Research Work

My main interest is in Algebraic Topology with a strong bias towards its applications in other areas of Mathematics. Some of my papers are listed below.

DIFFERENTIAL GEOMETRY

Several papers involving the use of topology to study questions in Differential Geometry were written with Duan Haibao. My favourite of these results is the fact that for every point x of a compact sub-manifold M of Euclidean space there is sphere tangent to M at x and also tangent to M at some other point y. (To avoid counterexamples, one has to interpret a plane as a sphere of infinite radius.)

• The existence of bitangent spheres. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), no. 1-2, 85--87.

The interaction between geometric properties and topological properties of manifolds is another interest. Einstein manifolds with certain positivity conditions on their curvature and with more complicated topology than expected are constructed in the paper written with Charles Boyer, Kris Galicki and Ben Mann :

• Compact $3$-Sasakian $7$-manifolds with arbitrary second Betti number. Invent. Math. 131 (1998), no. 2, 321--344.

LINEAR ALGEBRA

Since around 1960, results in linear algebra have been proved using algebraic topology but no `elementary' proofs have been found for them. The most famous is the result of Bott and Milnor on the non-existence of real division algebras of dimensions other than 1,2,4 and 8.

• Linear spaces of real matrices of given rank. Contemp. Math., 188, Amer. Math. Soc., Providence, RI, 1995, 219 -229.
• Linear spaces of real matrices of large rank. Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), no. 1, 147-151.
• On a question of Milnor concerning singularities of maps. Proc. Edinburgh Math. Soc. (2) 43 (2000), no. 1, 149-153.

MULTI-VALUED GROUPS

A rather different project is the development of a theory of n-valued groups and is being carried out with Victor M Buchstaber. This theory has many ramifications, in particular it has led to a result that generalises the classical result of I M Gelfand on maximal ideals of algebras of functions and generalises the Nullstellensatz.

• Multivalued groups, their representations and Hopf algebras. Transform. Groups 2 (1997), no. 4, 325--349.
• Multivalued groups, n-Hopf algebras and n-ring homomorphisms. Lie groups and Lie algebras, 85--107, Math. Appl., 433, Kluwer Acad. Publ., Dordrecht, 1998
• The Gelfand map and symmetric products. math.CO/0109122 Selecta Mathematica, 8 (2002), 523-535.
• Rings of continuous functions, symmetric products and Frobenius algebras. Russian Math. Surveys 59 (2004), no. 1. 125-144.

I have experience of several universities. I studied in Cambridge and Warwick and worked in Hull, Princeton, Swansea, Oxford, Berkeley, Sydney, Mexico, Paris, Edinburgh and Bonn.

I have supervised over 20 graduate students in Oxford and Edinburgh and they have worked on a wide range of topics.
A recent student worked on understanding how a robot hand can grasp an object firmly and another on classifying vector fields of a simple type on two-dimensional surfaces. Currently students are working on the topological properties of certain manifolds constructed using interesting differential geometric methods and on symplectic and contact geometry.