# FACES OF MATHEMATICS Professor Frances Kirwan University of Oxford Frances Kirwan, Professor of Mathematics at the University of Oxford, researches in algebraic and symplectic geometry. Her work endeavours to understand the structure of geometric objects by subtle investigation of their algebraic and topological properties. This work calls on facts and techniques from many other areas of mathematics, as well as requiring a specific ingenuity of its own.

 "The sort of things that I'm working with are very abstractly defined -- mainly so-called "moduli spaces" in geometry. The whole idea of a moduli problem is that you're trying to classify something in geometry, some geometrical objects. The moduli space is the set of equivalence classes of the objects that you want to classify, but with some natural geometrical structure on it which reflects how these objects vary. "Algebraic geometry is such a rich and highly developed subject, partly because it has a really long history, and partly because it takes pieces from all the different parts of pure mathematics, and incorporates them and uses them -- including in the last twenty or thirty years or so, from mathematical physics. "Mathematicians tend to divide into two groups: there are people who think algebraically and think in equations, and then the other half who think in pictures, and I'm one of them. If I'm thinking about a problem, then I need some sort of 2-dimensional picture to get me going, even if the actual problem is in some large number of dimensions, an arbitrary number of dimensions, or an infinite number of dimensions. If I'm just trying to get going on a problem, then what I need to do is to draw some diagrams, to draw some pictures. "Some of the things that I'm working on with some collaborators are problems involving singular spaces. If you're a geometer, you want to be able to differentiate and whatever, so you'd really like to have nice smooth manifolds to work with -- with extra structure, maybe, but certainly smooth. But that turns out to be a bit much to hope for, and algebraic geometers have to cope with things not being at all smooth, having all sorts of nasty singularities coming up all over the place. I'm interested in trying to understand the topology of these moduli spaces that arise in algebraic geometry and a lot of them will have singularities."