Research in Geometry and Topology

The Topology and Geometry Group plays a major role in the School's research programme. Our members have extremely broad interests, ranging from Algebraic Geometry and Topology to Theoretical Physics. We have an international network of colleagues and collaborators, with whom we often exchange visits in order to further our projects. We run the weekly Manchester Geometry Seminar, as well as the more specialised Topology Seminar, and Geometry and Mathematical Physics Seminar, and have links with neighbouring topology groups through the Transpennine Topology Triangle. Given the interdisciplinary nature of our work, we have common interests with several other research groups in the School; some of us share active membership.

Our members have a long history of successful supervision of PhD students, who are encouraged to involve themselves in the Group's activities from the earliest stage of their studies.

Our Research Interests

Our research interests are described in greater detail on our respective homepages. Here are some key areas:

Algebraic Geometry : geometry in characteristic p, combinatorial and enumerative geometry, computational algebraic geometry.

Algebraic and Differential Topology : bordism and cobordism, formal group laws, homology and stable homotopy theory, homotopy groups and braids, Hopf rings, immersions of manifolds, K-theory and bundles, the Landweber-Novikov and Steenrod Algebras, loop spaces and decompositions, model categories and rational topology, p-local finite groups, toric manifolds and varieties, unstable homotopy theory.

Differential Geometry : connection theory, singularities and completion of manifolds, universal connections, supermanifolds; symplectic geometry, Poisson and Dirac structures, momentum maps; applications to general relativity, cosmology, geometry of parametric statistical models and Hamiltonian dynamics.

Geometry: computational complexity, euclidean surfaces, geometric group theory, hyperbolic structures, optimisation, polytopes and their graphs.

Mathematical Physics: geometry of supermanifolds and its applications; algebraic structures in supermathematics and mathematical physics; mathematical problems of quantum physics; quantization; cohomology in physics; bracket structures and homotopy algebras; polynomial invariants of supermatrices; integrable systems and Frobenius manifolds.

Algebra: formal power series, Hopf algebras, quantum groups, representations of GL(n), symmetric functions, umbral calculus and sequences of polynomials

Staff Members:

PhD projects

Several of our members' homepages give details of the PhD projects for which they are offering supervision, together with information about recently graduated students and their theses. A list of projects is offered by Jelena Grbic and Nige Ray