am working in the area of automatic theorem proving: that is, trying to write computer programs that can solve mathematical problems. Some people believe that mathematics requires a characteristically human intelligence -- not for the routine calculational tasks but for the deeper insights that go into proving a complex theorem. I strongly believe that one day all this will be automated, and as a result I find it a fascinating and challenging problem to understand what is going on in the brains of people like me when we do research. With a colleague, Mohan Ganesalingam, I have written a program that solves some easy problems. Over the next few years we intend to develop it so that it will solve harder ones and begin to show what computers are capable of. Our ultimate aim, which is a long way off, would be to put ourselves out of business!