I shall continue to work in Flow-Induced Vibrations. The practical problems there have not yet been solved, and there is work for generations of mathematicians. The main problem to work on at the moment is to increase our understanding of fluid-structure interaction. Semi-empirical models might help with this, provided we can formulate one that actually is useful in predicting behaviour over wide ranges of geometries. There's more hope that semi-empirical models will simply be used, in control of vibrating structures or in prediction for design, in lieu of a good rational theory.
I shall also continue to work in Scientific Computing. We do not have a satisfactory theory of complexity of computational problems with backward error in mind---most of the mathematical theory of complexity ignores data error, when keeping it in mind can very significantly reduce the complexity. This overriding perspective is something I want to tackle, though it is perhaps too grand in scope.
Right now I will be pursuing the SVD algorithms for polynomial systems, as a step along that road (that has practical value also). I will also be pursuing symbolic generation of finite-difference methods.
But there is a more `bread and butter' issue. Mathematics has to be seen to be relevant to survive. I believe that `mathematical consulting', both within the University and without must be developed. I plan to expand my horizons by working on semi-empirical mathematical models for HIV transmission (and for viral populations in humans), for the practical reasons that such research is both interesting and possible, and regarded as being important at my University. That sounds a bit cynical, but it isn't meant in a cynical way---given the choice between working on equally interesting topics, why choose the one that's least regarded?
But the most important thing to keep in mind is to keep the enjoyment level up. I love my
work, and occasionally find it satisfying on a level deeper and more personal than would be
believed by any non-mathematician. ``I love my job so much, I'd do it for nothing. Unfortunately, they know that.''