Geometry
Research in Bath in these topics is divided between algebraic geometry and differential geometry with excursions into discrete differential geometry. The members of staff working in this area are Prof F.E. Burstall, Prof DMJ Calderbank, Dr U Hertrich-Jeromin, Prof A.D. King and Prof G.K. Sankaran.
There is a regular and lively Geometry seminar with a strong ecumenical flavour and, from time to time, Bath also plays host to the wandering COW seminar.
Algebraic Geometry
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 of Narasimhan-Seshadri, Kobayashi-Hitchin etc. provide differential geometric constructions of the moduli spaces and results, such as the Verlinde formula, from theoretical physics 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.
Differential Geometry
A broad view of differential geometry is taken at Bath encompassing (pseudo-)Riemannian, Kähler and Weyl geometry well as the less fashionable (and more difficult) parabolic geometries which include the classical conformal, projective and Lie sphere geometries.
Themes that interest us include submanifold geometry (and more generally, the variational geometry of maps between manifolds); classical differential geometry from a modern perspective; explicit constructions of special metrics in Riemannian and Kähler geometry; links with integrable systems (soliton theory, Darboux-Bäcklund transformations and twistor theory); discrete integrable geometries and links with theoretical physics (gauge theory, string theory and supersymmetry).
Among the topics of active research are harmonic maps of surfaces into homogeneous spaces (these solve a variational problem in Riemannian geometry and include the sigma-models of theoretical physics); integrable surface geometry (surfaces of constant mean curvature; Willmore surfaces in conformal geometry and their analogues in projective and Lie sphere geometry; isothermic surfaces); geometry of discrete nets (discrete principal nets; discrete isothermic and special isothermic nets; discrete nets of constant mean curvature; transformations and Bianchi permutability theorems); relative Cartan geometry: a uniform approach to submanifold geometry against a parabolic background (such as conformal, projective or Lie sphere submanifold geometry); scalar-flat Kähler metrics and K-stability; self-duality in conformal geometry.
All these activities are informed by a constant interest in the related areas of algebraic geometry, Lie theory and theoretical physics.
