The legacy of Jackson, Peterson, Erbe

Peterson Conference

A group of Peterson’s former students at the conference presented him with a framed copy of his mathematical family genealogy.

UNL’s prominence in differential equations began with University of Nebraska-Lincoln Professor Lloyd Jackson (1922-2009) who had a profound effect on not only research in ordinary differential equations, but also on the people of the Department of Mathematics.

Jackson, who was born in Fairbury, Neb., demonstrated an early aptitude for mathematics and by the time he arrived in Lincoln in September 1939 to begin his undergraduate studies at UNL, he had already mastered most of the material in the first two years of mathematics. Jackson received his A.B. degree in mathematics in 1943, and, after serving in the military, he returned to UNL and earned an MA in 1948. He went on to earn a Ph.D. in mathematics at the University of California, Los Angeles in 1950.

Jackson returned to UNL as a faculty member that same year. During his career at UNL, he established himself as one of the nation’s leading scholars in ordinary differential equations. Jackson received a distinguished teaching award from UNL in 1964 and became the department’s first named professor, UNL Regents Professor, in 1967.

It was in 1967 that Jackson had a life-changing effect on now UNL Professor Allan Peterson, who was then a University of Tennessee, Knoxville doctoral student. Jackson was one of four main speakers at an eight-week NSF-sponsored conference at the University of Colorado. Another of the main speakers was John Barrett, Peterson’s adviser at Tennessee. It was at this conference that Peterson met Jackson and decided he would like to teach at UNL and have Jackson as his mentor.

Thanks to Peterson’s adviser, Jackson arranged for him to give a talk at UNL in the spring of 1968. As it turned out, Peterson’s talk wasn’t the only one on campus that day.

“The same day I gave my talk, Bobby Kennedy was giving a talk in the Coliseum,” Peterson said. “Lloyd Jackson really liked Bobby Kennedy, but he came to my talk instead of going to see him.” An overflow crowd of 12,000 heard Kennedy’s anti-war message that day at UNL.

Peterson also met another mathematician at the 1967 conference who would later play a major role in his career: Lynn Erbe.

“Lynn graduated the same year I did [1968], and he was one of Lloyd’s Ph.D. students whom I met at the conference,” Peterson said. “I liked every one of Lloyd Jackson’s students that I met.”

While Erbe went on to teach at the University of Alberta, Peterson graduated from Tennessee and began teaching at UNL. Nearly 30 years later, in 1997, the pair began advising Ph.D. students together at UNL.

Both Erbe and Peterson’s research on differential equations was impacted by the work of Jackson.

In a tribute to Jackson in the Journal of Mathematical and Computer Modelling on his 75th birthday, Erbe and Peterson dedicated the papers to Jackson’s “kindness, dedication to mathematics, and influence on others” and said “it was the case, as pointed out by one of his former students, that whatever was written down was always correct and precisely stated.”

Jackson published his notes from that illustrious conference in 1967. The paper, “Subfunctions and Second Order Differential Inequalities,” was published in Advances in Mathematics, one of the most prestigious mathematics journals, and it continues to have a profound effect on the work done in this and related areas by numerous researchers. In a sense, this paper was the culmination of much of the work that had occupied Jackson up to that time.

Over the next two decades, many results were obtained for solutions of second-, third-, and higher-order nonlinear boundary value problems, by Jackson and many of his students.

One of Jackson’s greatest legacies is the large number of quality Ph.D. students that he produced at UNL. Of his 14 students, at least seven have become a chair of a mathematics department: Elwood Bohn, Ohio University; Jerrold Bebernes, University of Colorado; Lynn Erbe, University of Alberta; Kenneth Heimes, Iowa State University; Ronald Mathsen, North Dakota State University; Klaus Schmitt, University of Utah; and Keith Scrader, University of Missouri.

Between 1997 and 2013, Erbe and Peterson have co-advised 12 Ph.D. students at UNL. Erbe advised another four students while at Alberta, and Peterson has advised another 15 UNL students on his own.

“I really like working with Ph.D. students; that’s one of my favorite things,” Peterson said. “You’re teaching them something, but of course they are teaching you something too. They are taking the problems you give them, and they solve things that you didn’t even know about.”

Peterson, a Charles Bessey Professor of Mathematics, has published 168 research papers and authored and/or co-authored seven textbooks (with another two in progress) that are frequently cited in the field. Peterson also participates on the editorial review boards of 14 mathematical journals and has been involved with the Research Experiences for Undergraduates (REU) site at UNL, which was just renewed for another three years. He also has served on the board of the International Society of Difference Equations since it was formed. Erbe has published about 250 research papers and has written one textbook.

Erbe said that Peterson has shown his dedication to mentoring students by never taking a sabbatical and by making a point of being available to his students during the summers.

Several of his former students came back to honor him at the Peterson Conference, held at UNL in October 2013 (see page 2). Peterson had several fond memories of the conference: “First, I was able to thank my family for their support during these 45 years of teaching at UNL. Then I had my attending Ph.D. students stand, and I told them it was a privilege for me to work with them. I consider my students my extended family. Finally, I was glad to have the opportunity to thank my colleagues for the honor of working with them.”

While Peterson’s research started in boundary value problems with Jackson, it evolved to difference equations after 1988, then to time scales, and now to fractional calculus. Erbe and Peterson have collaborated to build one of the foremost research centers in the country in the emerging field of dynamic equations on time scales.

Conference honors Peterson’s achievements

The Peterson Conference celebrated the impact its namesake, Professor Allan Peterson, has had on mathematics and recognized his 45 years of teaching service at UNL.

Held Oct. 25-27, 2013, the three-day conference was organized around the subject of the calculus of time scales with a goal of creating collaborations between recent Ph.D.s, graduate students and established leaders in the field.

Participants at the Peterson Conference came from around the world, including Sydney, Australia; Ankara, Turkey; and more than 24 different colleges and universities located across the U.S.

Five distinguished guest speakers gave plenary talks: Dr. Ravi Agarwal, Texas A&M University-Kingsville; Dr. Martin Bohner, Missouri University of Science and Technology; Dr. Saber Elaydi, Trinity University; Dr. John Graef, University of Tennessee at Chattanooga; and Dr. Johnny Henderson, Baylor University. In addition to the talks by invited plenary speakers, 18 talks were selected to be given by recent Ph.D.s and current grad students.

The Peterson Conference was a very timely event, set to coincide not only with the 45-year anniversary of Peterson’s teaching at UNL (Peterson began teaching at UNL in 1968), but also the 25-year anniversary since the emergence of time scales analysis. Conference organizers saw the fulfillment of their goals for the conference in the discussion of nonlinear oscillation theory, integral transform theory, fixed point theory, and particularly fractional calculus, which has had recent important developments.

Speakers were encouraged to explore the applications to economics, engineering and biology. Other research areas explored for interaction included differential equations, difference equations and q calculus.

A group of Peterson’s former students who attended the conference presented him with a framed copy of his mathematical family genealogy given to him by the department. This genealogy traces Peterson’s roots back to well-known mathematicians such as Leonhard Euler in 1726 and Jacob and Johann Bernoulli in 1690.

“It is a privilege to have Al Peterson as a colleague,” said Judy Walker, professor and chair of the math department. “He is a wonderful mentor, teacher, and scholar, and he has made a tremendous impact on our department and in his field. I am thrilled that the department was able to host this conference in his honor.”

– Lindsay Augustyn and Stephanie Vendetti