An EPSRC Doctoral Training Centre in Geometry and Number Theory

LSGNT

London School of Geometry & Number Theory

What Are We Looking For In An Applicant?

LSGNT has a diverse cohort; our students come from over all over the globe and from all kinds of different backgrounds. Even if you think you’re not a typical maths PhD student, we encourage you to think again and apply. What we’re looking for might not be what you expect.

Moving from undergraduate to research mathematics

Research is very different from undergraduate mathematics. Many people who don’t do well parroting lecture notes in undergraduate exams excel at the creative problem-solving we’re after. Conversely, many people who are brilliant undergraduate problem-solvers find they do less well when confronted with open-ended problems that do not fit well into one narrowly-defined area of mathematics. Those who are brilliant at working alone, perhaps driven by competitiveness, can struggle to harness the greater power of collaborative work with their peers. Those who are good at forming relationships and working well with others can thrive. Some students come with huge background knowledge in pure maths but can find it makes their thinking rigid; those with less sometimes bring new perspectives and ideas from other fields.

So we are not looking for students who necessarily know everything or aced every exam (though that can be good too).

We are looking for people with curiosity, who like to explore ideas, solve problems, and see connections between different areas. We want people with a dogged determination to get to the bottom of things, to really understand what’s going on. We like imagination, creativity and improvisation. We’re looking for enthusiasm and potential, not the finished article.

We like to see that you’ve demonstrated an active interest in the topics you’ve studied – thinking beyond the syllabus, about links to other areas for instance. That you’ve taken some responsibility for your learning. Of course there are also more prosaic criteria, like your academic fit to the programme (do we have a potential advisor with similar interests to yours?) as well as your fit to the ethos of LSGNT (would you like working in a communal mathematical environment?)

We are also interested in what you expect from a PhD and a supervisor. You may want a 3-year plan with milestones and a pre-determined outcome, but our vision of a successful PhD is more free-flowing. It is the result of a relationship between student and supervisor which does not follow a script. It is more of a shared improvisation, discussing promising ideas and working together at the board. A spirit of adventure and curiosity certainly help.

What are we looking for in an interview?

It can be hard to identify the qualities we’re after. Our approach is to offer you as many opportunities as possible to show us your potential: reference letters, your CV, undergraduate results, your personal statement and an interview. We are looking for some of these to impress us, not all of them.

So you should think of the interview as another opportunity, not a test you must pass. If it does not work out that certainly does not doom your application – it means we will be focussing on other aspects of it. We fully understand interviews can be a nervy, stressful experience (though of course we will do all we can to put you at ease). We fully understand that afterwards some people will feel they “messed up” – it is completely normal to find it impossible to think clearly in such a situation. It doesn’t matter nearly as much as you might think, because we’re looking for people with a talent for solving problems, seeing patterns and making connections, not confident people who are good at interviews. We take many applicants who feel their interview did not go well.

What happens in an interview?

We’ll ask you to suggest a subject you like and feel comfortable working with. Then we’ll ask a few questions about it. We’re not trying to catch you out. We want you to do well; we will give you every opportunity to show us your potential.

Pick a topic that you really like, don’t choose an advanced topic just because you feel that it will impress us. You will impress us more by talking about an elementary topic that you really own. Something simple – if you pick something hard it will be harder to solve problems in that area.

We aren’t examining the quantity of your knowledge (of this area or any other). We won’t be asking you questions to demonstrate how much you know. But we want to see you’ve thought about the topic, and followed your curiosity. Maybe you’ve made connections to other areas, or wondered about them.

So if you tell us you studied the classification of orientable 2-dimensional closed manifolds, and that you’ve also taken a course on algebraic curves, we might ask you to explain the genus and discuss the relations between a topological definition (in terms of Betti numbers or Euler characteristic, say) and one via holomorphic or algebraic differential forms.1 We’d be impressed if you’d done some research into their equality, or had some thoughts about why they’re the same, or how to prove it. We would be less impressed if you just stated that they’re equal because a lecturer said so, or you’d never thought about what algebraic curves look like when the ground field is the complex numbers. We might then ask you why a cubic curve has genus 1.Or to think about how you might prove that a genus 0 curve is P1. We don’t normally expect you to solve these problems – or certainly not at once or without hints from us. We just want to chat with you about how you think about it, what your initial reactions are, what occurs to you, what techniques you might bring to bear on it, or how you would go about thinking about it: would you try some simpler examples first, for instance?

Don’t hold back; try not to be shy and certainly don’t be deferential or worry about saying something wrong. We like open debate and ideas. We’ll encourage the promising ideas to help lead you in the right direction.

The ability to answer basic questions in a topic you know well is a much better indicator of research ability than knowing lots of statements of difficult theorems. If we can see evidence that you’ve internalised a subject, made it your own, and can now solve problems with it, we’ll be more impressed by that than by you having taken more advanced courses and learned the theorems.

Basic questions are the bread-and-butter of day-to-day mathematics; it’s what we all do. It is crucial to be able to test ideas quickly before committing to them long-term, and we do that by asking basic questions. Simple examples are surprisingly powerful in studying, connecting, guiding development, testing, and understanding complex problems and ideas.

1. If you have no idea what any of these things are, don’t worry, we’ll be asking you about something closer to your expertise.