In October 2020 I became a professor at the Collège de France. My official title is Professeur titulaire de la chaire Combinatoire, which translates as the holder of the combinatorics chair. Chairs are not fixed for all time, and this is the first time that there has been one in combinatorics. Before that I was Rouse Ball Professor of Mathematics at Cambridge. Now, I still live in Cambridge and hold a part-time position in the department, which means that I continue to lecture here (typically one Part III course per year) and to supervise research students.
I am a combinatorialist in a fairly broad sense: first and foremost I am drawn to problems with statements that I can understand without having to do too much work. And within that category I tend to prefer problems that promise to involve other areas of mathematics, especially analysis, either by using tools from those areas or by providing tools to those areas.
Several years ago, before I had a blog, I wrote a number of these, for a variety of related reasons. One was to provide a more thorough discussion of definitions and basic results than I could normally hope to give in a lecture course from the Cambridge Mathematical Tripos . Another was to try to indicate, in the spirit of George Polya , how certain well-known proofs and definitions might have been discovered by anybody with just a few basic mathematical instincts. A general index to these discussions can be found here . Some of them, in updated form, became posts on my blog. Recently I have been blogging less, but have a social media presence on Twitter, where I often discuss mathematics.
This is the title of a book I wrote that came out in 2002. It is aimed at a reader who is probably comfortable with GCSE mathematics and maybe has gone a bit further than that as well, and who would like to know more about how the subject changes at university level. The main ideas discussed are how we use mathematics to model the world, the process of abstraction, what proofs are, how mathematicians generalize in surprising ways (such as making sense of shapes whose dimension is not an integer) and various notions such as limits, geometry in high dimensions, and curved space. A final chapter attempts to answer questions that mathematicians often get asked by non-mathematicians.
This is a book I edited (and partly wrote) with the help of June Barrow-Green and Imre Leader. It came out in 2008 and could be thought of as "Mathematics: A Very Long Introduction". It had the aim of being a genuinely useful reference work in mathematics. That was a difficult aim, since mathematics is hard enough to explain at the best of times, and even more so if one has limited space. Is there any point in trying to summarize algebraic geometry in ten pages, for example? Probably not, but the articles in the PCM don't try to summarize , so much as to provide an initial overview and perspective. I like to think of them as "prequels" to textbooks -- things you would read to get an idea of why you were bothering to learn some concept that your lecturer seems to take for granted is interesting. Were I undertaking the project now, I would probably not include as many of the shorter articles, because the natural place to get a quick introduction to a mathematical topic is now Wikipedia, which was less developed back then. But the longer articles offer something that is harder to find online.
Discrete Analysis is a mathematics journal that I set up with various colleagues from around the world. It is an arXiv overlay journal, which means that it does not host the papers it publishes, but instead links to them on arXiv. (However, the papers have a uniform appearance thanks to a style file kindly supplied to us by László Babai and Alex Russell.) Our first paper was published in early 2016.
The journal had a mathematical aim and a non-mathematical one. The mathematical aim was to be a natural home for high-quality papers in additive combinatorics and related fields, as previously it could be quite hard to find an appropriate specialist journal for such papers. The non-mathematical aim was to demonstrate that a well-respected mathematics journal could be run very cheaply -- our costs per article are orders of magnitude smaller than the costs of journals run by major commercial publishers. We meet these costs ourselves from a small grant: we do not charge authors (and obviously cannot charge readers given that the papers are on arXiv).
One other feature of the journal that we are proud of is that each article is accompanied by an editorial introduction, which attempts to describe the paper for a wider audience than the core audience of the paper itself. The result is that the Discrete Analysis website is a lot more than just a list of titles that are linked to arXiv. If you want to know what is going on in additive combinatorics, you may enjoy browsing it.
My more recent papers are on arXiv. Some older ones that aren't there can be found here . There are also some survey articles and a videoed lecture, in which I explain my general attitude to mathematics.
In the late 90s I gave a Part III course which included a section on the K-theory of Banach algebras, for which I produced a set of printed notes . I found Blackadar's book too compressed and Wegge-Olsen's not compressed enough, and was aiming at something like the geometric mean. What I produced has its faults, but may be useful when read in conjunction with those books. I should mention also that I gained greatly from reading some (I think still unpublished) notes of Bernard Maurey on this and related topics.
Similarly, I produced printed notes on Vinogradov's three-primes theorem for a Part III course last year. Again, I was aiming at something between two existing treatments: this time I found Vaughan (The Hardy-Littlewood Method - CUP) too compressed and Nathanson (Additive Number Theory Vol. I - Springer) not compressed enough, though this did not stop me finding both books very useful. As will be clear to anybody who reads them, my notes were aimed at those who had attended the lectures.
Further course notes can be found linked to from my blog. You may also be interested in an online masters-level course I gave in 2020 and put on YouTube, entitled Topics in Combinatorics. I tried to design it in such a way that many of the videos would be short and fairly self-contained. Accompanying notes are linked to from the video descriptions.