# Prof Tim Dokchitser

"Maths is a language; it’s a little bit like learning Chinese poetry – you can’t do it in five minutes, you have to learn the culture, get a feel for the language, then finally you will be able to appreciate what’s behind it."

Prof Tim Dokchitser is a Royal Society Research Fellow in the School of Mathematics. His fascination with mathematics began during his school years in Russia and over the past twenty years since, has drawn him closer into the world of number theory.

Growing up in a distinctly scientific family – albeit one that veered more towards physics than maths – he was supported throughout his education and spent time at a number of institutions across Europe developing his specialism.

Yet his closest collaboration, and the platform for his most notable achievements to date, has been with his brother, a fellow number theorist at Cambridge University where Tim was based until he took up post at Bristol in 2011.

Tim’s research is focused on arithmetic geometry and its connections to group theory and representation theory. Most of his work concerns L-functions of arithmetic varieties, elliptic curves and the Birch-Swinnerton-Dyer conjecture, one of several “million dollar questions” in mathematics.

I was 12 years old when personal computers first came to Russia; that was quite a revolutionary experience. They were the kind of computers that you could completely pick apart to see what they could do; they were a major source of fascination for me.

I’d always dreamt of doing something that combined maths and computers. I went to a school in Russia where they had very good mathematics and physics teachers, it was quite specialist in that sense.

I found out later that two of my teachers were actually well known number theorists although they never really spoke about their own work.

The idea of creating something new really appealed to me. Before the PCs you had just calculators but those were completely boring by comparison, so when we were suddenly presented with the possibility of being able to write computer games, that was completely amazing to me.

Maths is a language; it’s a little bit like learning Chinese poetry – you can’t do it in five minutes, you have to learn the culture, get a feel for the language, then finally you will be able to appreciate what’s behind it.

There’s a lot of terminology in maths and a lot of things which can seem to hide the creativity from someone looking at it for the first time.

When I teach students I always try to explain the language that is behind the formulas. I pick a subject and try to explain the technical bits so students know the complicated aspect but also appreciate why we do it and why it’s important.

Pure maths is usually used for foundational things to build up mathematics that will help in other areas that will finally get applied. In pure mathematics we have a feeling for what is a fundamental problem and if we can solve it, it can open up a neighbouring problem.

It’s very good as a scientist that you’re not stuck in one place – it’s important to see what people are doing around the world and the problems they are thinking about.

I started my undergraduate studies in Sweden. I then did my PhD in Holland and then I went to a few places, to Germany, to Sydney a few times – where I still go to keep up the connections I have with computational number theorists there – and after that I came to the UK, to Durham for several years, Edinburgh for six months, then Cambridge for six years and finally Bristol.

Some of my choices have been a little bid random – we went to Sweden as a family because my father is a physicist and he had to go there for several years. Me and my brother (he is seven years younger than me) both went to Sweden; my brother is also a mathematician and we do most of our work together.

He is still in Cambridge where our academic paths converged. That was obviously a good collaboration; it’s very convenient working on papers when your offices are next door to each other!

Sometimes when you collaborate it can be difficult to explain to someone what you are thinking but with us, I just open my mouth and he already knows what I am going to say and that helps a lot. I think there are at least another four pairs of brothers in number theory all over the world so it’s apparently not that unusual.

I wonder if there is a gene somewhere inside of me and my brother that pushed us both towards number theory. My mother was also a physicist – although she is now more into the history of art - so science has always been in the family.

I’m not sure if there’s anything genetic but it definitely depends on education. My parents and some good teachers pushed me, personally, and I believe if you’re interested in something and you’re encouraged to do it, that helps.

After a while in science, you realise there are many more questions out there that are just as important as the ones you’re working on but which you don’t have the time to even think about, so there is no worry that you will run out of queries. Normally, while I am working on a particular project other things will occur to me that I think are interesting.

There are a couple of things that I am really proud of having worked on, technical problems in number theory. There was one problem that was formulated in the 1960s and which me and my brother solved just six years ago; that’s probably the one I’m most happy about.

It’s related to a famous problem, the Birch-Swinnerton-Dyer conjecture, one of the million dollar problems in number theory and we solved one small special part of that.

It’s not that we started out intending to solve it; we found that various pieces to this jigsaw came together over two years or so and we realised we had the answer.

I’m not sure we believed it at first, I don’t remember a specific moment but I do recall it was an exciting time in our careers.

Another discovery was an algorithm for computing L-functions, that is a problem that we solved which is probably more useful to the mathematical community.

Back in my school years I didn’t know what life as a scientist would be like but if I knew then how it would turn out, I’d say this was as close to my childhood dream as it might have been.

Of course my dream wasn’t fully formulated back then but I liked playing with computers and maths problems so in that sense, I would have chosen this as a profession.”

## Related links

- Elliptic Curves, Hilbert Modular Forms and Galois Deformations
*Birkhäuser*, 2013 - Brauer relations in finite groups, with A. Bartel
*arXiv*, 2011 - On the Birch-Swinnerton-Dyer quotients modulo squares (PDF), with V. Dokchitser
*Annals of Math*, 2010 - Elliptic curves with all quadratic twists of positive rank, with V. Dokchitser
*Acta Arithmetica*, 2009