The Geometry and Topology 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 our activities from the earliest stage of their studies.
- 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: Symplectic and Poisson geometry, momentum maps and their applications to Hamiltonian systems, Dirac structures. Supermanifold geometry: Forms and integration on supermanifolds. Classical supermanifolds and analogs of classical differential-geometric structures. Super linear algebra. Berezinians, their properties and applications. Generalizations of algebra homomorphisms such as n- and p|q-homomorphisms. Geometric and algebraic structures having origins in modern physics, in particular, quantum field theory and string theory. Odd symplectic and odd Poisson geometry. Odd Laplacians. Q-manifolds. Lie algebroids, their analogues and generalizations. Derived brackets and homotopy algebras. Geometry associated with differential operators.