COMMUNICATION The “Wide Influence” of Leonard Eugene Dickson Della Dumbaugh and Amy Shell-Gellasch Communicated by Stephen Kennedy ABSTRACT. Saunders Mac Lane has referred to “the noteworthy student. The lives of wide influence” exerted by Leonard Dickson on the these three students combine with mathematical community through his 67 PhD students. contemporary issues in hiring and This paper considers the careers of three of these diversity in education to suggest students—A. Adrian Albert, Ko-Chuen Yang, and Mina that the time is ripe to expand our Rees—in order to give shape to our understanding of understanding of success beyond this wide influence. Somewhat surprisingly, this in- traditional measures. It seems fluence extends to contemporary issues in academia. unlikely that Leonard Dickson had an intentional diversity agenda for his research program at the Introduction University of Chicago. Yet this This paper raises the question: How do we, as a mathe- contemporary theme of diversity Leonard Dickson matical community, define and measure success? Leonard adds a new dimension to our un- produced 67 PhD Dickson produced sixty-seven PhD students over a for- derstanding of Dickson as a role students over a ty-year career and provides many examples of successful model/mentor. forty-year career. students. We explore the careers of just three of these students: A. Adrian Albert, Ko-Chuen Yang, and Mina Rees. A. Adrian Albert (1905–1972) Albert made important advances in our understanding of When Albert arrived at Chicago in 1922, the theory of algebra and promoted collaboration essential to a flour- algebras was among Dickson’s main research interests. ishing research community. Yang returned to China with As Irving Kaplansky observed, Dickson’s “considerable” ideas and problems from the mathematical frontier and influence on Albert manifested itself in his 1927 mas- helped build the educational structures necessary to begin ter’s thesis—where he determined all two-, three- and a research focus in his homeland. Rees played a funda- four-dimensional associative algebras over a nonmodular mental role in creating the interface between government field—as well as in his 1928 dissertation “Algebras and and academic research that has been crucial to the United their radicals and division algebras” [2, p. 246]. In his States’ preeminence in mathematics since World War II. And yet Albert is typically celebrated as Dickson’s most dissertation Albert proved that every central division algebra of dimension 16 is not necessarily cyclic but is Della Dumbaugh is professor of mathematics and associate dean always a crossed product. Albert’s thesis placed him at of the School of Arts & Sciences at the University of Richmond. Her the center of activity in the field of linear associative e-mail address is [email protected]. algebras. In particular, he, along with the German math- Amy Shell-Gellasch is currently associate professor of mathe- ematicians Richard Brauer, Helmut Hasse, and Emmy matics at Montgomery College. Her e-mail address is amy.sg@ Noether, sought to determine all central division algebras. earthlink.net. In 1931 the German trio established the principal theorem For permission to reprint this article, please contact: that every central division algebra over an algebraic num- [email protected]. ber field of finite degree is cyclic. One year later, Albert DOI: http://dx.doi.org/10.1090/noti1552 and Hasse published a joint work that gave the history 772 NOTICES OF THE AMS VOLUME 64, NUMBER 7 COMMUNICATION of the theorem and described Albert’s near influence in mathematics through his stu- miss. Although Albert would go on to make dents. In his memorial article in Scripta significant contributions to the theory of Mac Lane Mathematica, I. N. Herstein observed, Riemann matrices and to introduce single- “Adrian was extremely good at working handedly the American school of nonasso- referred to with students. This is attested by the thirty ciative algebras, he maintained an interest in Albert as mathematicians who took their PhDs with associative division algebras throughout his him. In their number are many who are well more than forty-year-long career. He wrote Leonard known mathematicians today. His interest more than one hundred forty papers and in his students—while they were students eight books, was invited to deliver a plenary Dickson’s best and forever afterwards—was known and lecture at an International Mathematical appreciated by them.” Daniel Zelinsky, in Congress, and received the AMS Cole Prize student particular, described Albert as an advisor in 1939. who treated his PhD students “almost like The scope of Albert’s talents extended members of his family” [4, p. 665]. beyond the production and publication of mathematical A. A. Albert’s success clearly reflects the influence of results. He, like Dickson and E. H. Moore, made signifi- Dickson. He pursued a vigorous research program that cant contributions to both the University of Chicago and emphasized collaboration, helped launch the careers of the AMS. During Albert’s tenure as a faculty member at numerous students, and served the profession and his the University of Chicago, he participated in a variety of institution in varied and important ways. Any advisor committees, organized conferences, chaired the mathe- would consider a student like Albert an unqualified matics department, and served as dean of the Division of success. In fact, Saunders Mac Lane referred to Albert as Physical Sciences. While chair he “skillfully” found support Leonard Dickson’s best student [3, p. 131]. But are there to maintain a steady flow of visitors and research instruc- other ways of measuring influence and success? Would tors. Albert used his influence to persuade the university a career dedicated solely to bringing the next generation to make an apartment building available, affectionately of researchers to maturity or striving to bridge the gap known as “the compound,” to house visitors. Kaplansky between academia and the larger world be worthy of this claims that the compound became the “birthplace of many description? a fine theorem” [2, pp. 251–2]. Albert realized that the infiltration of new ideas frequently encouraged a fresh Ko-Chuen Yang (1896–1973) perspective on mathematics. In addition to Albert, four other students completed their Albert’s career reflected a strong commitment to the PhDs under Dickson’s guidance in 1928. Yet we rarely hear mathematical community at large. He served the AMS of them. Dickson had recently turned his mathematical in a variety of capacities: as a committee member, as an attention, in part, to Waring’s Problem. Ko-Chuen Yang editor of the Bulletin and the Transactions, and, like Dick- (克純 楊) earned his PhD that year with a dissertation on son and Moore, as president of the Society, in 1965–66. “Various Generalizations of Waring’s Problem.” This thesis The concerns of American mathematicians in the middle not only represented the first of many dissertations that two quarters of the twentieth century were, however, reflected Dickson’s recent investigations in this area, but somewhat different from those in the early years when it also marked the first and only Chinese student to earn Moore and Dickson made their contributions, and Albert’s a PhD under Dickson. service quite naturally addressed the changing needs of Yang’s journey to Dickson actually began as a small boy American mathematicians. In particular, Albert helped at the very beginning of the twentieth century, when China establish government research grants for mathematics was still in the Qing Dynasty. In June 1900 the Boxer upris- comparable to those existing ing spread to Beijing and, in particular, to the area where in other areas of science. He foreign diplomats lived and worked. On June 21, 1900, the apparently found satisfaction Qing declared a war on all foreign nations with diplomatic in this nationally oriented work ties to China. This Boxer Rebellion was suppressed in Au- for “he was always pleased to gust by a coalition army of soldiers from eight countries use his influence in Washing- (Russia, Britain, Germany, France, the United States, Italy, ton to improve the status of Japan, and Austria-Hungary). Consequently, in 1901 the mathematicians in general, and Qing government was forced to sign the Boxer Protocol, he was willing to do the same which demanded that China pay an indemnity valued for individual mathematicians at about US$337 million at the time to the eight foreign whom he considered ‘worthy’” governments over the course of thirty-nine years. About [4, p. 665]. This latter category US$24–$25 million was paid to the United States. A. Adrian Albert served included his students. This amount was not only deemed excessive by many the AMS in a variety of Beyond his service to Chi- American government officials, but it also exceeded the capacities, including cago, the AMS, and the math- actual expense for losses incurred. President Theodore as president in 1965– ematical community at large, Roosevelt proposed to Congress that the United States 1966. Albert exerted considerable return the indemnity funds to China with the stipulation AUGUST 2017 NOTICES OF THE AMS 773 COMMUNICATION that the money be used for Chinese students to study demia Sinica and, in July 1952, in the United States. The proposal was implemented in was appointed its first director. 1908 with about US$12 million returned to China in this Through it all, Hua emerged manner. These students subsequently became known as as an important mathematician, Boxer Scholars. According to The Cambridge History of leader, scholar, and teacher in China, this sum “created a potent mechanism for support China in the mid-to late twen- of Chinese higher education.” tieth century. As the British The Chinese government also used the Boxer Indem- number theorist Heini Halber- nity Funds to create a college preparatory school in 1911 stam put it, “If many Chinese to prepare students for study in the United States.