Research in algebraic geometry at Bath has as a unifying theme the geometry of moduli spaces, especially over the complex numbers.
Moduli spaces parametrise other geometric or algebraic objects and are of basic importance in algebraic geometry. Particular moduli problems studied involve abelian varieties, K3 surfaces and other symplectic manifolds, vector bundles, and representations of associative algebras and quivers.
Abelian varieties include Jacobians and elliptic curves and are important in geometry, number theory and complex analysis. K3 surfaces form one of the basic types of algebraic surface and are examples both of Calabi-Yau manifolds and of symplectic manifolds. The study of these moduli spaces, whose construction typically involves a quotient by a discrete group action, has strong links with number theory and especially modular forms. The study of the structure at infinity of the moduli spaces involves toric geometry.
Vector bundles have close links with gauge theory. Theorems and conjectures of Narasimhan-Seshadri, Kobayashi-Hitchin, Yau, Tian and Donaldson et al. provide differential geometric constructions of the moduli spaces and nonlinear analogues such as Calabi's extremal Kähler metrics. Results from theoretical physics, such as the Verlinde formula, give powerful tools for computing cohomology.
Moduli of representations of quivers add geometry to classification problems in representation theory and have recently also found links with gauge theory and string theory. Such moduli spaces are useful in geometric problems, such as the McKay correspondence, where they are involved in `non-commutative' resolutions of quotient singularities and, now, more general toric singularities.