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Algebra, geometry, and number theory research

Our research includes algebraic geometry, differential geometry, group theory, representation theory, homological algebra, and analytic number theory.

Algebra, geometry, and number theory are interlinked in multiple ways: roughly speaking, algebra provides the language and geometry the intuition.

Our research 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, homological algebra, and moduli spaces, while work in smooth and discrete differential geometry ties in with algebraic and Lie groups, discrete groups, and combinatorics.

There are also interactions with algebraic geometry and analysis in number theory, with geometric analysis and ergodic theory in differential geometry, and with integrable systems in mathematical physics.

We have a regular and lively seminar with a strong outward-looking flavour and, from time to time, Bath plays host to the famous wandering COW algebraic geometry seminar. We also host the Group Pub Forum community pages.

Our research can be arranged under the following broad headings.

Algebraic geometry

Research staff: David Calderbank, Alastair Craw, Alastair King, Daniel Loughran, Johannes Nordström, Gregory Sankaran

Research in algebraic geometry at Bath has as a unifying theme the geometry of moduli spaces. Moduli spaces parametrise other geometric or algebraic objects and are fundamental in algebraic geometry. Particular moduli problems studied involve abelian varieties, K3 surfaces and other symplectic manifolds, Fano varieties, vector bundles, and representations of associative algebras and quivers.

Abelian varieties are important in geometry, number theory and complex analysis. K3 surfaces are one of the basic types of algebraic surfaces and are examples 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. Work 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 toric singularities.

Differential geometry and Lie theory

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

A broad view of differential geometry is taken at Bath encompassing (pseudo-)Riemannian and Kähler geometry, as 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.

Number theory

Research staff: Daniel Loughran, Gregory Sankaran

Number theory is the study of the integers and their properties. Many of the motivating problems come from the study of Diophantine equations, which are polynomial equations where one is interested in solutions in the whole numbers. Whilst originally viewed as a curiosity from antiquity, in recent times they have found spectacular applications to modern society and information security through cryptography.

Research in number theory in Bath covers the areas of arithmetic geometry, analytic number theory, and algebraic number theory. Specific topics include: existence and distribution of rational points on varieties, distribution of number fields, applications of modular forms, and the arithmetic of moduli stacks. We have close links with the algebraic geometry group and use a range of tools from additive combinatorics, sieve theory, harmonic analysis, étale cohomology, class field theory, algebraic groups, lattices, and the theory of automorphic forms.

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 of 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.