DEPARTMENT of MATHEMATICS   Pure Research Topology, Geometry and Combinatorics

Welcome to our Homepage!

|  Research Interests  |  Seminars  | Staff Members  |  Postgraduate Opportunities  |

The Topology, Geometry and Combinatorics Group plays a major role in the Department's research activities. We have an international network of colleagues and collaborators, with whom we often exchange visits in order to further our programmes. We share the weekly Manchester Geometry Seminar with out neighbours at UMIST, and have other links through the Transpennine Topology Triangle. We also organise occasional seminars of a more specialised nature. We offer supervision in many projects leading to a postgraduate degree, within the framework of the Department's overall postgraduate program.

TOPOLOGY is often referred to as "rubber sheet geometry", and is one of the largest and most important areas of mathematics developed this century. Increasingly, it is being applied in fields such as algebra, geometry, and dynamics, and in other disciplines such as computer science, economics, and theoretical physics. GEOMETRY itself is a much older subject, but modern research is no less exciting, and has links with almost every other branch of mathematics. For example, the infamous "travelling saleperson problem" is based on the geometry of polytopes, yet involves ultramodern theories of computational complexity. COMBINATORICS is basically the study of structures associated to finite sets, and underpins much of theoretical computer science. Nevertheless, its methods are assuming increasing importance wherever complicated algebraic or geometric problems arise, and are becoming an indispensable part of the modern mathematician's toolkit.

Because of these links, most of us have common interests with other Departmental Groups. 

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Our Research Interests

Our research interests span a wide range of topics, and are detailed on our respective homepages. Here are some keywords, and a sample article (in dvi or postscript form) by each of us:

ALGEBRAIC & DIFFERENTIAL TOPOLOGY: bordism and cobordism, formal group laws, homology and homotopy theory, Hopf rings, immersions of manifolds, K-theory and bundles, the Landweber-Novikov and Steenrod Algebras, toric manifolds and varieties

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

ALGEBRAIC COMBINATORICS: chromatic polynomials, formal differential operators, formal power series, Hopf algebras, partially ordered sets, quantum groups, representations of GL(n), symmetric functions, umbral calculus and sequences of polynomials 

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Seminars

For details of the joint Manchester-UMIST geometry seminar, click here.

Details of the Transpennine Topology Triangle are also available.

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Staff Members of the Group

Peter Eccles (Algebraic and Differential Topology)

Ron Ledgard (Algebraic Geometry and Coding Theory)

Nige Ray (Algebraic Topology and Combinatorics)

Grant Walker (Algebraic Topology and Combinatorics)

Reg Wood (Algebraic Topology and Combinatorics)

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Postgraduate Opportunities
Topology, Geometry and Combinatorics make up a large and exciting area of modern mathematics, in which we have supervised many MSc and PhD students over the last few decades; some of our ex-students now hold academic positions in a variety of mathematics departments around the world. Others have used their training with us as a stepping-stone to succesful careers in commerce, finance, industry, and teaching, for example.

We are keen to continue attracting postgraduate students to work in our group, and currently offer MPhil and PhD supervision in many different projects relating to our interests.

Please contact us individually by email if you wish, or check out the Department's postgraduate programmes. 

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