Department of Mathematics

Conformal geometry, submanifolds, and natural partial differential equations


Natural partial differential equations (PDE) are equations whose coefficients are geometrically determined by the underlying structure  on which they are defined. They provide the primary tool for studying  global structure via analysis, and also a vast array of physical and natural phenomena from particle equations to biological systems.

Submanifolds are also ubiquitous in mathematics and the physical sciences; these are mathematical structures which formalise and generalise the notion of surfaces in Euclidean 3-space. For example minimal surfaces are currently generating interest in nanotechnology. But these structures, and their higher dimensional analogues, are also important for the classical mathematics of gravity, and are the basic geometric objects of string and brane theory, which attempts to unify gravity with the other forces of nature. Biological cell membranes are known to arrange themselves in a way which minimises the Willmore energy, a quantity which measures the extent to which a surface differs from a round sphere.

The structures, quantities, and PDE mentioned are unified by conformal geometry and its generalisations. This research is based around exploring this observation to develop new theory and results for the understanding of these structures.

The research project is supported by the Royal Society of New Zealand Marsden Fund.

Researchers at The University of Auckland: