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# A Memorial Tribute to David B. Wales

Communicated by *Notices* Associate Editor Han-Bom Moon

David Bertram Wales was born on July 31, 1939, in Vancouver, British Columbia, and died on July 17, 2023, in London, England. His father was a high school physics and math teacher, and later a school principal. His mother had a degree in library science and spent her time raising him and his two younger brothers.

David began working on the theory of finite groups when he was a PhD student at Harvard. In 1967, he started teaching and doing research at Caltech, where he spent more than 50 years. David contributed significantly to results like the construction of one of the 26 sporadic simple groups and the determination of the finite subgroups of complex Lie groups of types and Besides group theory, David worked in combinatorics and representation theory of algebras. .

David loved teaching mathematics and working with students. He enjoyed collaborating with other mathematicians and relaxing with them over a glass of wine. David also enjoyed serving in other roles at Caltech including administrative positions in the math department and the office of student affairs. His interests outside mathematics included hiking and international travel. David was fortunate to be active, healthy, and high on life until his unexpected death due to pneumonia.

The contributions below shed more light on David’s mathematical interests and what it was like to work with him. This article was organized by the second author.

Canadian by birth, David Wales received his undergraduate and masters degrees from the University of British Columbia. He obtained a PhD from Harvard, working under the direction of Richard Brauer, with a thesis on the representation theory of finite groups. Wales then took a position at Caltech, where he remained until his death. Marshall Hall, one of the most distinguished group theorists of the time, was then at Caltech, making it a good fit for Wales.

In his early years at Caltech, Wales worked on several projects involving the representation theory of finite groups: He determined the finite groups with a faithful representation of small degree. He and his students determined the simple groups of order for a prime and with cyclic Sylow Wales and his student Huffman determined the quasiprimitive linear groups of degree -subgroups. generated by the conjugates of an element with equal eigenvalues.

During the ten-year period beginning in 1965, finite group theory was rocked by the discovery of 21 of the 26 sporadic groups, that is, those groups that appear in the classification of finite simple groups and are not members of an infinite series. In the sixties, Marshall Hall (and independently Janko) discovered the sporadic Hall-Janko group (HJ). Hall and Wales proved the existence and uniqueness of HJ; Wales proved the uniqueness (subject to suitable constraints) of HJ, using a pretty argument applicable to other sporadics. Later in the seventies, John Conway and Wales proved the existence of the sporadic Rudvalis group.

In later years, Wales turned to questions involving representation theoretic aspects of algebraic combinatorics, particularly Brauer centralizer algebras. He also worked with Arjeh Cohen toward a determination of the finite subgroups of exceptional Lie groups that are maximal among closed subgroups. And he collaborated with numerous people on smaller projects.

Wales had a distinguished record of service at Caltech, particularly with the undergraduate student population. He served terms as master of student houses and dean of undergraduate students, and two terms as executive officer for mathematics. He was faculty secretary for a number of years.

During the seventies Caltech was a center of activity in finite group theory, with many young mathematicians holding positions as research instructors. Examples include Steve Smith, Richard Foote, Joe Carroll, and Bob Guralnick. There were also many visitors, including extended stays by John Conway and John MacKay, and a year-long conference funded by NSF with scores of visitors including Danny Gorenstein and John Thompson. Wales mentored the younger mathematicians and collaborated with older visitors.

There was of course a weekly group theory seminar, but there was also an informal meeting on Friday afternoons in the basement bar of the Athenaeum (the Caltech faculty club) convened over pitchers of margaritas. It was at one of these meetings that David Wales met his wife Kathy.

During David’s tenure as master of student houses, he and Kathy lived in the Master’s House, a large old California house situated on the Caltech campus with various features, such as a large backyard, that made it well suited for entertaining students. The house contained a pipe organ that was two stories high. The Waleses liked cats, and usually owned one or more Abyssinian cats. I recall one week my wife and I were recruited to walk to the Master’s House each day to feed the cats and play with them in the Waleses’ absence. It was during this time that our younger daughter Meredith died in an automobile accident. My family decided to hold a get together for friends of Meredith and the family, to celebrate her life. The Waleses volunteered the yard of the Master’s House as a venue for the event. We’ve always felt a special connection to David and Kathy for this kindness.

David Wales died of a lung infection in London during a trip to England which included the chance to see one of his grandchildren play in an international tournament in ultimate frisbee. He is survived by his wife Kathy, his children Jon and Jennifer, and his grandchildren.

Working with David was a great pleasure and led to more joint papers than I had with anyone else. This is largely due to the many positive aspects of David’s character, which I greatly enjoyed. He was thorough in his treatment of mathematics and at the same time positive in his attitude and joyful to people in general. Where I was the impulsive person with wild ideas, David would provide stability and reassurance that there was something good in what I did. Where I would try to explain an idea in too many words, David would reshape our texts into an understandable format. Where I worked late at night and sent my immature results to him before going to bed, David would check them early in the morning and set the ground for further discussion. When we got stuck, David would present the simplest example that should be examined before trying further bold theories. On a personal level, we got along very well. I highly enjoyed David and Kathy’s warm welcoming and interested attitude and the many trips we made together to painting exhibitions, nature (mostly beaches), castles, and the beers in Old Town Pasadena to celebrate a new result. David was a great friend and my ultimate math companion.

Let me describe the main parts of our mathematical work. In 1977, David Wales published the article

My thesis advisor, Tonny Springer, suggested that I also classify finite groups generated by homologies (the projective version of reflections) over the octaves (also often called the octonions). The first order of business would then be to determine the finite groups of automorphisms of Such groups have a faithful linear representation in seven dimensions over the reals, and, since David was an expert on these, he joined in. This led to our first joint publication. Finite groups generated by homologies over the octaves are subgroups of the complex Lie group of type . and so we proceeded by trying to classify all finite subgroups of this Lie group and, while we were at it, those of type , as well. Early in 1989, we got a chance to work on it during a stay at Caltech that David had organized for me; the result

Around that time, our attention shifted to two new developments. First, during David’s sabbatical at Eindhoven University of Technology, we dived into extremal elements in Lie algebras (an element of a Lie algebra is called extremal if is contained in the subspace spanned by The ). spanned by such an element in a simple Lie algebra is the Lie algebraic counterpart of a long root subgroup of a group of Lie type. Anja Steinbach and Rosane Ushirobira helped out. The resulting paper -space

From 2015 on, I started writing interactive mathematics courses and tests for a private enterprise, which was set up by my son and a friend of his. David got involved as my conscience since he checked the English language version and the soundness of proofs. As was the case with our joint writing of papers, when he would report that he did not understand a proof, I could expect there to be an error in it. The construction of the courses was concluded shortly before David died.

I first met David in fall, 1976, when I began my two-year tenure as a Bateman Research Instructor at Caltech, although I was familiar with his work on groups of order before then. David soon became a mentor, colleague, and friend. Caltech was one of the epicenters of the Classification of Finite Simple Groups (CFSG) program, with David, Michael Aschbacher, Marshall Hall, myself and, in 1977, Robert Guralnick on the faculty. As an expert in representation theory, David was an invaluable resource to me, a newly minted PhD; and since he was associate dean of students, I could always count on him to support my undergraduate students with expert advice and compassion.

David was vital to the weekly Caltech group theory seminars. To typify his participation, I remember that we once worked through some notes on the recently published Lusztig-Deligne theory of characters of the groups of Lie type. Not satisfied with the theory alone, David insisted that we work through an example. So we spent a couple of hours after the seminar computing the complete character table of a substantially large linear group using the Lusztig-Deligne formulas, and then checked all the orthogonality relations by hand. That was the kind of mathematician David was: expert and knowledgeable in the theory, but down-to-earth and practical in its workings.

My sole coauthored paper with David, *strongly closed* subgroup of a Sylow 2-subgroup of such that is quaternion of order 8, and then we used the CFSG to force to be trivial. [For any prime, a subgroup of a Sylow -subgroup of any finite group is called *strongly closed* if for each every , of -conjugate that lies in must belong to This property of . does not depend on the Sylow subgroup containing it, so it is intrinsic to -subgroups of The theory of strongly closed 2-subgroups in the special case when .] is abelian is a cornerstone of the CFSG proof itself, but for nonabelian it had been largely unstudied at this point in time. David’s and my work on this particular configuration became the template for a series of papers by myself and my doctoral students, as well as Ramon Flores, culminating in the complete classification of groups possessing a nontrivial strongly closed for any prime -subgroup (where we exclude the trivial cases when or and applications thereof to both Artin’s conjecture and homotopy theory. This work also dovetailed nicely with Aschbacher’s extensive program on ),*fusion systems*, with its concomitant applications to algebraic topology—a very active area of contemporary research. So this paper with David, which we were happy to dedicate to our mutual hero, Walter Feit, on the occasion of his 60th birthday, was a seminal contribution to an active field of mathematics, as described in greater detail in the survey

David was a truly genuine, jovial, and congenial friend; a gifted raconteur; and always a joy to interact with. He was also an avid adventurer, hiker, and camper. One of my most enduring happy memories of David was camping with him by the shadow of Half Dome in Yosemite National Park. His mathematics and his life as an educator and a friend are personified to me by that pinnacle.

I first met David in September 1977 when I started my postdoctoral position at Caltech. I had just received my degree the previous June. It was a quite exciting year. Caltech was having a special year in group theory and had quite a number of visitors such as Danny Gorenstein, Graham Higman, and many others. Caltech also had a very strong group theory presence with Michael Aschbacher, Marshall Hall, and David. Richard Foote was another group theory postdoc who had started a year earlier.

The abstract algebra class at Caltech was a sophomore-level class and there were three sections taught in the fall and winter terms. They were taught by Richard, David, and myself. Of course, David gave us a lot of advice (although Richard and I were quite enthusiastic and so I think we covered more material than David; I had Peter Shor in my class). Richard’s notes were the birth of the Dummit-Foote book on abstract algebra.

I got to know David well during my two years at Caltech and he became a lifelong friend in addition to a mathematical colleague. We had many lunches at the Athenaeum. During my first year, the membership dues was only $1 for me but were considerably higher for Michael and David and so I always signed for the lunches.

I continued to come over to Caltech for the group theory seminar and other events for many years after leaving Caltech for USC and my friendship with David grew. During the pandemic, I did not go to Caltech but fortunately in February and March of 2022, Tim Burness was visiting Caltech and he and I had a shared office. I was coming to Caltech at least twice a week for that period and saw David quite a lot. We had lunch several times as well as dinner and many conversations. Currently, David’s last PhD student, Claire Levaillant, works at USC under the joint supervision of Aaron Lauda and myself.

David had a very strong mathematical career. He was involved in the construction of one of the sporadic simple groups (a group discovered independently by Hall and by Janko of order and much of his early work was on classifying simple groups under various assumptions and studying low-dimensional representations. )

Over his long career, he wrote papers on many different topics (including many joint works with Arjeh Cohen and Phil Hanlon). The ones that were most relevant to my work were his papers mostly with Cohen about subgroups of exceptional groups. In

In three papers

Although David and I often discussed mathematics, we had only one joint paper,

I feel very fortunate to have had David Wales as a mentor and collaborator. He was an outstanding algebraist and I learned much from our mathematical work together. Perhaps more importantly, he was someone whose emotional intelligence rivaled his mathematical intelligence. He never lost sight of the fact that mathematics is ultimately a human endeavor, and one which is challenging and often solitary. And so he generously supported those around him with kindness, humor, and empathy. This was an important part of what made him such a special colleague and friend.

I was a PhD student at Caltech from 1977 to 1981. At the time, Caltech’s graduate program in Math was small and cohesive. We bonded over coursework, teaching experiences, and Friday-evening libations in the basement of the Athenaeum. By then a full professor, David was always there for us. Encouraging us, sharing our frustrations, joining in our evening gatherings at the Athenaeum. For us, as PhD students, David represented the faculty with kindness and support. And his care for others extended well beyond our community of graduate students. For many years, he served as dean of students assisting countless Caltech undergraduates navigate challenges big and small. And he was first in line to recognize the contributions of the Math Department’s hard-working staff.

David had passions outside of mathematics. He loved his family—Kathy, Jonathan, and Jennifer—and often spoke with great pride about their accomplishments. He relished his Canadian heritage. And he had a playful side and a sense of humor that helped put all things into perspective. For many years, he and I bet $1 each year on the outcome of the Harvard-Dartmouth football game. Sadly, I found myself on the losing end of this bet most years. And then one year, as an apparent act of consolation, David offered me “a deal”—a chance to get back the dollar I had just lost. He would bet me $1 that there was at least one rouge during the Grey Cup—the Canadian Football League championship game. I asked “what in the world is a rouge”? He went on to explain it in terms that made it seem that a rouge was very unlikely to happen. And so, I enthusiastically agreed to this offer only to lose the bet and find out later that rouges are a common occurrence in Canadian Football games.

David was also a first-rate mathematician with an avid curiosity. After earning my PhD in 1981, and two years at MIT, I returned to Caltech as a Bantrell Fellow in 1983. At the time, I was interested in the Brauer centralizer algebras, a family of algebras that can be defined via a combinatorial construct indexed by two parameters—a positive integer and a multiplication factor My immediate interests were to identify those values of . and for which the Brauer algebra is semisimple and the structure of the radical for those values of where semisimplicity fails. David was excited by these questions and immediately jumped in to help.

In some sense, this was an ideal collaboration. David brought deep expertise in algebra generally, and semisimplicity specifically, to the table. And I brought combinatorial methods as well as advanced computational tools to bear—simplifying the discriminant calculations by applying the representation theory of the symmetric groups—to make vast calculations just within the reach of what were at the time cutting-edge computational tools: the CRAY 1 and CRAY 2 machines at the Minnesota Supercomputer Center. Our collaboration resulted in a series of papers that included conjectures (most notably that the Brauer algebras were semisimple for noninteger values of later proved by Hans Wenzl using Jones’ tower construction. What I remember most about that work was what it was like to have David as a collaborator. He had a passion for mathematical discovery that was infectious. David loved doing concrete examples. But in our work on the Brauer algebras, examples quickly became so large and complex that hand calculations were impossible. And so David felt a true excitement at what could be gleaned by combining high end computational methods with the tools of algebra and combinatorics. )

I’ve recently been working on a family of algebras (the Okada algebras) that share many of the properties of the Brauer algebras. This has caused me to look back at those papers that David and I wrote together all those many years ago. And I can’t help but imagine how much David would have enjoyed being part of this new project. I miss having him as a collaborator and as a friend.

David and I met for the first time in 2001 when I was 21 years old and a summer intern in neuroscience at Caltech. At that time I was a math major at the Ecole Normale Supérieure de Cachan in France, thus I quite naturally went to visit the math department. There I was asked in which field of mathematics I was interested and my answer was “algebra.” This led me to David’s office. I found him standing, looking towards the ground and in very deep mathematical thoughts. He simply said to me “Come back at 11.” It turned out simplicity, kindness and professionalism described him well. David started advising my PhD work two years later.

During my graduate studies, David’s door was always open to me, except when his close collaborator Arjeh Cohen was visiting. At these times, both of them were so focused on mathematics that an earthquake could very well occur without them noticing. During and after my studies with him, David always enjoyed wishing me a happy Bastille Day on July 14, which I found nice. Conversely, if I missed wishing him a happy Canada Day on July 1, I fortunately had a backup three days later. As an advisor, David was serious and demanding. Studying the bibliography before my candidacy exam was taking more than a week and he was upset as I came into his office for my weekly meeting showing him only a half page. A week later, I came back with 7 pages and as he saw the first page numbered as 1 slash 7, he exclaimed to me: “You have written 117 pages, you have made good progress.”

For my PhD, David had me work on representations of diagrammatic algebras named after Joan Birman, Jun Murakami, and Hans Wenzl. These algebras contain the braid group. This was shortly after the linearity of the braid group had been shown by Daan Krammer algebraically and by Stephen Bigelow topologically, using a representation of Ruth Lawrence. The latter representation became known as the Lawrence-Krammer representation of the braid group and to this date, it is the only known faithful representation of the braid group. As part of my work, David had me build my own representation of the braid group, namely one equivalent to the Lawrence-Krammer representation but of simpler expression. Later on, I worked on representations of Artin groups of types and based on the beautiful constructions for tangles of type by Cohen, Gijsbers, and Wales.

David was an open-minded person with a good sense of humor. He was a happy person and had a balanced life. His happiness was highly communicative. His personal honesty was particularly well conveyed through mathematics. His mathematical knowledge was immensely broad but there were always more things that we did not know than things that we knew. It always made the interactions true and pleasant.

David was always very supportive and encouraging to the students. He was specifically involved in the undergraduate students’ well-being. He was also generous to the graduate students who were not his own students. David’s overall generosity to the students was an important part of his life. The postdocs also liked him. I recall talking with a postdoc who said to me about David closing both eyes “he is extremely nice.”

## Acknowledgments

The organizer owes much to Kathy Wales for input to this article. We are grateful to Jonathan Hall and an anonymous referee for some helpful remarks.

## References

- [1]
- Arjeh M. Cohen and David B. Wales,
*in a -orbits module for characteristic -dimensional*, J. Algebra**185**(1996), no. 1, 85–107, DOI 10.1006/jabr.1996.0314. MR1409976,## Show rawAMSref

`\bib{CW}{article}{ author={Cohen, Arjeh M.}, author={Wales, David B.}, title={$\mathrm {GL}(4)$-orbits in a $16$-dimensional module for characteristic $3$}, journal={J. Algebra}, volume={185}, date={1996}, number={1}, pages={85--107}, issn={0021-8693}, review={\MR {1409976}}, doi={10.1006/jabr.1996.0314}, }`

- [2]
- Arjeh M. Cohen and David B. Wales,
*Finite subgroups of and*, Proc. London Math. Soc. (3)**74**(1997), no. 1, 105–150, DOI 10.1112/S0024611597000051. MR1416728,## Show rawAMSref

`\bib{CWEF}{article}{ author={Cohen, Arjeh M.}, author={Wales, David B.}, title={Finite subgroups of $F_4(\mathbf {C})$ and $E_6(\mathbf {C})$}, journal={Proc. London Math. Soc. (3)}, volume={74}, date={1997}, number={1}, pages={105--150}, issn={0024-6115}, review={\MR {1416728}}, doi={10.1112/S0024611597000051}, }`

- [3]
- Arjeh M. Cohen, Anja Steinbach, Rosane Ushirobira, and David Wales,
*Lie algebras generated by extremal elements*, J. Algebra**236**(2001), no. 1, 122–154, DOI 10.1006/jabr.2000.8508. MR1808349,## Show rawAMSref

`\bib{CSUW}{article}{ author={Cohen, Arjeh M.}, author={Steinbach, Anja}, author={Ushirobira, Rosane}, author={Wales, David}, title={Lie algebras generated by extremal elements}, journal={J. Algebra}, volume={236}, date={2001}, number={1}, pages={122--154}, issn={0021-8693}, review={\MR {1808349}}, doi={10.1006/jabr.2000.8508}, }`

- [4]
- Robert M. Guralnick, Martin W. Liebeck, Dugald Macpherson, and Gary M. Seitz,
*Modules for algebraic groups with finitely many orbits on subspaces*, J. Algebra**196**(1997), no. 1, 211–250, DOI 10.1006/jabr.1997.7068. MR1474171,## Show rawAMSref

`\bib{GLMS}{article}{ author={Guralnick, Robert M.}, author={Liebeck, Martin W.}, author={Macpherson, Dugald}, author={Seitz, Gary M.}, title={Modules for algebraic groups with finitely many orbits on subspaces}, journal={J. Algebra}, volume={196}, date={1997}, number={1}, pages={211--250}, issn={0021-8693}, review={\MR {1474171}}, doi={10.1006/jabr.1997.7068}, }`

- [5]
- Robert M. Guralnick and David B. Wales,
*Subgroups inducing the same permutation representation. II*, J. Algebra**96**(1985), no. 1, 94–113, DOI 10.1016/0021-8693(85)90041-9. MR808843,## Show rawAMSref

`\bib{GW}{article}{ author={Guralnick, Robert M.}, author={Wales, David B.}, title={Subgroups inducing the same permutation representation. II}, journal={J. Algebra}, volume={96}, date={1985}, number={1}, pages={94--113}, issn={0021-8693}, review={\MR {808843}}, doi={10.1016/0021-8693(85)90041-9}, }`

- [6]
- W. C. Huffman and D. B. Wales,
*Linear groups of degree containing an element with exactly equal eigenvalues*, Linear and Multilinear Algebra**3**(1975/76), no. 1-2, 53–59, DOI 10.1080/03081087508817092. Collection of articles dedicated to Olga Taussky-Todd. MR401937,## Show rawAMSref

`\bib{HW1}{article}{ author={Huffman, W. C.}, author={Wales, D. B.}, title={Linear groups of degree $n$ containing an element with exactly $n-2$ equal eigenvalues}, note={Collection of articles dedicated to Olga Taussky-Todd}, journal={Linear and Multilinear Algebra}, volume={3}, date={1975/76}, number={1-2}, pages={53--59}, issn={0308-1087}, review={\MR {401937}}, doi={10.1080/03081087508817092}, }`

- [7]
- W. C. Huffman and D. B. Wales,
*Linear groups containing an involution with two eigenvalues*, J. Algebra**45**(1977), no. 2, 465–515, DOI 10.1016/0021-8693(77)90338-6. MR435243,## Show rawAMSref

`\bib{HW2}{article}{ author={Huffman, W. C.}, author={Wales, D. B.}, title={Linear groups containing an involution with two eigenvalues $-1$}, journal={J. Algebra}, volume={45}, date={1977}, number={2}, pages={465--515}, issn={0021-8693}, review={\MR {435243}}, doi={10.1016/0021-8693(77)90338-6}, }`

- [8]
- Richard Foote and David Wales,
*Zeros of order of Dedekind zeta functions and Artin’s conjecture*, J. Algebra**131**(1990), no. 1, 226–257, DOI 10.1016/0021-8693(90)90173-L. MR1055006,## Show rawAMSref

`\bib{FW}{article}{ author={Foote, Richard}, author={Wales, David}, title={Zeros of order $2$ of Dedekind zeta functions and Artin's conjecture}, journal={J. Algebra}, volume={131}, date={1990}, number={1}, pages={226--257}, issn={0021-8693}, review={\MR {1055006}}, doi={10.1016/0021-8693(90)90173-L}, }`

- [9]
- Richard Foote, Hy Ginsberg, and V. Kumar Murty,
*Heilbronn characters*, Bull. Amer. Math. Soc. (N.S.)**52**(2015), no. 3, 465–496, DOI 10.1090/bull/1492. MR3348444,## Show rawAMSref

`\bib{FGM}{article}{ author={Foote, Richard}, author={Ginsberg, Hy}, author={Kumar Murty, V.}, title={Heilbronn characters}, journal={Bull. Amer. Math. Soc. (N.S.)}, volume={52}, date={2015}, number={3}, pages={465--496}, issn={0273-0979}, review={\MR {3348444}}, doi={10.1090/bull/1492}, }`

- [10]
- David B. Wales,
*Linear groups of degree containing an involution with two eigenvalues II .*, J. Algebra**53**(1978), no. 1, 58–67, DOI 10.1016/0021-8693(78)90204-1. MR480770,## Show rawAMSref

`\bib{W}{article}{ author={Wales, David B.}, title={Linear groups of degree $n$ containing an involution with two eigenvalues $-1$. II}, journal={J. Algebra}, volume={53}, date={1978}, number={1}, pages={58--67}, issn={0021-8693}, review={\MR {480770}}, doi={10.1016/0021-8693(78)90204-1}, }`

- [11]
- David B. Wales and Hans J. Zassenhaus,
*On -groups*, Math. Ann.**198**(1972), 1–12, DOI 10.1007/BF01420495. MR338020,## Show rawAMSref

`\bib{WZ}{article}{ author={Wales, David B.}, author={Zassenhaus, Hans J.}, title={On $L$-groups}, journal={Math. Ann.}, volume={198}, date={1972}, pages={1--12}, issn={0025-5831}, review={\MR {338020}}, doi={10.1007/BF01420495}, }`

## Credits

Figure 1 and Figure 4 are courtesy of Kathy Wales.

Figure 2 is courtesy of Caltech and appears at https://digital.archives.caltech.edu/collections/Images/10.43-12/.

Figure 3 is courtesy of Dan Roozemond.

Photo of Michael Aschbacher is courtesy of Michael Aschbacher.

Photo of Arjeh M. Cohen is courtesy of Arjeh M. Cohen.

Photo of Richard Foote is courtesy of the University of Vermont.

Photo of Robert Guralnick is courtesy of USC.

Photo of Philip Hanlon is courtesy of Dartmouth College.

Photo of Claire Levaillant is courtesy of Claire Levaillant.