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.
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.