# Albert Nijenhuis

### Quick Info

Born
21 November 1926
Eindhoven, The Netherlands
Died
13 February 2015
Seattle, Washington, USA

### Biography

Albert Nijenhuis' high school studies were at the Gymnasium in Arnhem where it appears he was a fairly average student. When World War II broke out in September 1939, Nijenhuis was only twelve years old. For a few months life in The Netherlands went on much as normal but on 10 May 1940 Germany invaded and the country was quickly occupied. Nijenhuis was able to continue his studies at the Gymnasium where, at the age of fourteen, he became fascinated by mathematics. In June 1944 Allied troops landed on the French coast and began to fight their way south and east. By the end of August, Paris was liberated and by early September Brussels was also liberated. The Allies planned a major airborne drop of troops to take bridges at Eindhoven and Arnhem. This was known as Operation Market Garden and was intended to open a route for the Allies to drive towards Berlin. Due to a number of different causes, the operation was defeated by the end of September 1944 after a violent battle lasting for over a week. The Germans evacuated Arnhem, the Gymnasium there was closed down and Nijenhuis' schooling came to an end. The Nijenhuis family were forced to leave the town.

When the Arnhem Gymnasium closed, Nijenhuis was seventeen years old and had only one year of schooling left. His grandparents lived in a small Dutch village and, since it was comparatively safe, he went there. He lived for a year with his grandparents and during this time he continued his studies, working on his own. He was able to take the State examinations in 1945 and, having obtained good passes in these, he entered the University of Amsterdam in the autumn of 1945. L E J Brouwer was a professor at Amsterdam at this time. Nijenhuis was awarded his Candidat degree, equivalent to a Bachelor's degree, in 1947. He progressed rapidly, taking only half of the normal length of study to pass some of the examinations, and was awarded his Doctorandus degree, equivalent to a M.Sc. degree, in 1950. Undertaking research for his Ph.D. at the Mathematisch Centrum (now the Centrum Wiskunde & Informatica) advised by Jan Arnoldus Schouten, he was an associate there during 1951-52.

Nijenhuis summarises Schouten's life in the following words [1]:-
A descendant of a prominent family of shipbuilders, Schouten grew up in comfortable surroundings. He became not only one of the founders of the "Ricci calculus" but also an efficient organiser (he was a founder of the Mathematical Centre at Amsterdam in 1946) and an astute investor. A meticulous lecturer and painfully accurate author, he instilled the same standards in his pupils.
In fact Schouten, almost certainly because of the pressures of World War II, had become ill in 1943, and had at that time resigned his position in Delft. From 1948 until 1953 he was professor of mathematics at the University of Amsterdam but he did not teach. He was director of the Mathematical Research Centre at Amsterdam for five years. While still a research student advised by Schouten, Nijenhuis published papers such as $X_{n-1}$-forming sets of eigenvectors (1951) and An application of anholonomic coordinates (1951).

In 1952 Nijenhuis was awarded a Ph.D. for his 238-page thesis Theory of the geometric object which was published by the University of Amsterdam in that year. Václav Hlavaty (1894-1969) writes in a review [3]:-
The work is divided into three chapters. The first chapter is based on two fundamental concepts: "transformation element" and "allowable coordinate systems". ... The second chapter deals mainly with the deformation theory of fields of geometric objects (which may even be functionals of some other geometric object) and is based on the concept of "group-germ". ... While the second chapter deals with components of objects, the third chapter deals with their transition induced by the transformation element which characterizes the change of coordinates. Among many topics considered by the author we mention only the formulation of criteria that an object be a differential geometric object.
In this work Nijenhuis solved an open question in the theory of deformations, the tool he introduced to solve it now being known as the 'Nijenhuis Tensor'. After the award of his Ph.D., Nijenhuis travelled to the United States where he spent a year at Princeton University on a Fulbright Fellowship. In August of the following year he became a member of the Institute for Advanced Studies at Princeton. He remained there until September 1955. While at Princeton he published the papers: On the holonomy groups of linear connections. IA, IB. General properties of affine Marianne Nijconnections (1953), A theorem on sequences of local affine collineations and isometries (1954), On the holonomy groups of linear connections. II. Properties of general linear connections (1954) and Jacobi-type identities for bilinear differential concomitants of certain tensor fields. I, II (1955).

After three years at Princeton, Nijenhuis was appointed as an Instructor in Mathematics at the University of Chicago in 1955. Also in the year 1955 he married Marianne; they had four daughters Erika, Karin, Sabien and Alaine. After spending the academic year 1955-56 in Chicago, Nijenhuis was appointed as an assistant professor at the University of Washington and later was promoted to professor there. During his time at the University of Washington, he was an invited speaker at the International Congress of Mathematicians at the University of Edinburgh, Scotland in August 1958. At the Congress he gave the lecture Geometric aspects of formal differential operations on tensor fields which was published in the Conference Proceedings. He became a U.S. citizen in 1959 then was awarded a J S Guggenheim Fellowship in 1961 which allowed him to spend the months September 1961 to April 1962 at the Institute for Advanced Study. Ramesh Gangolli writes [2]:-
Albert's association with our department [at the University of Washington] goes back to 1956, when he joined the department as an Assistant Professor. This was when Carl Allendoerfer was Chair, and was pushing to build up the department's strength in differential geometry. He was a Professor when I arrived as a naive Assistant Professor here in 1962. I was becoming interested in analysis on symmetric spaces at that time. When I started in this direction, I began to feel keenly that I knew very little differential geometry and algebra, both of which were essential to where I wanted to go. Albert was on a Guggenheim Fellowship then, and was spending the second year of the fellowship in Seattle. He was running a seminar on Differential Geometry that I attended and he attended a seminar on Lie algebras that I started in order to study Harish-Chandra's work, and we became friends.
He was a Fulbright Professor spending some time at the University of Amsterdam in 1963-64 before taking up an appointment as professor of mathematics at the University of Pennsylvania. Ramesh Gangolli writes [2]:-
... in 1963, Albert received an offer from the University of Pennsylvania which he accepted. Richard Kadison had moved there from Chicago and was charged with refreshing the lustre of the Mathematics department's ivy. He conducted several raids (some friends began to call him the Viking), and we suffered by losing both Albert Nijenhuis and Michael Fell, who were recruited by him. Fortunately, although Albert accepted the Penn appointment as of Fall 1963, he stayed on in Seattle, on leave from Penn, until 1964, which enabled me to continue our seminar for another year.
He held this position the University of Pennsylvania until he retired in 1987 but he spent two periods away, first as a visiting professor at the University of Geneva in 1967-1968, and later at Dartmouth College in 1977-1978.

At the University of Pennsylvania, his excellent teaching was recognised with the "Good teaching award - Fall 1974" and the "Good teaching award - Spring 1975". His research interest in deformations changed over the years towards combinatorics and, in 1975, he published his most famous work namely the book Combinatorial algorithms written in collaboration with Herbert S Wilf. They wrote in the Preface:-
In the course of our combinatorial work over the past several years, we have been fond of going to the computer from time to time in order to see some examples of the things we were studying. We have built up a fairly extensive library of programs, and we feel that other might be interested in learning about the methods and/or use of the programs. This book is the result.
Hale Freeman Trotter, reviewing the 1978 second edition writes [9]:-
The form of this book is highly appropriate to its subject. Problems and their solutions are first discussed in mathematical terms, followed by formal algorithms with commentary for the solutions. ... The mathematical sections can be read by themselves with pleasure. The well-organized combination of the two, however, is what gives the book its special value. The authors have provided convincing support for their belief that studying the interplay between computer programs, algorithms, and mathematics can illuminate all three.
For fuller extracts from Prefaces and reviews of this classic text, see THIS LINK.

Although working at the University of Pennsylvania, Nijenhuis continued his love of Seattle and [2]:-
Nijenhuis was honoured with being elected a corresponding member of the Royal Netherlands Academy of Arts and Sciences in 1966 and as a Fellow of the American Mathematical Society in 2012.

Around 2013 his health began to deteriorate and he died following several months of failing health.

### References (show)

1. A Nijenhuis, J A Schouten, in Dictionary of Scientific Biography (New York 1970-1990). See THIS LINK.
2. R Gangolli, Albert Nijenhuis November 21, 1926 - February 13, 2015, University of Washington (2015). https://www.math.washington.edu/Newsletter/2015_newsletter.pdf
3. V Hlavaty, Review: Theory of the geometric object, by Albert Nijenhuis, Mathematical Reviews MR0050364 (14,320a).
4. A Nijenhuis, J A Schouten: A Master at Tensors, Nieuw archief voor wiskunde 20 (1972), 1-19.
5. Albert Nijenhuis: 1926-2015, In Memoriam, University of Pennsylvania (2015). https://www.math.upenn.edu/about/department-history/in-memoriam
6. Albert Nijenhuis, Institute for Advanced Study, Princeton, New Jersey. https://www.ias.edu/scholars/albert-nijenhuis
7. Albert Nijenhuis, 1926-2015, The Seattle Times (22 February 2015).
8. Albert Nijenhuis, Mathematics, University of Pennsylvania (2015). http://www.upenn.edu/emeritus/memoriam/Nijenhuis.html
9. H F Trotter, Review: Combinatorial algorithms (2nd edition), by Albert Nijenhuis and Herbert S Wilf, American Scientist 67 (5) (1979), 620.
10. E G Whitehead, Jr., Review: Combinatorial algorithms, by Albert Nijenhuis and Herbert S Wilf, Bull. Amer. Math. Soc. 82 (6) (1976), 870-871.
11. E G Whitehead, Jr., Review: Combinatorial algorithms (2nd edition), by Albert Nijenhuis and Herbert S Wilf, SIAM Review 22 (2) (1980), 238-239.