Skip to Main Content

Robert Israel “Bob” Jewett (1937–2022)

Walter R. Bloom
Richard J. Gardner
Al Hales
Joel Spencer
Terence Tao
Benjamin Weiss

Communicated by Notices Associate Editor Emilie Purvine

Robert Israel “Bob” Jewett was born on December 14, 1937 in Providence, Rhode Island. His father, Abraham, had emigrated from Poland/Ukraine to Canada in 1921 and then to the USA in 1923, while his mother, Mame (Mary) née Katz, was born in Providence to parents from Russia. In 1946, Bob’s family, including his older sister Rosalie, moved to Venice, California, a relocation partially motivated by Bob’s problems with hay fever. Bob’s much younger brother Phil was born in 1948.

Bob attended the local public schools and was very interested, while at Venice High School, not only in physics and mathematics, but also animals and insects. In 1955, he started college at Caltech. There was no zoology major, so he focused on physics and mathematics, eventually the latter. In 1958, he received Honorable Mention for his individual performance in the nationwide Putnam competition in mathematics, helping the Caltech team to place third in the nation. Bob also stood out as a volleyball and track and field star.

Bob graduated from Caltech in 1959 and went to the University of Oregon for graduate work in mathematics. Before this move, however, he worked for the summer at Caltech’s Jet Propulsion Laboratory (JPL), in the coding theory section headed by Solomon Golomb. He was also employed there in the summers of 1960 and 1961. During that time he wrote several research papers, one containing the combinatorial Hales–Jewett theorem HJ63.

Figure 1.

Bob (center) with the “Diggers” volleyball team, 1968.

Graphic without alt text

Bob received his PhD in 1963, with a thesis on analysis on locally compact abelian groups written under the direction of Karl Stromberg. He spent the next year as a postdoc at the Institute of Advanced Study in Princeton. This was followed by two years teaching at Uppsala University in Sweden. In 1966–69, Bob was an assistant professor at the University of Washington, during which period he wrote his important paper Jew69 in ergodic theory leading to the Jewett–Krieger theorem. For the academic year 1969–70, Bob taught at the IMPA in Rio de Janiero, returning to the USA in 1970 to take a position at Western Washington University (WWU) in Bellingham.

Shortly after arriving at WWU, in 1972, Bob shared SIAM’s first Pólya Prize in Combinatorics, along with cohonorees Ron Graham, Al Hales, Klaus Leeb, and Bruce Rothschild, for his part in the Hales–Jewett theorem. In 1975, Bob’s fundamental paper Jew75 on convos (a.k.a. topological hypergroups) appeared.

Except for visits to UCLA (Spring 1974), the University of Auckland (1976), the University of Oregon (1982–83), and the University of Wisconsin (1984–85), Bob spent the remainder of his career at WWU, retiring in 2010.

At the suggestion of Ron Graham, in 2016 Árpád Bényi, Steve Butler, Amites Sarkar, and Jozsef Solymosi organized a celebratory meeting at WWU, titled “50 Years of the Hales–Jewett Theorem.” This very successful event involved three generations of researchers and attracted distinguished speakers from far and wide: Vitaly Bergelson, Fan Chung, David Conlon, Ron Graham, Neil Hindman, Imre Leader, Dhruv Mubayi, Jaroslav Nesetril, and Gabor Tardos.

In 2014, Bob was hit by a car at a crosswalk. He survived the resulting operation and long rehabilitation with his trademark stoicism and wonderful sense of humor intact. Despite this setback, he was generally healthy until the last few months of his life. He died peacefully on July 30, 2022.

The Hales–Jewett Theorem

By Joel Spencer

The Hales–Jewett theorem HJ63, which appeared in 1963, occupies a central place in the development of Ramsey theory. Indeed, it was this result that turned a collection of Ramsey-type theorems into Ramsey theory; see GRS90, p. 35. The mantra: complete disorder is impossible.

To take a basic case, Ramsey showed in 1930 that for all , there exists a sufficiently large such that every red/blue coloring of the complete graph contains either a red or a blue . The rediscovery of this result by Erdős and Szekeres and the countless extensions and conjectures of Erdős truly advanced the subject. Several other results are in the same spirit. For example, much earlier, in 1916, Issai Schur showed that if is finitely colored, there is a monochromatic solution to the equation . As with Ramsey, Schur’s interests were elsewhere and his result was greatly extended by many, especially Richard Rado and Walter Deuber. In 1950, R.P. Dilworth showed that appropriately large partially ordered sets must contain either large chains or large antichains. The most important result in this telling was van der Waerden’s theorem from 1927, stating that for all , there exists an such that if is -colored, there must be a monochromatic arithmetic progression of terms. This was long considered a beautiful number-theoretic result, but (as Al Hales recalls) Bob Jewett felt it could be placed in a more general setting. Their final result was purely combinatorial, removing the algebraic structure. Indeed, it can be formulated in terms of the classic children’s game Tic-Tac-Toe!

Define , the -cube over elements, by

(For , this is the Hamming cube.) By a combinatorial line in we mean a suitably ordered set of points in such that each coordinate is either constant or runs through in that precise order. For example, in ,

form a combinatorial line. Combinatorial lines lie on geometric lines in -space but the converse is not true. For example, in classic Tic-Tac-Toe played on , lie on a geometric line (and represent a winning position in the game) but do not form a combinatorial line.

The Hales–Jewett theorem states that for all , there is an such that if is -colored, there necessarily exists a monochromatic combinatorial line. For the game enthusiasts: -player Tic-Tac-Toe with combinatorial lines of length , played in sufficiently high dimensions, cannot end in a draw!

The Hales–Jewett theorem immediately implies van der Waerden’s theorem; just associate with its base value . But the Hales–Jewett theorem is combinatorial; instead of , we may take coordinate values in any space and still get a monochromatic combinatorial line.

Let denote the minimal such that the Hales–Jewett theorem holds. The asymptotic upper bounds on as are enormous, even for . The original proof of Hales and Jewett gave an upper bound on somewhat like the Ackermann function. In 1988, Shelah She88 found an upper bound which is primitively recursive. Let denote an exponential tower of twos of height . Let denote the -times iterated Tower function, beginning at . Shelah’s bound was roughly . The lower bound is exponential, so there remains a very large gap.

The Hales–Jewett theorem lies at the heart of a group of related results. The general theme is that for any fixed structure and , a sufficiently large -colored structure necessarily contains a monochromatic substructure . For example, Gallai’s theorem from 1943 states that if is any finite subset of and is finitely colored, then there exists a monochromatic homothetic to . Indeed, we may find a with and nonzero . Further, by compactness, for any we need only -color a suitable finite .

Gallai’s theorem is a fairly easy consequence of the Hales–Jewett theorem. Another is the following extended Hales–Jewett theorem, proved by Ron Graham and Bruce Rothschild in 1969: For all , there is an such that if is -colored, there exists a monochromatic combinatorial -space (a natural generalization of a combinatorial line).

Figure 2.

Caltech Big T yearbook photo, 1959.

Graphic without alt text

Gian-Carlo Rota felt that Ramsey’s theorem should hold in general lattices and, specifically, conjectured the following vector space Ramsey theorem: For any finite field and , if is sufficiently large and the -dimensional subspaces of (considered as a vector space) are -colored, then there exists a -dimensional subspace , all of whose -dimensional subspaces are the same color. This was proved by Graham & Rothschild and independently by Klaus Leeb, a joint paper appearing in 1972.

The Hales–Jewett theorem and its variants have spawned extensions and applications in many directions. For example, the polynomial Hales–Jewett theorem, proved by Bergelson and Leibman BL99 in 1999, leads to topological dynamics. It is slightly too complicated to state here, but the reader may consult Walters Wal00 for short combinatorial proofs of it and a consequence, the polynomial van der Waerden theorem. The latter states that if are polynomials with integer coefficients and no constant term, then whenever is finitely colored, there exist such that and , , all have the same color.

Joel Spencer is a professor emeritus of mathematics and computer science at NYU Courant. His email address is spencer@cims.nyu.edu.

Graphic without alt text

The Density Hales–Jewett Theorem

By Terence Tao

Van der Waerden’s theorem can be equivalently phrased as a statement about infinite colorings: whenever the natural numbers are finitely colored, one of the color classes must contain arbitrarily long arithmetic progressions. However, the known proofs of this theorem did not shed much light on which color class had this property, and why. In 1936, Erdős and Turán conjectured what we would now call a density version of the van der Waerden theorem: the assertion that in fact any set of natural numbers of positive (upper) density contains arbitrarily long arithmetic progressions. This of course would imply van der Waerden’s theorem, since by the pigeonhole principle whenever one colors the natural numbers into finitely many classes, at least one of them must have positive density; but the claim is far stronger, and formalizes the intuition that it is simply the size of a set of natural numbers that forces the existence of patterns such as arithmetic progressions contained inside it.

The conjecture of Erdős and Turán was famously demonstrated in 1975 by Szemerédi Sze75 in a remarkable tour de force of combinatorial reasoning, and the conjecture is now known as Szemerédi’s theorem. This theorem and its generalizations have had many applications in combinatorics and number theory; for instance, in 2004 Green and I used it to show that the primes contain arbitrarily long arithmetic progressions (despite having density zero). In 1977, Furstenberg Fur77 gave a new proof of Szemerédi’s theorem that was both conceptual and highly influential, using the tools of ergodic theory, developing what is now known as the Furstenberg correspondence principle to convert the problem to one of understanding the recurrence properties of dynamical systems. Roughly speaking, the point was that every dense set of integers could be interpreted as the set of return times for some dynamical system—the set of times in which a particle traversing some state space returns to a given set of states of positive measure. Furthermore, the dynamics of this system could be made to be measure-preserving, allowing the techniques of ergodic theory to come into play.

Spurred by the success of this ergodic theoretic approach, Furstenberg and his coauthors and students began locating and proving density versions of many of the other landmark results of Ramsey theory. Some density versions were false, and others could be established by variants of Furstenberg’s methods, but establishing a density version of the Hales–Jewett theorem (which would be a sweeping generalization of Szemerédi’s theorem) turned out to be particularly challenging. It was only in 1991 that Furstenberg and Katznelson FK91 finally managed to establish this result by pushing the methods of ergodic theory to their limits, applying them to systems of families of sets which have only the barest hint of dynamical structure. The proof is fearsomely complicated; for instance, just one step of the argument relies on a Ramsey theorem of Carlson and Simpson which is in turn a significant strengthening of the Hales–Jewett theorem in which one requires a color class to contain an infinite-dimensional combinatorial subspace, rather than simply a combinatorial line. It was also purely qualitative: while it does show that a dense subset of a sufficiently high-dimensional cube will necessarily contain a combinatorial line, it does not specify at all the precise relation between the density of the set and the dimension required.

In 2009, inspired in part by his previous work on Szemerédi’s theorem, Gowers proposed to locate a purely combinatorial proof of the density Hales–Jewett theorem by a creative new paradigm—an online crowdsourced effort, where dozens of mathematicians, mostly communicating through blogs and wikis, would contribute and debate possible attack strategies. After a very intensive seven-week effort involving thousands of comments by many mathematicians, such a combinatorial proof was finally obtained in 2010, with the results eventually being published in Pol12 under the pseudonym “D.H.J. Polymath;” the initials here stand for “Density Hales Jewett.” The Polymath project continued to solve a number of other problems in the same format; it has kept the same pseudonym ever since, even though the other problems were no longer directly related to the density Hales–Jewett theorem.

The Polymath proof of the density Hales–Jewett theorem was simplified in 2014 in a twelve-page paper of Dodos, Kanellopoulos, and Tyros DKT14, which as one corollary gives what is arguably the shortest and most elementary proof of Szemerédi’s theorem, and also gives the currently best-known quantitative bounds for the density Hales–Jewett theorem. Research in this area is still ongoing; for instance, there is a conjectural “density polynomial Hales–Jewett theorem” (a density version of the polynomial Hales–Jewett theorem of Bergelson and Leibman) that should also be true, and would imply a staggering number of other density Ramsey theorems already known in the literature, but remains open at this time of writing. Gowers Gow22 provides further information about these developments.

Terence Tao is a professor of mathematics at UCLA. His email address is tao@math.ucla.edu.

Graphic without alt text

On the Jewett–Krieger Theorem

By Benjamin Weiss

The Jewett–Krieger theorem is one of the fundamental theorems lying on the interface between topological dynamics and ergodic theory. In topological dynamics one studies the properties of the iterations of a homeomorphism of a compact space . In classical ergodic theory the setting is that of a probability space and a measurable invertible mapping of that preserves the measure . The basic building blocks of measure preserving transformations are those that are indecomposable, in the sense that one cannot find a set with such that . These are called ergodic systems.

It is a basic fact that a homeomorphism of a compact space always has at least one invariant measure. Just as the Borel probability measures on form a compact convex set in the weak* topology, so too the -invariant probability measures form a compact convex set and it turns out that its extreme points are exactly the invariant ergodic measures for the transformation . In particular, if there is only one invariant measure, say , for the system is ergodic. Such topological dynamical systems are called uniquely ergodic. This property has significant topological consequences. Indeed, if denotes the closed support of this unique measure then is -invariant and is a minimal system, which means that all orbits are dense.

A rotation of the unit circle by an irrational multiple of is an example of a uniquely ergodic system. Prior to the work of Bob Jewett most of the examples of uniquely ergodic systems were extensions of various kinds of these simple systems. When Kolmogorov defined entropy as a numerical invariant for measure preserving systems it turned out that all the known examples of uniquely ergodic systems had zero entropy. It was only in 1967 that Frank Hahn and Yitzhak Katznelson gave an involved construction of a uniquely ergodic system with positive entropy. It came as a complete surprise to all of the experts in the field when Jewett published his result. For its statement, the following definition is needed. The product of a system with itself is the mapping on the product probability space where . A system is called weakly mixing if its product is ergodic. Examples of weakly mixing systems are automorphisms of the torus with no eigenvalue that is a root of unity.

Jewett’s theorem Jew69 can be stated as follows.

For every weakly mixing system there is a uniquely ergodic topological system , where is the Cantor set, with a unique invariant measure such that the systems and are isomorphic.

In one stroke the family of uniquely ergodic systems was enlarged to include models of all weakly mixing systems. The natural question arose as to whether one could drop the additional assumption of weak mixing. This was accomplished in the following year by Wolfgang Krieger, who proved the same theorem under only the (necessary) condition of ergodicity. Krieger’s result appeared only in 1973 Kri72 and in the meantime a different proof was given by Georges Hansel and Jean-Pierre Raoult. It then became clear that the property of unique ergodicity was as prevalent as possible. The original proof of Jewett was extended in 1979 to cover the ergodic case by Bellow and Furstenberg BF79. Jewett needed a certain lemma which was easy to prove under the hypothesis of weak mixing. Bellow and Furstenberg showed that this property is true for all ergodic systems by a clever use of Neil Hindman’s famous combinatorial theorem.

The Jewett–Krieger theorem was extended to the setting of measure preserving actions of the real line; this was the original setting for ergodic theory, which started from questions in statistical mechanics. This was done, first by Konrad Jacobs assuming weak mixing, and then, in 1974, by Denker and Eberlein DE74 in the general ergodic case.

More generally one can consider actions of any locally compact group . To begin with one asks when every action of by homeomorphisms of a compact space fixes some probability measure. The answer is if and only if the group is amenable. This class contains all solvable groups but does not contain the free group on two or more generators. Indeed much of the classical ergodic theory was extended to this class of groups and in particular, the Jewett–Krieger theorem was established for all discrete elementary amenable groups in Wei85. In joint work with Alain Rosenthal this was extended to all discrete amenable groups. He wrote a series of papers containing this and many other refinements in the following years.

In that same paper I outlined a different kind of extension. If and are two systems and there is a measurable mapping such that and then the second is called a factor of the first. I showed that every uniquely ergodic model of the factor can be continuously extended to a uniquely ergodic model of the larger system. This “relative” version has been applied by Huang, Shao and Ye HSY19 to prove new results in the study of multiple ergodic averages.

Jewett’s result motivated many other results in the spirit of finding topological models for measurable systems with special properties. His work spawned an entire branch in the interplay between measure and topology which is still growing.

Benjamin Weiss is a professor emeritus of mathematics at the Hebrew University of Jerusalem. His email address is weiss@math.huji.ac.il.

Graphic without alt text

Jewett’s Hypergroups

By Walter R. Bloom

The concept of a group-like structure where the product of two elements results in a set rather than another element has been around since the first half of the twentieth century, but the tie-up with the topology of the underlying space and Borel measures first appeared in the early s when Charles Dunkl Dun73, Robert Jewett Jew75 and René Spector Spe75 independently created locally compact hypergroups with the view to developing standard harmonic analysis on these spaces. There were also precursors by Jean Delsarte in 1938, Boris Levitan in 1945, and Salomon Bochner in 1956 with the study of generalised translation operators.

There are technical differences between the various definitions, but in the setting of analysis on topological group-like structures, the basis for much subsequent research was Jewett’s lengthy paper Jew75, a remarkable work that developed the main harmonic analysis of what he termed convos. It is Jewett’s rather extensive axiom scheme that has subsequently proved most influential. In a nutshell, we write for a locally compact Hausdorff space acting as base space, for the Dirac (point) probability measure at and for the convolution algebra of bounded complex-valued regular Borel measures on , where is a probability measure with compact support . In the locally compact group case we have , but we are in unfamiliar territory when is a subset of containing more than one point.

Figure 3.

Passport photo, 1982.

Graphic without alt text

We could list all the Jewett axioms and discuss the nuances of their relationships, but this is better left for the readers of his paper. One principal hypergroup axiom is that we have a continuous map , with a suitable topology on . There is also an involution on respecting this convolution operation in the sense that if and only if . Here, involution takes the place of group inverse and is the neutral element of , taking the place of the group identity. We then pass from the sparsely structured base space to the more richly structured measure algebra . For example, the group operation of (left) translation is replaced by the hypergroup operation of generalised (left) translation on suitable functions on :

At this stage we illustrate the theory with the important case of double coset hypergroups , where is a locally compact group with left Haar measure and is a compact subgroup with normalised Haar measure , and indeed this formed the basis of Jewett’s theory Jew75, Section 8.2. The double coset space doesn’t inherit a multiplication from if isn’t normal, but the space of measures on does inherit a convolution from the measure algebra .

Jew75, Theorem 8.2B: The space with the quotient topology and convolution

is a hypergroup with neutral element . (This equality of Radon measures and similar equalities below are best understood by evaluating both sides at continuous functions ; the integrand above then simplifies to .) If then . A left Haar measure is given by

The significance of (left) Haar measure is its invariance under (left) translation.

Jew75, Theorems 3.3F, 3.3G: Let . For every -finite (with respect to ) nonnegative Borel-measurable function on and , is also -finite,

where .

The representation theory of hypergroups has been well developed in Jew75, Chapter 11 and is vastly simplified in the commutative case. Let be a commutative hypergroup with neutral element , involution and Haar measure . Bounded measurable functions are called characters when , and for all . The essential difference between characters on groups and characters on hypergroups is that on groups it is easy to “shift around” the character through . For hypergroups the argument isn’t so easy; the problem is that is not the evaluation of at a point, but rather is in general an integral.

A locally bounded measurable function is said to be positive definite if

for all choices of and . All continuous characters are automatically positive definite and the dual of is just the set of continuous characters with the compact-open topology in which case must be locally compact.

For we have the Fourier transform

Lev64, Jew75, Theorem 7.3I (Levitan–Plancherel theorem): There exists a unique nonnegative measure on such that

for all , and is dense in .

The inverse Fourier transform of is given by

Jew75, Theorems 12.3A, 12.3B (Bochner’s theorem): For , is continuous bounded positive definite on . For every continuous bounded positive definite function on there exists a unique such that .

Back to double coset hypergroups, an interesting example is given by the Naimark hypergroup BH95, Section 3.5.66, Jew75, Sections 9.5 and 15.2 which arises from a solution of a particular Sturm–Liouville boundary value problem over ( ) or by analysing the geometry of random walks on the hyperbolic plane . We obtain the convolution

whenever , and Haar measure is given by . Here with characters (indexed by )

for all . Note that and . The Plancherel measure on is just

and .

The mark of an excellent paper is not only that it is well written, novel, and makes a substantial contribution to the field, but also that it lends itself to further developments across several areas. Here, these include harmonic analysis, operator algebras, and differential equations. In particular, we highlight follow-up studies of negative definite functions, the Lévy continuity theorem, the Lévy–Khintchine formula and convolution semigroups, all forming the basis of probability theory on hypergroups and related structures; see BH95 and the many papers that have appeared since.

Walter R. Bloom is a professor emeritus of mathematics at Murdoch University, Australia. His email address is w.bloom@murdoch.edu.au.

Graphic without alt text

By Al Hales

Bob and I were fellow students at Caltech, he a year ahead of me. We met in about 1958, either at the Math Club or on a volleyball court. By 1961 we were both in grad school, he at the University of Oregon and I at Caltech. But during the summers of 1959 and 1960 we were both working at Caltech’s Jet Propulsion Lab (JPL) under the supervision of Solomon Golomb. Bob remembered asking me for a good place to read about van der Waerden’s theorem, and apparently I suggested Khinchin’s “Three Pearls of Number Theory.” Later he told me that he thought he could see how to generalize the theorem to structures other than the integers. So he drew me into the project and we started considering possible generalizations. Did we need two operations? Commutativity? Associativity? Units/inverses? Etc. Eventually we decided that an arbitrary semigroup was the right setting. But then it would work for a free semigroup, and this is just the space of sequences! So we realized we had proved that -dimensional “Tic-Tac-Toe” has no tying positions if is large compared to the edge length! Using a known result, this meant that the first player has a forced win.

We could not resist considering the dual question, and soon realized that using Hall’s theorem on distinct representatives we could show that the second player could force a tie if the edge length was large compared to the dimension . We were pleased with our dual results, which first appeared in a JPL report, and decided to submit them for publication. But we had no idea of their future.

Bob and I were coauthors of several other reports at JPL, only one of which was published in the open literature: “Recent Results in Comma-free Codes.” This appeared under the pseudonym B. H. Jiggs, standing for coauthors Baumert, Hales, Jewett, Golomb, Gordon and Selfridge (“i” being a dummy initial). After that our research directions seemed to drift apart—his in the analytic direction and mine in the algebraic direction. We always had plenty of math to talk about, but no further joint papers.

After we received our doctorates Bob went to the East Coast and Europe for several years. Ginny and I were married and spent a year in England and three years in Cambridge, Mass. During this time we met with Bob on several occasions, in Princeton and in the Boston area. Then we both returned to the West Coast. I was at UCLA but took a year’s sabbatical at the University of Washington in 1970–71, by which time Bob was at Western Washington University (WWU) in Bellingham. So we had a number of chances to meet with him in the Northwest.

Shortly after Ginny and I returned to UCLA from sabbatical, Bob and I were each pleasantly surprised to receive phone calls from Gian-Carlo Rota telling us that we would be co-recipients (with Graham, Leeb, and Rothschild) of SIAM’s first “Pólya Prize in Combinatorics,” based on our joint paper. We traveled to Austin, Texas, in late 1972 for this presentation.

Since Bob’s family lived in Southern California, he often traveled to our area to visit them, giving us a chance to get together. In addition, I arranged for him to get a visiting position at UCLA for a quarter in 1974.

Twenty years or so later all this changed—directions reversed! I took early retirement from UCLA and we moved to La Jolla for me to take a new position at IDA/CCR. We bought land on Orcas Island in Washington, and then built a small vacation house there. So now we were traveling up to Bob’s area every two or three months, and we made a point of meeting him for dinner each time, usually in Bellingham though sometimes on “our” island.

Also, during this period, I gave several colloquium talks at WWU while passing through. And there was the wonderful 2016 conference at WWU on “50 Years of the Hales–Jewett Theorem.”

Bob recovered from his 2014 road accident, but his mobility was certainly affected. The series of health problems that followed did not seem to affect his wonderful sense of humor. We continued to see Bob on trips to Orcas until the pandemic began to affect all our travel plans. I think the last time was in May 2021. As should be clear from the above, he was more than a friend and colleague, essentially a member of our extended family. We miss him very much.

Al Hales is a professor emeritus of mathematics at UCLA and adjunct research staff at CCR-La Jolla. His email address is hales@ccr-lajolla.org.

Graphic without alt text

By Richard J. Gardner

By 2016, when the “50 Years of the Hales–Jewett Theorem” conference was held at WWU (Western Washington University), Bob Jewett had been retired for six years and by choice no longer drove a car, so he was chauffeured to and from his senior living home by volunteers. Still equipped with a sharp mathematical mind, he graciously accepted the attention, but to me seemed slightly bemused by all the fuss.

Before the conference, most of the WWU math faculty knew that Bob had done some fine research, but many were not fully aware of its significance. He was always ready to talk about mathematics, yet almost never mentioned his own work. I only recall Bob giving a single colloquium talk during the twenty years we overlapped at WWU, a beautifully presented and entirely elementary exposition of -adic addition and multiplication. In fact, although he continued to publish some nice joint work sporadically, Bob’s major results, addressed in other articles in this memorial tribute, were all in print by 1975. After the WWU conference, I asked him why, given his obvious talent. He said, “Nothing else turned up.”

Figure 4.

Pólya prizewinners at the 2016 Western Washington University conference. Left to right: Ron Graham, Bruce Rothschild, Al Hales, Bob Jewett.

Graphic without alt text

Behind this reply lies Bob’s curious, and to some extent unfathomable, personality. Early adventures described below notwithstanding, he generally preferred not to take any action unless it was necessary, and on occasion even if it was. He was a procrastinator and somewhat forgetful. Letters might remain unopened. Bob sensed that most forms could be ignored, and at some point while still employed even stopped completing his annual tax return, having discovered that the IRS would do it for him. (Dear reader, do not try this at home.) Like G.H. Hardy, he disliked gadgets; he never owned a laptop or cellphone, and never used email or the internet. His office PC was employed solely for exams and lecture notes produced with outdated T software. Telephone was the only means for long-distance communication, but during the thirty years I knew him well and saw him often, Bob phoned me only once.

Despite these peculiarities, Bob was an excellent and very popular colleague. He was genial, modest, considerate, witty, and always ready to have a chuckle. By chalking reminders to himself on his office blackboard, he somehow managed to arrive on time for committee meetings, where his intelligence and straightforward good sense were greatly appreciated. He attended most graduate oral exams and colloquium talks, and usually had insightful questions to ask the speaker. Bob’s wide knowledge of mathematics was a valuable departmental resource; his insight took him quickly to the heart of the matter, and I do not recall him being proved wrong in a mathematical discussion.

There is no PhD program at WWU, but Bob was very active in teaching master’s students. One recalls her classmates referring to Bob as “God,” because he seemed to know all mathematics and how the different areas connected. (In fact, Bob’s research straddles algebra and analysis, and discrete and continuous.) Others fondly remember his patience and sense of humor. In 2016, I used Bob’s notes to teach Math 523, Advanced Calculus of Several Variables. Designed for a quarter-long course, the notes are a masterpiece of efficient and almost error-free exposition, not based on any textbook but developed from scratch. Another set of Bob’s class notes, in a similar style, focused on random walks. Bob was also a good undergraduate teacher; some students might have preferred more leniency and less honesty—I don’t think Bob was capable of being dishonest—but even they often recognized his brilliance and fundamental kindness. An ex-student I know, now a successful teacher but rather lazy at the time, recalls asking Bob for a letter of recommendation for a PhD program. Bob gently replied, “I would have to tell them that you don’t work very hard.” To the student, this frank assessment from one of his favorite teachers was a much-appreciated wake-up call. One might think that Bob’s exams would be as clever as he was, but on the contrary, Bob always advocated for a completely straightforward approach to testing.

While young, Bob was athletic and adventurous. During his academic years in Sweden and Brazil, he learned the languages well enough to give lectures and exams. In Sweden, he was amused to be able to settle a tax dispute by discovering that the Swedish and American versions of the double-taxation treaty did not agree. At the end of his stay in Sweden, in May 1966, he bought a Volvo, drove it into Eastern Europe (a nontrivial matter at that time), shipped it from Gothenburg to New York, and motored across the Northern USA to Washington state, stopping to hike along the way.

Bob told me there were women he would have married, and those that would have married him, but none in both groups. While I knew him, he lived alone, apparently quite contentedly, in accommodation unadorned by decoration of any kind that always featured a recliner and a desk and tables piled with books and assorted paper. But Bob was not a loner. He welcomed company and could be counted upon to enliven the chat at dinner tables or pubs. More than one faculty spouse told me how much they enjoyed Bob at departmental social gatherings: “At least if Bob was there, there was somebody interesting to talk to.”

An observation of Virginia Hales helps reconcile the contradictory aspects of Bob’s personality. She wrote, “He seemed to only live in the moment, his mind was never somewhere else when he was with you…It may have been this last trait that made him absentminded when it came to doing necessary tasks.” I agree. Spending time with Bob was always fun, and it was a privilege and a pleasure to know him.

Richard J. Gardner is a professor emeritus of mathematics at Western Washington University. His email address is richard.gardner@wwu.edu.

Graphic without alt text

References

[BF79]
A. Bellow and H. Furstenberg, An application of number theory to ergodic theory and the construction of uniquely ergodic models, Israel J. Math. 33 (1979), no. 3-4, 231–240 (1980), DOI 10.1007/BF02762163. A collection of invited papers on ergodic theory. MR571532Show rawAMSref\bib{MR571532}{article}{ author={Bellow, A.}, author={Furstenberg, H.}, title={An application of number theory to ergodic theory and the construction of uniquely ergodic models}, note={A collection of invited papers on ergodic theory}, journal={Israel J. Math.}, volume={33}, date={1979}, number={3-4}, pages={231--240 (1980)}, issn={0021-2172}, review={\MR {571532}}, doi={10.1007/BF02762163}, } Close amsref.
[BL99]
V. Bergelson and A. Leibman, Set-polynomials and polynomial extension of the Hales-Jewett theorem, Ann. of Math. (2) 150 (1999), no. 1, 33–75, DOI 10.2307/121097. MR1715320Show rawAMSref\bib{MR1715320}{article}{ author={Bergelson, V.}, author={Leibman, A.}, title={Set-polynomials and polynomial extension of the Hales-Jewett theorem}, journal={Ann. of Math. (2)}, volume={150}, date={1999}, number={1}, pages={33--75}, issn={0003-486X}, review={\MR {1715320}}, doi={10.2307/121097}, } Close amsref.
[BH95]
Walter R. Bloom and Herbert Heyer, Harmonic analysis of probability measures on hypergroups, De Gruyter Studies in Mathematics, vol. 20, Walter de Gruyter & Co., Berlin, 1995, DOI 10.1515/9783110877595. MR1312826Show rawAMSref\bib{MR1312826}{book}{ author={Bloom, Walter R.}, author={Heyer, Herbert}, title={Harmonic analysis of probability measures on hypergroups}, series={De Gruyter Studies in Mathematics}, volume={20}, publisher={Walter de Gruyter \& Co., Berlin}, date={1995}, pages={vi+601}, isbn={3-11-012105-0}, review={\MR {1312826}}, doi={10.1515/9783110877595}, } Close amsref.
[DE74]
Manfred Denker and Ernst Eberlein, Ergodic flows are strictly ergodic, Advances in Math. 13 (1974), 437–473, DOI 10.1016/0001-8708(74)90075-9. MR352403Show rawAMSref\bib{MR352403}{article}{ author={Denker, Manfred}, author={Eberlein, Ernst}, title={Ergodic flows are strictly ergodic}, journal={Advances in Math.}, volume={13}, date={1974}, pages={437--473}, issn={0001-8708}, review={\MR {352403}}, doi={10.1016/0001-8708(74)90075-9}, } Close amsref.
[DKT14]
P. Dodos, V. Kanellopoulos, and K. Tyros, A simple proof of the density Hales-Jewett theorem, Int. Math. Res. Not. IMRN 12 (2014), 3340–3352. MR3217664Show rawAMSref\bib{MR3217664}{article}{ author={Dodos, P.}, author={Kanellopoulos, V.}, author={Tyros, K.}, title={A simple proof of the density {H}ales-{J}ewett theorem}, date={2014}, issn={1073-7928}, journal={Int. Math. Res. Not. IMRN}, number={12}, pages={3340\ndash 3352}, url={https://doi-org.ezproxy.library.wwu.edu/10.1093/imrn/rnt041}, review={\MR {3217664}}, } Close amsref.
[Dun73]
C. F. Dunkl, The measure algebra of a locally compact hypergroup, Trans. Amer. Math. Soc. 179 (1973), 331–348. MR320635Show rawAMSref\bib{MR320635}{article}{ author={Dunkl, C.~F.}, title={The measure algebra of a locally compact hypergroup}, date={1973}, issn={0002-9947}, journal={Trans. Amer. Math. Soc.}, volume={179}, pages={331\ndash 348}, url={https://doi-org.ezproxy.library.wwu.edu/10.2307/1996507}, review={\MR {320635}}, } Close amsref.
[Fur77]
Harry Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math. 31 (1977), 204–256, DOI 10.1007/BF02813304. MR498471Show rawAMSref\bib{MR498471}{article}{ author={Furstenberg, Harry}, title={Ergodic behavior of diagonal measures and a theorem of Szemer\'{e}di on arithmetic progressions}, journal={J. Analyse Math.}, volume={31}, date={1977}, pages={204--256}, issn={0021-7670}, review={\MR {498471}}, doi={10.1007/BF02813304}, } Close amsref.
[FK91]
H. Furstenberg and Y. Katznelson, A density version of the Hales-Jewett theorem, J. Anal. Math. 57 (1991), 64–119, DOI 10.1007/BF03041066. MR1191743Show rawAMSref\bib{MR1191743}{article}{ author={Furstenberg, H.}, author={Katznelson, Y.}, title={A density version of the Hales-Jewett theorem}, journal={J. Anal. Math.}, volume={57}, date={1991}, pages={64--119}, issn={0021-7670}, review={\MR {1191743}}, doi={10.1007/BF03041066}, } Close amsref.
[Gow22]
W. T. Gowers, The enduring appeal of Szemerédi’s theorem, Lond. Math. Soc. Newsl. 500 (2022), 37–44. MR0000000Show rawAMSref\bib{MR0000000}{article}{ author={Gowers, W.~T.}, title={The enduring appeal of {S}zemer\'edi's theorem}, date={2022}, issn={2516-3841}, journal={Lond. Math. Soc. Newsl.}, number={500}, pages={37\ndash 44}, review={\MR {0000000}}, } Close amsref.
[GRS90]
Ronald L. Graham, Bruce L. Rothschild, and Joel H. Spencer, Ramsey theory, 2nd ed., Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., New York, 1990. A Wiley-Interscience Publication. MR1044995Show rawAMSref\bib{MR1044995}{book}{ author={Graham, Ronald L.}, author={Rothschild, Bruce L.}, author={Spencer, Joel H.}, title={Ramsey theory}, series={Wiley-Interscience Series in Discrete Mathematics and Optimization}, edition={2}, note={A Wiley-Interscience Publication}, publisher={John Wiley \& Sons, Inc., New York}, date={1990}, pages={xii+196}, isbn={0-471-50046-1}, review={\MR {1044995}}, } Close amsref.
[HJ63]
A. W. Hales and R. I. Jewett, Regularity and positional games, Trans. Amer. Math. Soc. 106 (1963), 222–229. MR143712Show rawAMSref\bib{MR143712}{article}{ author={Hales, A.~W.}, author={Jewett, R.~I.}, title={Regularity and positional games}, date={1963}, issn={0002-9947}, journal={Trans. Amer. Math. Soc.}, volume={106}, pages={222\ndash 229}, url={https://doi-org.ezproxy.library.wwu.edu/10.2307/1993764}, review={\MR {143712}}, } Close amsref.
[HSY19]
Wen Huang, Song Shao, and Xiangdong Ye, Pointwise convergence of multiple ergodic averages and strictly ergodic models, J. Anal. Math. 139 (2019), no. 1, 265–305, DOI 10.1007/s11854-019-0061-3. MR4041103Show rawAMSref\bib{MR4041103}{article}{ author={Huang, Wen}, author={Shao, Song}, author={Ye, Xiangdong}, title={Pointwise convergence of multiple ergodic averages and strictly ergodic models}, journal={J. Anal. Math.}, volume={139}, date={2019}, number={1}, pages={265--305}, issn={0021-7670}, review={\MR {4041103}}, doi={10.1007/s11854-019-0061-3}, } Close amsref.
[Jew69]
Robert I. Jewett, The prevalence of uniquely ergodic systems, J. Math. Mech. 19 (1969/1970), 717–729. MR0252604Show rawAMSref\bib{MR0252604}{article}{ author={Jewett, Robert I.}, title={The prevalence of uniquely ergodic systems}, journal={J. Math. Mech.}, volume={19}, date={1969/1970}, pages={717--729}, review={\MR {0252604}}, } Close amsref.
[Jew75]
Robert I. Jewett, Spaces with an abstract convolution of measures, Advances in Math. 18 (1975), no. 1, 1–101, DOI 10.1016/0001-8708(75)90002-X. MR394034Show rawAMSref\bib{MR394034}{article}{ author={Jewett, Robert I.}, title={Spaces with an abstract convolution of measures}, journal={Advances in Math.}, volume={18}, date={1975}, number={1}, pages={1--101}, issn={0001-8708}, review={\MR {394034}}, doi={10.1016/0001-8708(75)90002-X}, } Close amsref.
[Kri72]
Wolfgang Krieger, On unique ergodicity, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Univ. California Press, Berkeley, Calif., 1972, pp. 327–346. MR0393402Show rawAMSref\bib{MR0393402}{article}{ author={Krieger, Wolfgang}, title={On unique ergodicity}, conference={ title={Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability}, address={Univ. California, Berkeley, Calif.}, date={1970/1971}, }, book={ publisher={Univ. California Press, Berkeley, Calif.}, }, date={1972}, pages={327--346}, review={\MR {0393402}}, } Close amsref.
[Lev64]
B. M. Levitan, Generalized translation operators and some of their applications, Israel Program for Scientific Translations, Jerusalem; Daniel Davey & Co., Inc., 1964. Translated by Z. Lerman; edited by Don Goelman. MR0172118Show rawAMSref\bib{MR0172118}{book}{ author={Levitan, B. M.}, title={Generalized translation operators and some of their applications}, note={Translated by Z. Lerman; edited by Don Goelman}, publisher={Israel Program for Scientific Translations, Jerusalem; Daniel Davey \& Co., Inc.}, date={1964}, pages={v+200}, review={\MR {0172118}}, } Close amsref.
[Pol12]
D. H. J. Polymath, A new proof of the density Hales-Jewett theorem, Ann. of Math. (2) 175 (2012), no. 3, 1283–1327, DOI 10.4007/annals.2012.175.3.6. MR2912706Show rawAMSref\bib{MR2912706}{article}{ author={Polymath, D. H. J.}, title={A new proof of the density Hales-Jewett theorem}, journal={Ann. of Math. (2)}, volume={175}, date={2012}, number={3}, pages={1283--1327}, issn={0003-486X}, review={\MR {2912706}}, doi={10.4007/annals.2012.175.3.6}, } Close amsref.
[She88]
Saharon Shelah, Primitive recursive bounds for van der Waerden numbers, J. Amer. Math. Soc. 1 (1988), no. 3, 683–697, DOI 10.2307/1990952. MR929498Show rawAMSref\bib{MR929498}{article}{ author={Shelah, Saharon}, title={Primitive recursive bounds for van der Waerden numbers}, journal={J. Amer. Math. Soc.}, volume={1}, date={1988}, number={3}, pages={683--697}, issn={0894-0347}, review={\MR {929498}}, doi={10.2307/1990952}, } Close amsref.
[Spe75]
René Spector, Aperçu de la théorie des hypergroupes (French), Analyse harmonique sur les groupes de Lie (Sém., Nancy-Strasbourg, 1973), Lecture Notes in Math., Vol. 497, Springer, Berlin, 1975, pp. 643–673. MR0447974Show rawAMSref\bib{MR0447974}{article}{ author={Spector, Ren\'{e}}, title={Aper\c {c}u de la th\'{e}orie des hypergroupes}, language={French}, conference={ title={Analyse harmonique sur les groupes de Lie}, address={S\'{e}m., Nancy-Strasbourg}, date={1973--1975}, }, book={ series={Lecture Notes in Math., Vol. 497}, publisher={Springer, Berlin}, }, date={1975}, pages={643--673}, review={\MR {0447974}}, } Close amsref.
[Sze75]
E. Szemerédi, On sets of integers containing no elements in arithmetic progression, Acta Arith. 27 (1975), 199–245, DOI 10.4064/aa-27-1-199-245. MR369312Show rawAMSref\bib{MR369312}{article}{ author={Szemer\'{e}di, E.}, title={On sets of integers containing no $k$ elements in arithmetic progression}, journal={Acta Arith.}, volume={27}, date={1975}, pages={199--245}, issn={0065-1036}, review={\MR {369312}}, doi={10.4064/aa-27-1-199-245}, } Close amsref.
[Wal00]
Mark Walters, Combinatorial proofs of the polynomial van der Waerden theorem and the polynomial Hales-Jewett theorem, J. London Math. Soc. (2) 61 (2000), no. 1, 1–12, DOI 10.1112/S0024610799008388. MR1745405Show rawAMSref\bib{MR1745405}{article}{ author={Walters, Mark}, title={Combinatorial proofs of the polynomial van der Waerden theorem and the polynomial Hales-Jewett theorem}, journal={J. London Math. Soc. (2)}, volume={61}, date={2000}, number={1}, pages={1--12}, issn={0024-6107}, review={\MR {1745405}}, doi={10.1112/S0024610799008388}, } Close amsref.
[Wei85]
Benjamin Weiss, Strictly ergodic models for dynamical systems, Bull. Amer. Math. Soc. (N.S.) 13 (1985), no. 2, 143–146, DOI 10.1090/S0273-0979-1985-15399-6. MR799798Show rawAMSref\bib{MR799798}{article}{ author={Weiss, Benjamin}, title={Strictly ergodic models for dynamical systems}, journal={Bull. Amer. Math. Soc. (N.S.)}, volume={13}, date={1985}, number={2}, pages={143--146}, issn={0273-0979}, review={\MR {799798}}, doi={10.1090/S0273-0979-1985-15399-6}, } Close amsref.

Credits

Figure 1 is courtesy of Seattle Parks and Recreation.

Figure 2 is courtesy of Caltech Archives and Special Collections.

Figure 3 is courtesy of Phil Jewett.

Figure 4 is courtesy of Stephanie Abegg and Amites Sarkar.

Photo of Walter R. Bloom is courtesy of Walter R. Bloom.

Photo of Richard J. Gardner is courtesy of Richard J. Gardner.

Photo of Al Hales is courtesy of Al Hales.

Photo of Joel Spencer is courtesy of Joel Spencer.

Photo of Terence Tao is courtesy of Terence Tao.

Photo of Benjamin Weiss is courtesy of Benjamin Weiss.