Rod Gover - Research

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Research | Publications | Supervision | Teaching

Professor of Mathematics, University of Auckland

Head of the Analysis, Geometry and Topology Group.

Projects offered for supervision

Research profile

My main interest is in the application of differential geometry and Lie representation theory to problems in analysis, complex analysis and mathematical physics. I have published work on a diverse range of topics including integral transforms and their applications to representation theory, quantum groups and their representation theory. My main area of specialisation is the class of parabolic differential geometries. A special calculus called tractor calculus is important for treating geometries in this class. A current theme of my work is the further development of this calculus, its relationship to other geometric constructions and tools, as well as its applications to the construction and understanding of local and global geometric invariants and natural differential equations.

Fields of interest

Specific research interests include the following.

• Foundational theory of parabolic geometries and tractor calculus.

Parabolic differential geometries form an important class of structures; conformal geometries, CR geometries, almost quaternionic geometries, and projective differential geometries are examples, but there are many more. Tractor calculus provides an approach to these structures which is conceptually powerful, uniform (similar ideas apply to many different structures) and calculationally effective. Most importantly it provides a link between Lie representation theory and the Elie Cartan approach to geometry.

• Local invariant theory and invariant differential operators.

These are the basic tools for studying geometric structures. Problems include the development of formulae, and algorithms for obtaining formulae, and classification issues.

• Special invariants and global structure.

Here the basic problem is to find and explore the applications of global invariants, in particular those which do not arise in an obvious way from local invariants.

• Geometry of partial differential equations (PDE).

The idea is to apply geometric ideas to the study of PDE and their solution space. Often this also draws on prolongations and Cartan connections to capture geometric structure.

• Special geometries.

Here the focus is the use of the theory of parabolic geometries, prolonged differential systems and related ideas to study and construct special geometric manifolds such as Poincare-Einstein structures.

• Conformal approaches to Riemannian geometry.

Classical questions linked to, or amenable to study by, conformal structure.

• Applications to mathematical physics.

Many of the problems in conformal geometry are motivated by or related to theoretical physics.