Academics

Welcome

Welcome to my webpage. I am Stipendiary Lecturer in Mathematics at Keble College where I teach first- and second-year undergraduates. I give tutorials and classes for a range of courses in the Pure Mathematics syllabus. In MT 2011, I teach Mods Linear Algebra I, Mods Geometry I, Mods Intro to Pure Maths, and Part A Algebra.

I am currently finishing my DPhil under the supervision of Professor Frances Kirwan. The title of my thesis is "Universal moduli of parabolic sheaves on stable marked curves" - please see the research section for more information.

My previous teaching experience includes College Lecturerships at Merton College (2010/11) and Jesus College (Jan 2009 - June 2010). The first- and second-year courses which I have taught (in tutorials and classes) include:

- Mods: Linear Algebra I and II; Geometry I and II; Intro to Groups, Rings, Fields; Analysis III (Riemann integration).

- Part A: Algebra (further linear algebra and ring theory); Analysis (basic topology of Euclidean space and complex analysis); Intro to Fields; Group Theory; Topology; Number Theory.

In addition, I have been class tutor or teaching assistant for the following third-year courses:

Michaelmas Term 2009: class tutor for B9a Galois Theory, teaching assistant for B3a Geometry of Surfaces.

Hilary Term 2009: teaching assistant for B3b Algebraic Curves.

Michaelmas Term 2008: teaching assistant for B3a Geometry of Surfaces.

Hilary Term 2008: teaching assistant for B3b Algebraic Curves.

Research Interests

My main research area is the moduli theory of (decorated, e.g. parabolic or Higgs) sheaves in algebraic geometry, geometric invariant theory (GIT), and symplectic geometry.

Roughly speaking, algebraic geometry is the study of geometric objects such as curves or surfaces which are defined by algebraic (i.e. polynomial) equations. Classic examples are the conic sections: these are curves defined as the zero set of a single (non-degenerate) quadratic equation in two variables. The beauty of algebraic geometry arises from the interplay between geometric intuition and algebraic precision: if we wish to answer geometric questions (such as "is this surface smooth?" or "in how many points do these two curves meet?"), there are powerful algebraic tools (such as the ring of polynomial functions on our curve or surface and the structure of its ideals) that help us find precise answers.

Moduli problems are classification problems, such as: when are two given curves the same, up to a linear change of coordinates, or up to a mapping preserving its algebraic structures? Whilst a lot of pure mathematics consists of classification questions (e.g. "classify finite simple groups up to isomorphism", or "are these two graphs just different ways of drawing the same combinatorial object?"), they take an added flavour in geometry: because geometric objects come in continuous families, the set of equivalence classes inherits a geometric structure and is a space itself: this is called a moduli space. For example, it makes sense to ask whether two given curves in the plane are "close" to each other, and if they are, then the two points of the moduli space corresponding to these two curves should be "close".

This may sound very abstract, but there are in fact important connections between moduli theory in algebraic geometry and modern physics. For example, the moduli space M of all Riemann surfaces (a.k.a. complex algebraic curves) of given genus plays an important role in string theory: a string moving through space-time X traces out a surface, called a worldsheet (which is almost holomorphic, i.e. almost a Riemann surface), so may be viewed as a map from a Riemann surface to X (which almost, but not quite preserves the holomorphic structure). Thus, thinking about a moduli space of maps from Riemann surfaces to X (satisfying `almost holomorphic' conditions) gives us information about all worldsheets. Finally, integrals over this moduli space sum up information about all possible trajectories and quantum actions (these are the `path integrals' of quantum mechanics). This illustrates how geometry and moduli theory in particular is inspired by physical ideas: the moduli space of maps just mentioned generalises the moduli space M of Riemann surfaces: M corresponds to the case when X is a point.

I am particularly interested in universal moduli of decorated sheaves on varying stable curves (i.e. moduli of pairs (C,E) of a curve C and a decorated sheaf E on C) and in moduli of (parabolic and Higgs) sheaves on a fixed nodal curve. I am working on constructions of such moduli spaces using GIT and I study their topology and geometry using (universal) matrix divisors.