Tribute to Oscar E. Lanford III by Arthur Jaffe

Oscar Lanford was not only a remarkable and deep mathematician; he was also a wonderful friend. We met in the fall of 1961 during our graduate years at Princeton; Oscar had arrived one year before me. Our friendship became marked by our office desks in Palmer 420 being within talking distance, and we did a great deal of that. I came to understand that Oscar had read and understood every mathematical text or paper that I was interested in, and Oscar routinely cleared away every cobweb in my puzzlements. Over time Oscar became my closest friend in Princeton.

He was already together with Regina, and eventually Lizabeth was also on the scene. Here are Oscar and Regina with Lizabeth in the 1965-6 academic year, visiting my home in Kingston, New Jersey, near Princeton. Years afterward, Lizabeth became my advisee when she concentrated in physics as a Harvard undergraduate, before becoming a pediatric cardiologist.

We both worked on our theses with Arthur Wightman on the question of the existence of non-linear quantum field theories. A successful resolution of this problem would show the mathematical (logical) compatibility of special relativity with quantum theory and interaction—a major open problem in theoretical science as well as in mathematics. Success would also give the first non-trivial example of a relativistic quantum field with non-trivial scattering, which also satisfied the axioms of Wightman.

For the 1963-4 academic year, Wightman helped IHES start in Bures, by advising Léon Motchane on the organization of a year devoted to mathematical quantum field theory. It was good fortune that Wightman invited both Oscar and me to join him in that journey. IHES had moved to Bois Marie only one year earlier. Louis Michel has just joined the institute, and David Ruelle would arrive the following year.

While today all the professors at IHES have offices in the scientific buildings, in that early year most professors had offices in what is now the administrative building; a few had offices in the separate building that housed the kitchen and lunchroom. Construction had only just begun on what is now the scientific building, with the upper floor being completed half-way through our year there. The music pavilion served multiple functions as library, seminar room, and workspace for visitors. In January 1964, we moved into offices on the upper floor of the first scientific building, while the lower level remained an empty shell.

While some permanent members lived in Bures, most of the visitors for that year were housed in apartments in the Résidence Chevreuse in Orsay. Those long-term visitors that year included several students: Oscar, Sergio Doplicher, Edwardo de Raphael, Klaus Pohlmeyer, and Klaus Hepp on occasion, as well as Jurko Glaser, Res Jost, Harry Lehmann, Jean Lascoux, François Lurçat, Henri Epstein, and Michael Artin. Annie Rolland put everything together, accomplishing what we thought comprised the work of four persons.

This was a wonderful era for mathematical physics. New connections arose between physics and mathematics, and Oscar was at the center of much of this work. Before Regina arrived, we often had dinner together at the Croix de Bures on the Route de Chartres, which served a traditional French menu that we enjoyed while discussing mathematical physics. I always took the crème de marron as dessert.

In our thesis work, Oscar studied on the case of quantum fields with Yukawa interaction between bosons and fermions, while I concentrated on non-linear boson self-interactions. Eventually Oscar and I published one joint paper in 1967 giving our early results with Wightman on cut-off quantum field theories [2]. The full solution required the removal of these cutoffs, which I did for two-dimensional space-time in a long series of papers with James Glimm, and later also with Thomas Spencer. Members of our research groups later obtained the Yukawa model and boson self-interaction in three dimensions. Whether one can find any complete example in four-dimensional space-time remains a fundamental open problem.

When Oscar and I left Princeton in 1966, we wanted to continue our scientific discussions. We both had offers of faculty positions at Berkeley, and we thought that would be the way to go on. Somehow, I got side-tracked and ended up at Stanford. It turned out that the distance between Palo Alto and Berkeley was practically further than what it appeared to be on a map. So our interactions were less frequent, and the next year I moved back to the east coast.

Some sixteen years after our student encounters, Oscar left Berkeley to return to IHES as a professor, where he spent seven happy years. He had major collaborations with David Ruelle and others during that time, leading to the Dobrushin-Lanford-Ruelle equations [3]. Oscar also became interested in understanding the mathematical structure of the Feigenbaum fixed point, also studied by Collet and Eckmann [1,4,5]. In Oscar’s work on using a computer computation as part of a mathematical proof, he insisted on understanding exactly what the computer did in performing a calculation. This meant that Oscar would only program his computations in machine language, in order to be able to rely on error bounds with complete certainty. I believe this may have been the first “computer-assisted proof” about which there could be no doubt of its veracity.

Oscar also moved from statistical physics to dynamical systems, where he made fundamental progress. Eventually he moved to ETH Zurich. My close friend Konrad Osterwalder, who had become Rektor there, confided to me that one of his proudest moments was to recruit Oscar. I often visited ETH where it seemed inevitable that I would run into computer problems. The person who always spent his generous time to provide the way to solve them was Oscar.

I had lunch together with Oscar and Regina on August 21, 2013, at their home in Thalwil, along the right side of the lake of Zurich; at the time Oscar was quite ill. At lunch I took this snapshot (the cover picture) and gave them a copy of the photo in Kingston from our Princeton student days almost 50 years earlier. That lunch was on a beautiful afternoon, and I remember, when walking to the train, reliving in my mind many of our previous interactions. While I miss him sorely, his wonderful contributions to mathematics and physics will live forever.

References

1. Pierre Collet, Jean-Pierre Eckmann, and Oscar E. Lanford III; Universal properties of maps on an interval. Comm. Math. Phys. 76 (1980), 211–254.

2. Arthur M. Jaffe, Oscar E. Lanford III, and Arthur S. Wightman; A general class of cut-off model field theories. Comm. Math. Phys. 15 (1969) 47–68.

3. Oscar Lanford and David Ruelle; Integral representations of invariant states on B* algebras. J. Mathematical Phys. 8 (1967) 1460-1463.

4. Oscar E. Lanford III; Remarks on the accumulation of period-doubling bifurcations. Mathematical problems in theoretical physics (Proc. International Conf. Math. Phys., Lausanne, 1979), pp. 340-342, Lecture Notes in Phys., 116, Springer, Berlin-New York, 1980.

5. Oscar E. Lanford III; A computer-assisted proof of the Feigenbaum conjectures, Bull. A.M.S. (New Series), 6 (1982) 427–434.

Arthur Jaffe
Cambridge, Massachusetts
September 25, 2024