Since the end of the seventies there has built up first around Atiyah and later Donaldson a school based on ideas of Penrose regarding connections between the Riemannian geometry of four-dimensional space and three-dimensional holomorphic geometry. These ideas usually go under the name of Twistor Theory and can be traced back to Weierstraß and his description of minimal surfaces in three-dimensional Euclidean space. The fundamental idea is that problems in differential geometry can be solved by using either holomorphic or algebraic geometry; an idea which also occurs in the theory of integrable systems.
Many of the differential-geometric problems one studies in this way have their origins in physics. This is true of the geometry of instantons, monopoles and other Yang-Mills fields and the methods are also effective for the Einstein equations. Perhaps the most remarkable outcome is that the study of these equations from physics have given new life to four-dimensional differential topology and lead to many new, deep and surprising results. In higher dimensions, some of these ideas can be applied to geometry based on the quaternions. Such theories are also related to physics: quaternionic manifolds are target spaces for certain super-symmetric fermionic and bosonic sigma-models and some are solutions of the Einstein equations.
The geometers at IMADA do research in the area described above. Using holomorphic tools, we have looked for and found new solutions to the Einstein equations. This is part of a project investigating the correspondence between quaternionic geometry and complex manifolds, and this knowledge has been applied to the study of field theories which are higher-dimensional generalisations of monopoles. Recently, Einstein-Weyl geometry has been attracting some special attention. A substantial amount of the international research in this area takes place at IMADA in collaboration with geometers from Oxford and Paris.
The Kähler-Einstein equations have been studied both via Twistor Theory and via a Hamiltonian approach. It has been shown that quotients of nilpotent orbits of complex semi-simple Lie groups are quaternionic Kähler manifolds and this work is being used as part of an attempt to classify the compact examples. Furthermore, a series of results in algebraic geometry have been obtained by studying self-dual structures, and this has lead to an equivariant connected sum construction. This construction is based on work of Donaldson and Friedman, which also has been generalised to quaternionic geometry. An important part of the research is concerned with the study of complex deformation theory, super-geometry and quantum field theory.