John Ralph Alexander was born Jan 2, 1935, and died October 27, 2019.
Ralph's main research was in integral geometry and convexity theory.
He was most proud of his work on irregularities of distribution and on Hilbert’s fourth problem.
His paper ``Principles of a new method in the study of irregularities of distribution'' appeared in the prestigious journal Inventiones Mathematicae. William Chen in his review says: ``One of the attractions of the subject is its difficulty, and important breakthroughs are rare. The present work will be considered as one such instance.'' He concludes with the words, ``Workers in [this] subject will no doubt find a great deal of inspiration from this work.''
Furthermore, in 1994, the late Czech mathematician Jirı Matousek showed that Ralph’s bound is sharp. Together, their contributions completely resolve the L∞-norm problem, an achievement in discrepancy theory that has not been matched before or since, apart from Schmidt's 1972 result in 2 dimensions.
This paper of Ralph is a significant item in the Springer Lecture Notes publication "Sequences, discrepancies and applications" of Drmota and Tichy. Also, the reviews of subsequent papers by several authors include words to the effect that "The proof essentially relies on the methods of J. R. Alexander."
At the International Congress of Mathematicians in Paris in 1900, Hilbert proposed 23 problems which he believed would be very important for 20th century mathematical research. Ralph solved Hilbert’s fourth problem, about 2-dimensional geometries in which lines give the shortest distance between two points. Two other authors found the same solution but slightly earlier, all three using different methods. Ralph’s method was powerful, making a completely new connection with the theory of convex bodies known as zonoids. As a result, he found the first solution of Hilbert’s problem in all dimensions.
Ralph was a brilliant, if self-effacing, mathematician. A former department head said that if he were to design his ideal mathematician, the person would have Ralph's brain and the mouth of a certain self-promoting member of the department.
It is striking that memorial remarks by friends from every period of Ralph’s life spoke of his kindness and interest in others. As well, friends found it a real pleasure to talk with Ralph for the wide range of his interests and knowledge, whether on the many deficiencies of Jaguar cars, the writings of the Roman Pliny the Elder, or the threats to our environment.
John Ralph Alexander was in Anniston, Alabama, and moved in 1949 with his parents, J. Ralph Sr. and Gladys and his sister Martha Jane, to Kirkwood, Missouri, where he attended high school. He earned his undergraduate degree in mechanical engineering and Ph.D. in mathematics both from Purdue University. Ralph came to the University of Illinois as an assistant professor in 1963 and soon met Stephanie Taylor, and married her in 1965. The Alexanders lived most of their married lives in their lovely rural home in Mahomet. Ralph retired in 1993, and then pursued photography, taking beautiful nature pictures, particularly of the plants and wildlife near the banks of the Sangamon River. At the end of his life, Ralph received much aid from his friends and the staff at the Meadowbrook Health Center in Urbana in dealing with his poor health.
Ralph's surviving relatives are his wife Stephanie and his sister Martha Jane Alexander, as well as his in-laws Gillian and Carman Bradley, his nieces Mary Bradley, Lucy Bradley and Amy (Colin) Madsen, his great niece Samantha Madsen, and his great-nephew Cristopher Madsen.