May 16, 2013

Obituary: Donald L. Burkholder, 1927–2013

Donald Lyman Burkholder died in his sleep on April 14, 2013, in Urbana, Illinois. He was born January 19, 1927, in Octavia, Nebraska, the fourth of five children of Elmer and Susan (Rothrock) Burkholder. His mother had been a schoolteacher, and his father was a farmer who served on the community school board for many years. Education became the family business: of the four boys, the oldest was a superintendent of schools, the three youngest were college professors, and many in the next generation are educators.

In 1945, Don graduated from high school, where he was captain of the basketball team and senior class president, an honor (as he loved to relate) that came his way because his three classmates had already been president. He was drafted and entered the Civilian Public Service (CPS) as a conscientious objector, serving as a cook at a camp for fighting forest fires in Oregon and as an orderly at a mental hospital in New Jersey.

Following his discharge in December 1946, he acted on the recommendation of a friend and enrolled at Earlham College, a predominantly Quaker college in Richmond, Indiana. There he met his wife-to-be, Jean Annette Fox, and they were both drawn to the field of sociology by the vision and intellectual rigor of a new faculty member who had also served in the CPS, Bill Fuson.

After their wedding in June 1950, Don and Jean attended the University of Wisconsin in Madison as graduate students in sociology. In 1953, they went to the University of North Carolina at Chapel Hill, where Don had a fellowship to study sociological statistics. He soon discovered that his real interest lay in mathematics, and he completed a PhD in mathematical statistics in 1955 under the guidance of Professor Wassily Hoeffding. That summer, Don joined the Mathematics Department at the University of Illinois, Urbana-Champaign. In 1978, he was appointed professor in the Center for Advanced Study, allowing him to devote more time to research. He retired as professor emeritus in 1998.

Soon after he came to Illinois, Don, influenced by his eminent colleague Joseph Doob, turned to the study of martingales. It is now apparent that the two mathematicians who most advanced martingale theory in the last seventy years were Joseph Doob and Donald Burkholder. Martingales as a remarkably flexible tool are used throughout probability and its applications to other areas of mathematics. They are central to modern stochastic analysis. And martingales, which can be defined in terms of fair games, lie at the core of mathematical finance. Burkholder’s research profoundly advanced not only martingale theory but also, via martingale connections, harmonic and functional analysis.

In their 1970 Acta Mathematica paper, which followed Burkholder’s seminal 1966 paper “Martingale Transforms” in the Annals of Mathematical Statistics, Burkholder and Gundy introduced a remarkable technique which shows how certain integral inequalities between two nonnegative functions on a measure space follow from inequalities involving only parts of their distribution. This seemingly simple but incredibly elegant technique, now referred to simply as “the good–λ method”, revolutionized the way probabilists and analysts think of norm comparison problems. It is now widely used in areas of mathematics which involve integrals and operators. Burkholder’s outstanding work in the geometry of Banach spaces arose from his extension of martingale inequalities to settings beyond Hilbert spaces where the square function approach used in his earlier work fails. His work in the eighties and nineties on martingale inequalities with emphasis on identifying best constants has become of great importance in recent years in the investigations of two well known open problems, one concerning optimal $L^p$ bounds of certain singular integrals operators and their ramifications in quasiconformal mappings and the other related to a longstanding conjecture in the calculus of variations dealing with rank-one convex and quasiconvex functions. These problems come from fields which on the surface are far removed from martingales.

The paper of Burkholder and Gundy mentioned above and the 1971 Transactions of the American Mathematical Society paper of Burkholder, Gundy, and Silverstein, are exceptionally important. The first paper includes, in addition to the good–λ inequalities, fundamental integral inequalities comparing the maximal function and the square function of martingales. The second paper strikingly improved, and completed, work of Hardy and Littlewood on the characterization of the Hardy $H^p$ spaces via the integrability of certain maximal functions. While probabilistic techniques had already gained the respect of many analysts studying harmonic functions and potential theory, due to earlier work of Doob, Kakutani, Wiener and others, this landmark paper had a profound influence in harmonic analysis and propelled many analysts to learn probability.

In his five-decade career, Don gave hundreds of invited lectures and lecture series at universities all over the world. He lectured in England, France, Germany, Switzerland, Israel, Denmark, Sweden, Poland, Hungary, Japan, Singapore, Italy, Scotland, Spain, and Canada and at universities across the United States. He was editor of the Annals of Mathematical Statistics (1964–67), president of the Institute of Mathematical Statistics (1975–76), and a member of many councils, advisory committees, and governing boards. He was a dedicated teacher and mentored 19 PhD students. He was elected to the National Academy of Sciences in 1992, and was a Fellow of the American Academy of Arts and Sciences, the Society for Industrial and Applied Mathematics, and the American Association for the Advancement of Science. In December 2012, he was among the first class named as Fellows of the American Mathematical Society.

Don will be remembered not only for his profound contributions to mathematics, but also for the kind and decent ways in which he interacted with everyone he met, and for his encouragement and support to so many young mathematicians who had the great fortune of crossing paths with him.

Don was predeceased by his brothers Robert and Wendell Burkholder and his daughter Kathleen Linda Burkholder; and is survived by his wife of almost 63 years, Jean Annette (Fox) Burkholder; his son J. Peter Burkholder and son-in-law P. Douglas McKinney of Bloomington, Indiana; his son William F. Burkholder, daughter-in-law Joanne (McLean) Burkholder, and granddaughter Sylvie Kathleen Burkholder of Singapore; his sister Helen Dale and brother-in-law Ernie Dale of Auburn, Washington; his brother and sister-in-law John and Donna Burkholder of McPherson, Kansas; his sisters-in-law Anne Burkholder of McPherson, Kansas, and Leona Burkholder of Madison, Wisconsin, and 17 nieces and nephews

Written by Peter Burkholder, Indiana University; William Burkholder, Institute of Molecular and Cell Biology, Singapore; Rodrigo Bañuelos, Purdue University; Burgess Davis, Purdue University; and Renming Song, University of Illinois at Urbana-Champaign

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