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Research Plans

I'm currently scrambling. I now have more than six essentially
finished but as-yet unwritten papers, on topics ranging from
heat flow through concentric cylinders through interval defect-controlled
Taylor series methods for ordinary differential equations to the symbolic
solution of linear systems of equations with parameters. My research on
flow-induced vibration has been taken over by my graduate students
(I'll include links to their home pages as they become available).
Nonetheless I remain interested in this, particularly in the automation of
perturbation series methods, in computer-aided bifurcation studies (including
the use of Groebner bases and parametric solution of polynomial systems),
and in numerical (CFD) work supporting or complementing the semi-empirical
modelling work that has been done and is being done.

Another student is working on an empirical assessment of the quality of
numerical methods for the solution of initial-value ODE's, using some ideas
from defect control and optimization theory.

Yet another student is looking at compact methods for partial differential
equations.

In addition I have a book in preparation that is suffering from a serious
stall---there is a difficulty with making my Maple code compatible with the
new release.

Nonetheless I have found time to complete some work on the solution of
systems of overdetermined multivariate homogeneous
polynomial equations (still the
zero-dimensional case, and there must only be a finite number of
solutions at infinity). The work remaining to be done on that is to find
an effective and efficient algorithm for the simultaneous upper
triangularization of (nearly) commuting matrices.

Another project is to investigate the dynamics of the Lambert W
function. This work is in fact nearly complete.

I want to look seriously at the question of the complexity of the solution
of a differential equation system using the defect-controlled approach; I
believe that this could give useful insights and at the same time yield
provable results.

I also want to investigate arbitrary-precision numerical analysis.

Recently I have become very interested in what we call
Organic Mathematics.