Professor John Wood

Professor John Wood

Profile

I work in differential geometry studying harmonic maps: transformations of Riemannian manifolds which extremize a natural energy functional and include geodesics and minimal surfaces; and harmonic morphisms: mappings of Riemannian manifolds which preserve solutions of Laplace's equation and relate to complex structures and the shear-free ray congruences of Mathematical Physics.

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Research interests

My main interests are harmonic maps and harmonic morphisms in differential geometry.

Harmonic maps are mappings (`transformations') between Riemannian manifolds which extremize a natural `Dirichlet' energy functional. They include geodesics (paths of shortest distance such as great circles on a sphere), minimal surfaces (soap films) and non-linear sigma models in the physics of elementary particles. They also have applications to the theory of liquid crystals and robotics. I work in the construction and classification of harmonic maps, especially from surfaces to symmetric spaces, and, with others, have given a completely explicit construction for the important case of maps from the Riemann sphere to the unitary group and other classical groups and symmetric spaces.

Harmonic morphisms are mappings of Riemannian manifolds which preserve solutions of Laplace's equation; elementary examples are conformal transformations of the complex plane. The first serious study was by Jacobi who, in 1848, wanted to find all complex-valued solutions f to Laplace's equation on Euclidean 3-space such that any analytic function of f is still a solution---such maps are precisely the harmonic morphisms. This problem was also posed by probabilists working in stochastic processes, harmonic morphisms being the Brownian path-preserving transformations, see, for example [A. Bernard, E.A. Campbell and A.M. Davie, Brownian motion and generalized analytic and inner functions, Ann. Inst. Fourier (Grenoble) 29 (1979), 207--228]. Harmonic morphisms can be characterized as harmonic maps which satisfy an additional condition called `horizontally weakly conformality' or `semiconformality' [B. Fuglede, Harmonic morphisms between Riemannian manifolds Ann. Inst. Fourier (Grenoble) 28 (1978), 107--144] and [T. Ishihara, A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ. 19 (1979), 215--229]. My work has been to understand these mappings, starting with a complete solution to Jacobi's problem. I have also found links with other concepts, such as complex (`Hermitian') structures and the shear-free ray congruences of Mathematical Physics; in particular, I found a twistor theory for harmonic morphisms with values in a surface. In 2003, I wrote with P. Baird the first account in book form of this subject, which has become the standard text.