John W. Neuberger, 1934-2020

Remembrance of Professor Alfonso Castro

I first learned of Professor John W. Neuberger when studying nonlinear semigroups as a graduate student. His formula for the exponential of a nonlinear semigroup got me very interested in following his work.

I first met John in person at a meeting in Oklahoma State University. When one of the organizers of that meeting introduced me to John I could not believe how kind he could be to a recent Ph.D. considering his mathematical stature. Right after his talk I asked him about connections of his work with symmetric positive systems that I was studying at that time. His encouraging reply was a big boost for my beginning career as an independent researcher. At that time, I was working in Mexico which made it convenient for John to visit me. Actually, he was invited to Mexico by universities other than mine which allowed me to interact with John on several occasions. In one of those occasions my boss, Samuel Gitler, invited John, I and the corresponding wives for dinner to a very fancy restaurant in the Zona Rosa. It only took a quick exchange of platitudes for the conversation to turn into a dialog between John and Samuel because John knew quite a bit about a family of polynomials that show up in cohomology theory which was Sam’s specialty. After dinner we walked around the Zona Rosa while their mathematical conversation went on and on.

On another occasion, while John was visiting Mexico invited by a university other than mine and accompanied by his wife Barbara and his daughter Sandra, I invited them to go to Puebla. An archeological feature of the Puebla area is the Cholula pyramid. This pyramid is so large that Spaniards preferred to cover it with dirt to make it look like a mountain rather than destroying it. The pyramid was eventually discovered and a tunnel was built to allow people to see the different stages of construction of the pyramid. The tunnel is about five feet tall and rather narrow. When we arrived at the pyramid, I proceeded to get into the tunnel followed by Barbara, Sandra, Miryam (my wife), others and John. I felt very bad to have suggested that part of the excursion when I exited the tunnel and saw John coming last almost crawling. His height not always worked in his favor but he was a truly good-sport.

My time at UNT was made many times more pleasant and productive because of John’s presence. He really knew the meaning of mentor and colleague. By the time I arrived at UNT John had developed a large group of graduate students interested in the general area of Analysis and Partial Differential Equations.

Immediately John found a way for me to start a seminar that included graduate students McCabe, Dawson, Lee, Kurepa and Unsurangsie. The first three students completed their doctoral dissertations with John and the last two with me. From that point on there was a constant stream of well qualified students that have gone on to pursue successful careers. John’s contagious energy and optimism made every event so much more exciting. Two consecutive grants from the Texas Advanced Research Program, new results, papers published, etc., every day John renewed the energy of the least optimistic in the group.

My last in person reunion with John was in November of 2017 when Miryam and I visited the Neuberger’s in their retirement place. By that time John’s physical health had deteriorated quite a bit but his mind was sharp as a tack. The three days we spent together our conversation was 99% devoted to understanding the connection between his amazing characterization of local semigroups in his paper “A linear condition determining local or global existence for nonlinear problems” (2013) with blow up phenomena. John’s approach to determining when a local flow is global is by appealing to the linear semigroup defined in the phase space in the spirit of Lie group theory. His approach, for example, addresses precisely The Clay Institute Millennium Problem on the Navier-Stokes equation. It is too bad that life did not give us John for a few more years for maybe we would be celebrating the solution to one of the main questions in mathematics.

Alfonso Castro
McAllister Professor, Harvey Mudd College
January 2nd., 2021