Algebra and Geometry
Algebra and geometry stem from the study of equations and shapes, and their associated properties, structures and symmetries. The subjects are interlinked in multiple ways: roughly speaking, algebra provides the language and geometry the intuition.
Research in Bath strongly reflects these interconnections, relying upon advanced algebraic methods in geometry as much as geometrical ideas in algebra. Our research in algebraic geometry meets with quiver representation theory and homological algebra, while work in smooth and discrete differential geometry ties in with algebraic and Lie groups, discrete groups, and combinatorics. There are also interactions with number theory in algebraic geometry, with geometric analysis and ergodic theory in differential geometry, and with integrable systems and mathematical physics.
Bath hosts the Group Pub Forum community pages. We have twice hosted the conference "Groups St Andrews" in recent years (1997 and 2009); this 4-yearly conference is the largest regular international meeting of group theorists in the world.
Our research in algebra and geometry can be arranged under four broad headings:
- Algebraic geometry
- Differential geometry and Lie theory
- Group theory
- Representation theory and homological algebra
Related areas of our research include:
Further details of our research are given below.
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 conjuctures 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, fgive 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.
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.
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.
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.