I am a geometer and topologist, but for some time my main interest has been in applying these subjects in quantum field theory and string theory – trying to understand the role of the concept of space in fundamental physics.
We organize our perceptions of the world in terms of space and time, and, by analogy, classical physics attributes a ‘space of states’ to every physical system. Quantum theory forces us to see these state-spaces as approximations to less intuitive structures whose geometrical nature we should like to understand better. Quantum field theory shows us how to organize them in terms of conventional space-time, but to understand gravitation – the quantum dynamics of space-time itself – we need to go beyond this hybrid picture. String theory offers us the only hints we have in this direction.
- Emeritus Fellow, All Souls College (from 2009)
- Senior Research Fellow, All Souls College (from 1999 to 2009)
- Lowndean Professor of Astronomy and Geometry, and Fellow of St John’s College, Cambridge (from 1990 to 1999)
- Oxford University: Junior Lecturer in Mathematics (1965-6); CUF Lecturer (1966-70); Science Research Council Research Associate (1971-5); University Lecturer (1975-8); Reader (1978-89); Professor of Mathematics (ad hominem) (1989-90) (from 1965 to 1990)
- Fellow, St Catherine’s College, Oxford (from 1966 to 1990)
- Junior Research Fellow, Worcester College, Oxford (from 1964 to 1966)
- Graduate student, Balliol College, Oxford (from 1963 to 1964)
- Graduate student, St. John’s College, Cambridge (from 1962 to 1963)
- Undergraduate, University of Sydney (from 1958 to 1961)
- Geometry: algebraic topology and the structure of manifolds
- Quantum field theory: its conceptual and mathematical structure; geometric applications of field theory; two-dimensional conformal field theory
- Global analysis: index theory, geometry and representations of finite- and infinite-dimensional Lie groups; integrable systems
- Space and spaces, Presidential Lecture to the London Mathematical So-
ciety, November 2013.
- 'A geometric perspective on quantum field theory', Contemporary Mathematics, 620 (2014), pages 281-294.
- 'Reflections on Quillen's work on algebraic K-theory', Quarterly Journal of Mathematics 64 (2013), pages 645-664.
- 'Locality of holomorphic bundles, and locality in quantum field theory', in The Many Facets of Geometry, ed. O. Garcia-Prada et al., (Oxford: Oxford University Press, 2010), pages 164-176.
- ‘The definition of conformal field theory’, in U. Tillmann (ed.), Topolology, Geometry, and Quantum Field Theory (Cambridge: Cambridge University Press, 2004).
- (with A. Pressley), Loop groups (Oxford: Oxford University Press, 1986).
- ‘Loop groups and equations of KdV type’, IHES Publ. Math., 61 (1985), pages 5-65.
- ‘Unitary representations of some infinite-dimensional groups’, Comm. Math. Phys., 80 (1981), pages 301-42.
- ‘The topology of spaces of rational functions’, Acta Mathematica, 143 (1979), pages 39-72.
- ‘Categories and cohomology theories’, Topology, 13 (1974), pages 293-312.
- Publications (PDF)
- President of the London Mathematical Society (from 2011 to 2013)
- Editor, Topology (from 1970 to 1990)
- Member, Institute for Advanced Study (from 1969 to 1970)