In recent years I have divided my academic efforts between traditional research and teaching efforts, and a growing interest in the education and development of school mathematics teachers. My main research interests have been in probability and classical analysis. I have been especially attracted by problems in analysis that can be effectively approach using probabilistic methods or where new and unusual insights are obtained through probabilistic interpretations.

My primary research activity at this time concerns the efficient observation and extraction of information from observations of random fields, i.e. random data {j (x)} where x varies over a higher dimensional space such as R² or R³. If, for example, one wishes to predict a particular value j (x₀) of {j (x)} based on a finite set of observations {j (x_i): 0< i< n } where the sites { x_i } are restrained be in set D, an natural question is where should the sites { x_i } be placed in order to extract the maximal amount of useful information for predicting the value j (x₀). This is a problem of optimal design, and any effective solution of the problem will depend on the particular dependence structure of the field {j (x)}. Surprisingly, in a number of cases of interest, such problems can effectively be addressed using the tools of functional analysis, especially Sobolev spaces and spaces of Bessel potentials. Not only does the theory of these spaces shed light on the questions of probability, but the design problems lead, in turn, to new questions and results about approximation (known as spectral synthesis) in the function spaces and new results concerning the numerical integration of Sobolev functions. This work on this project is joint work with my former graduate student Raina Robeva at Sweet Briar College.

Another long-term joint research effort with the complex analysts J. Milne Anderson at University College London has dealt with problems of analysis closely linked to the boundary behavior of conformal mappings of the unit disc D in the complex plane into domains W with non-differentiable boundaries. A number of questions that arise here have a strong probabilistic component and this connection has proved to be fruitful to both analysis and probability. Two easily stated special results that have come from this collaboration concern the nowhere differentiable Weierstrass function

We have shown that for almost all a, if we define f_a(x)= a x + f(x) there is a subset E of the real line that has Lebesgue measure 0 and for which the image of the complement f_a(E^c) is a Lebesgue null set. We do not know if the result is true for all a or for a = 0.

It is known that the series for g(x) diverges for almost all x. We have strengthened this result and have shown that for almost all x the partial sums of g(x) are dense. These results continue to fascinate me and I hope to return to them in the near future.

One last research area of interest that I should mention is that of probabilistic inequalities. More than 25 years ago I showed that P(A Ç B) £ P(A) P(B) if A and B are symmetric convex sets in the plane and P is the standard mean 0 normal probability measure on R². Extending this to 3 or more dimensions has frustrated the best efforts of many mathematicians in the intervening years. After having set it aside for years, I have begun again to think on this problem.