Research

I have conducted research into a number of topics in mathematics since my coursework Masters project in 2001. Research in mathematics is somewhat like scientific research in general, but rather than extending what is known, it is more like discovering what can be known. Mathematics research involves proof as well as hypothesis and experiment.

My most successful method of working has been a combination of persistent curiosity, a willingness to entertain new fields of study while drawing on my breadth of knowledge, computer experimentation to discover patterns, conjectures based on these patterns, and literature searches and collaboration to connect the discovered patterns to known mathematics. Other useful methods have included finding the connections between two similar fields, and extending known methods to a different setting. I plan to continue along these lines and to explore new strategies.

My research topics include:

  • Constructive approximation: Sparse grids, approximation and quadrature on the sphere and compact manifolds.
  • Combinatorics and statistics: Random number generation and testing, combinatorics and statistics of words in sequences, new constructions for Hadamard matrices.
  • Clifford algebras: Numerical computation with Clifford algebras, numerical Clifford analysis.
  • Mathematical software: Object-oriented numerical analysis, parallel linear algebra using ScaLAPACK. Software to ensure reproducibility of computed results.

Presentations on other topics