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Computational Geometry: Professor T.N.T. Goodman

This research has resulted in advances in the theory of B-splines and their multivariate generalizations. Theoretical investigations are being carried out into the decomposition of spline spaces under subdivision, such as the study of orthogonal decomposition (i.e. wavelets), particularly in the multivariate case. The study of asymptotic properties of refinable functions, which generalises and unifies known work on wavelets and on probability, has also led to research on uncertainty principles, in particular the first general uncertainty principles on general spheres. A joint programme with computer science has involved algorithmic development with regard to shape preserving interpolation, and the use of adaptive approximation in Computer Aided Geometric Design.

Differential Equations: Dr Fordyce Davidson and Dr Niall Dodds

This research is concerned with the study of ordinary and partial differential equations and related analyses. Bifurcation theory and the related analysis of nonlinear eigenvalue problems forms a major part of this work with interest in both the underlying theory and its application to boundary value problems arising from the study of physical and biological systems. This abstraction allows for the use of powerful tools from nonlinear functional analysis. Related interests include travelling wave phenomena and complex (chaotic) dynamics in parabolic pdes, integro-differential equations and m-point, p-Laplacian boundary value problems (where the boundary condition relates solution values on the boundary to solution values in the interior of the domain). Over recent years, the study of the relatively new topic of calculus on measure chains has been undertaken. This forms a rigorous framework for the study of processes which take place on complex time scales, such as nonintersecting intervals. Difference calculus and differential calculus form two special cases of this more general unifying theory. The properties of dynamic equations on these "time scales" can be very different to those derived via standard calculus and basics, but central technical questions remain unanswered.