by Rodrigo Bañuelos and Burgess Davis
Don Burkholder had a long and distinguished career in mathematics. He
is recognized worldwide for his deep and lasting contributions to
martingale theory and its applications to other areas of mathematics.
As a person, colleague and friend, he was kind and generous, serving
as mentor and role model for many young people entering the field.
Burkholder published many of his foundational papers in IMS journals,
particularly The Annals of Mathematical Statistics and The Annals of Probability. His many contributions to probability and analysis have
been reviewed in three recent publications: (1)
“Don Burkholder’s
work on Banach spaces,” (2) “Donald Burkholder’s
work in martingales and analysis,” and (3)
“The
foundational inequalities of D. L. Burkholder and some of their
ramifications.” The first two articles, one written by
Gilles Pisier
and the other by
Rodrigo Bañuelos
and
Burgess Davis,
respectively,
appeared in Selected Works of Donald L. Burkholder, published
by Springer in 2010. The third article, written by Bañuelos,
appeared in the special volume Don Burkholder: A collection of
articles in his honor, of the Illinois Journal of
Mathematics, 2011. The reader is referred to these articles for an
in-depth look at Burkholder’s work, its applications and connections
to different fields in mathematics where his work has had, and
continues to have, an impact. Here, we briefly touch upon a few of his
papers.
In 1966, Burkholder published his celebrated
“Martingale transforms”
paper in The Annals of Mathematical Statistics. In the 1930s,
Marcinkiewicz
and
Paley
(in separate papers) proved an inequality for
the Haar system of functions in the unit interval, which is equivalent
to the boundedness of dyadic martingale transforms with the
predictable sequence taking values in
For the next several years, Burkholder continued to work on applications of martingale theory and produced many other influential papers. His 1977 publication in Advances in Mathematics, for example, examines the exit times of Brownian motion from cones and other domains in Euclidean spaces and applies this to the behavior of harmonic functions. This paper influenced many subsequent papers on this and related subjects.
In 1984 Burkholder returned to boundedness of martingale transforms.
In his Annals of Probability paper,
“Boundary
value problems and sharp inequalities for martingale transforms,” he
introduced the now called “Burkholder method.” This was a new
approach that reproved his 1966 boundedness of martingale transforms results with optimal constants and provided extensions to the
setting of Banach spaces. This allowed Burkholder and others to answer
many questions about the geometry of Banach spaces for which both
martingale transforms and Calderón–Zygmund singular integral
operators (most notably the Hilbert transform) acting on functions
taking values in those Banach spaces, are bounded on
Burkholder served as president of the IMS from 1975 to 1976 and was editor of The Annals of Mathematical Statistics from 1964 to 1967. He served on multiple editorial boards and scientific advisory committees for both the IMS and other scientific societies. He supervised 19 Ph.D. students and several postdocs. He was elected member of the National Academy of Sciences in 1992. He was a fellow of the IMS, the American Academy of Arts and Sciences, the Society for Industrial and Applied Mathematics, the American Association for the Advancement of Science and an inaugural fellow of the American Mathematical Society. In 1970 he was an invited speaker at the ICM held in Nice, France.
Don Burkholder grew up on a farm in Nebraska during the years of the Great Depression and the Dust Bowl, which perhaps had something to do with his ferocious work ethic. After he retired from his distinguished professorship position at the University of Illinois he was asked if he was now taking it easy, to which he replied “I am, I no longer work on Sunday afternoons.” His character also showed in how kind and helpful he was to others and specially to young mathematicians. He was always encouraging and never intimidating.
Burkholder attended Earlham College where he met Jean, his wife and the mother of their three children. He received his Ph.D. in mathematical statistics from the University of North Carolina at Chapel Hill in 1955 where he worked under the mathematical statistician Wassily Hoeffding, one of the founders of nonparametric statistics. His thesis, “On a certain class of stochastic approximation processes,” published in The Annals of Mathematical Statistics, extended work of Julius Blum, Kai Lai Chung, Joseph Hodges, and Erich Leo Lehmann on “asymptotic normality.” Immediately after graduating he took a job at the University of Illinois at Urbana-Champaign where the legendary Joseph L. Doob, working on martingales and applications of probability to potential theory and complex analysis, was a professor. Almost certainly influenced by Doob, Burkholder soon turned his attention to these subjects.
During his entire career, Burkholder wrote only six joint research papers. His closest collaborator, and the only mathematician with whom he wrote more than one paper, was Richard Gundy. Their work had a deep and lasting impact in probability and its applications. Burkholder’s later work in the eighties on martingales taking values in Banach spaces led him to new techniques to prove sharp martingale inequalities. The latter have had applications in areas which on the surface are far removed from martingales. The wide use of Burkholder’s ideas, results and techniques, shows the universality of his mathematics, although he seemingly “only” touched martingales.