## Moduli spaces

Moduli spaces are families of geometric objects, such as families of conformal structures on surfaces, or families of solutions to partial differential equations such as magnetic monopoles. They are often physically motivated. Many of the major issues in geometry today have their roots in physics. In his discovery of anti-particles, Dirac realised that important information is stored in the vacuum of a physical system. On the simplest level, this says that particles and anti-particles can spontaneously create from energy, and annihilate and release energy. One can study a geometric object, such as a sphere, by treating it as a possible ``universe", i.e. by confining the particles and anti-particles to live on the sphere. Similarly, we confine electromagnetism and gravity to live on geometric objects such as the sphere and study the physical laws of the system such as Maxwell's equations or Einstein's equations. By studying the vacuum of the system one hopes to get information about the underlying geometric object, and conversely further insight into more sophisticated physical systems.

Such systems in geometry are often nonlinear. Nonlinearity is a significant phenomenon widespread throughout mathematics and science often detected by the sensitivity of a system to small changes. My research focuses on specific nonlinear systems over manifolds such as magnetic monopoles, which are solutions to a nonlinear generalisation of Maxwell's equations, minimal surfaces, which are surfaces that minimise area and more generally are stationary with respect to area, and Einstein metrics, solutions to the nonlinear Einstein equation. Each of these three systems has solutions that belong to a special class of nonlinear systems known as integrable. An integrable system is a nonlinear system with conserved quantities, that allow one to find solutions using deep techniques from complex analysis and algebraic geometry such as classical results on elliptic functions.

The paper
• Weil-Petersson volumes and cone surfaces
describes relations between volumes of moduli spaces of hyperbolic surfaces. The paper
• Morse field theory
studies the moduli space of maps of graphs into a manifold using Morse functions to make the spaces of maps finite dimensional. The following papers are about magnetic monopoles that live in hyperbolic space. My most recent paper describes how Riemann surfaces that contain special divisors correspond to magnetic monopoles. This is an application of the general philosophy of integrable systems that algebraic information, basically using polynomials, enables us to understand solutions to a nonlinear partial differential equation. The two next most recent papers study the boundary values of hyperbolic monopoles. A Dirac monopole in hyperbolic space is determined by its limit on the 2-sphere at infinity. A nonlinear smoothing of Dirac monopoles, SU(2) monopoles, were also known to be determined by their limit on the 2-sphere at infinity in the case that the mass is half-integral. The boundary algebras paper and the holomorphic spheres paper generalise this to any mass.
• Spectral curves and the mass of hyperbolic monopoles
• Boundary algebras of hyperbolic monopoles
• Hyperbolic monopoles and holomorphic spheres
• Compactification of hyperbolic monopoles
• Zero and infinite curvature limits of hyperbolic monopoles
The following papers study instantons which are related to monopoles.
• Periodic instantons and the loop group
• Degenerating metrics and instantons on the four-sphere
• Real instantons and quaternionic classifying spaces

Embedded closed geodesics that are not shortest curves are difficult to locate. On a two-sphere with an incomplete metric such geodesics should exist and we show this for a large class of metrics. This gives examples of minimal surfaces. The papers
• Closed geodesics on incomplete surfaces
• Minimal spheres of arbitrarily high Morse index
study minimal surfaces in 3-manifolds.

## Topology of algebraic singularities

Algebraic geometry supplies many powerful tools for the study of geometry. It consists of a special class of geometric objects, algebraic varieties, defined as the zero set of complex polynomials. A point of a variety is an algebraic singularity if under all magnifications of a microscope a neighbourhood of the point does not resemble the neighbourhood of a point in Euclidean space. An example of an algebraic singularity is a point that lies on the intersection of two components of the variety. Techniques from knot theory supply useful information about algebraic singularities. A focus of my research is to study topological properties of algebraic singularities such as the ability to detect them via the distortion of nearby varieties, known as the monodromy. My most recent paper studies stable reduction of families of curves. Families of algebraic curves generally contain curves with singularities worse than double points. Stable reduction replaces the singular curves with curves with double points. This process is related to a natural decomposition of 3-manifolds. The papers
• Stable reduction and topological invariants of complex polynomials
• The Orevkov invariant of an affine plane curve
• Rational polynomials of simple type
• Unfolding polynomial maps at infinity
• Vanishing cycles and monodromy of complex polynomials
• Nontrivial rational polynomials in two variables have reducible fibres
study families of algebraic curves, the singularities that arise on the curves, and monodromy in the family. The paper
• A proof of Atiyah's conjecture on configurations of four points in Euclidean three-space
solves the four point case of an elementary conjecture regarding any number of points in space. The papers
• Algebraic Analysis of Linear Precoding for Blind Channel Identification
• Global Identifiability of Channel Identification Problems
apply algebraic geometry to the following problem that can arise, say, from mobile phone signals: Suppose you receive a message that has undergone an unknown linear transformation. How much test information should be sent to enable decoding of messages?