# Algebraic geometry

Research staff: David Calderbank, Alastair Craw, Alastair King, Jesus Martinez Garcia, Gregory Sankaran

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.

# Differential geometry and Lie theory

Research staff: Fran Burstall, David Calderbank, Veronique Fischer, Mark Haskins, Jesus Martinez Garcia, Roger Moser, Johannes Nordström

A broad view of differential geometry is taken at Bath encompassing (pseudo-)Riemannian and Kähler geometry well as (the more challenging) 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, Bäcklund-Darboux transformations and twistor theory) and dynamical systems (ergodic 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.

# Group theory

Research staff: Geoff Smith, Gunnar Traustason

The focus of research in group theory at Bath has been on discrete groups and their geometry.

Specific research interests include Engel groups and other generalized nilpotent groups, Engel Lie algebras, problems of Burnside type, and the cycle structure of infinite permutation groups. Notable developments include a structure theory for 4-Engel groups, which represents a significant contribution to the local nilpotence theorem for 4-Engel groups. Other recent work includes groups with all subgroups subnormal of bounded defect and a characterisation of group varieties of which the class of locally nilpotent (nilpotent-by-finite) groups form a subvariety, a property shared by the Burnside varieties.

# Representation theory and homological algebra

Research staff: Alastair Craw, Alastair King, Xiuping Su

The main focus of research in homological algebra is on applications to deformation theory and moduli problems for representations of algebras, which may be related to similar problems for sheaves on varieties through equivalences of derived categories, such as the Beilinson equivalence for projective spaces and the McKay correspondence. The role of derived categories in moduli problems and other areas of geometry is one of the exciting developments of recent years, which is closely followed at Bath. This leads on to the study of A-infinity structures, a tool from topology, whose importance in geometry and theoretical physics is beginning to be more widely appreciated.