Jonny Evans

Room 501, Department of Mathematics,
UCL, Gower Street, London, WC1E 6BT
j.d.evans • ucl.ac.uk

We have many bright undergrads here at UCL and many of you approach me, wanting to make a start on research mathematics.

Here's how you can make a start on research:

• Try and synthesise different things you are reading and figure out if there are connections between them.

This is how most research starts. The only difference between doing it as an undergrad and doing it as a professional mathematician is that as a professional, you have a clearer idea of (a) whether the answers to your questions are already known/who might know, (b) if the answer is not known, whether it is worth spending your time figuring it out (will it be of interest to other people? will it be something you can actually solve?)

It is likely that most of the questions you ask this way will have answers which are "well-known" (i.e. contained in a more advanced textbook or published in an obscure Russian journal). This does not in any way diminish the question - it means that it was interesting enough for someone else to figure out the answer. It's also worth finding the answer, either looking in the literature or thinking by yourself (both are good).

As an undergrad, don't worry if the questions you end up solving have well-known answers. Maths has been around a long time and it usually takes a while to get up to speed on the language. There's no rush to prove something "new" - as long as it's new to you, it's valuable. This is why PhDs take 3-4 years (on top of 4 years of undergrad studies).

In some cases, the answer to your question will turn out to be "we don't know (yet)". That's great - it means you're starting to poke at the nameless slumbering horrors that lurk at the margins of space and time. When you find a question like this, it's often ill-formulated or too big (and you risk being devoured). It's a really good exercise to try and find a precise formulation of a simpler question which is still open. Always look to understand the first nontrivial case which hasn't yet been understood. It may help to isolate crucial features where the difficulty lies.

The most useful things a supervisor can do, in my opinion, are:

• to point you towards fruitful things to read or think about,
• to help you decide which of your questions are worth pursuing and how you might pursue them,
• to turn your naive questions into more sophisticated ones which might be tractable,

For these reasons, I don't want to set formal summer research projects for undergrads: the questions I find interesting in my own research are the product of me thinking about stuff for years and probably seem unmotivated to someone who hasn't gone through this process. I could conceivably come up with problems that an undergrad could solve, but they would probably be contrived and uninteresting (and it's a bad idea to work on questions you find uninteresting).

What I am (more than) happy to do is to offer suggestions to undergrads of stuff to read, to meet and talk, and to listen to your questions.

Last updated 2nd April 2015.