Louis Auslander (1928–1997)

S. S. Chern; Thomas Kailath; Bertram Kostant; Calvin C. Moore, Coordinator; and Anna Tsao

Louis Auslander died on February 25, 1997, from He was an unusually effective advocate for the complications following a cerebral hemorrhage. development of applications of , and He was sixty-eight. Born July 12, 1928, in Brook- in his two years of government service as a DARPA lyn, , he received his Ph.D. under S. S. program manager he was very effective in shap- Chern at the in 1954. He ing and enhancing funding for applied mathematics held faculty positions at Yale University, Indiana programs. Indeed, he aggressively sought out op- University, , University of Cali- portunities for mathematics and mathematicians fornia at Berkeley, and Yeshiva University before and in an active manner defined, shaped, and en- joining the faculty at the City University Graduate couraged research areas and connections. His suc- Center in 1965. Since 1971 he had been Distin- cesses at DARPA were driven by his scientific in- guished Professor of Mathematics and Computer sights, his vision, and his political astuteness (all Science at CUNY Graduate Center, receiving the of which he had in abundance). President’s Medal there in 1989. He was a Guggen- Following are reflections by five people who heim Fellow in 1971–72 and was a member of the worked closely with Lou at different times in his career. Institute for Advanced Study in Princeton for the —Calvin C. Moore, Coordinator years 1955–56, 1956–57, and 1971–72. He was also a frequent consultant at the U.S. Naval Re- search Laboratory and for IBM, AT&T, and Hughes Laboratories during his career, and for two years, S. S. Chern 1989–91, he served as program manager for the Applied and Computational Mathematics Program I joined the University of Chicago in the summer at DARPA (Defense Advanced Research Projects of 1949, and Louis was one of my first students. Agency). An interesting thing happened: Chicago was a large Equally at home in pure and applied math- department and offered many mathematical top- ematics, Lou was an insightful mathematician with ics of choice, and he chose Finsler geometry. I had just written a paper on the subject and had sug- wide-ranging interests who made contributions to gested he write what might be called the first paper many different areas. Over the years he worked in on global Finsler geometry [1]. He was a courageous Finsler geometry, in the geometry of locally affine man. His thesis is now of historical significance in and locally Euclidean spaces, in the theory of solv- Finsler geometry. I recently came into contact with manifolds and nilmanifolds, on the representa- the subject again. I have great difficulty in talking tion theory of solvable Lie groups, on many aspects to people about a great landscape; Louis was of harmonic analysis (both in continuous and dis- unique. crete situations), on development of new fast al- Louis was a talented and devoted mathemati- gorithms for discrete Fourier transforms, and on cian. His later works are in the areas of functional the design of signal sets for communications and radar, among other areas. He was the author of over Calvin C. Moore is professor of mathematics at the Uni- one hundred papers and ten books, and he su- versity of California, Berkeley. His e-mail address is pervised the dissertations of eighteen doctoral [email protected]. students (see the accompanying box.) S. S. Chern is director emeritus at MSRI, Berkeley, CA.

390 NOTICES OF THE AMS VOLUME 45, NUMBER 3 analysis, and I leave it to others to comment on this. We maintained a continuous social and math- ematical contact, and I always enjoyed talking to him. He never lost interest in mathematics, and we were generally in agreement.

Calvin C. Moore I first met Lou in the fall of 1963 when he came to Berkeley as a visiting professor for the 1963–64 aca- demic year. Lou was by then an established math- ematician, and I was a newly appointed and rather junior assistant professor. Lou was full of ideas, and we started to talk. The results of Kirillov de- scribing the dual space of a connected and simply connected nilpotent had recently burst upon the mathematical scene and were being ab- sorbed. Lou suggested that we work on the case of solvable groups. Lou brought with him a pro- found knowledge and intuitive feeling for solv- able Lie groups from his previous work in differ- ential geometry, especially work on locally affine and locally Euclidean spaces. Lou was just learn- Louis Auslander, 1995. ing unitary , in which I had some background, so we were well matched, and a one parameter subgroup of a solvable group G the collaboration was extremely fruitful and ex- acts ergodically on the manifold G/H. Insightful citing. Lou taught me a lot about doing math- conversations with Lou during this year inspired ematics and about the process of collaborating my own subsequent work in this direction over the with others, which was especially important to next several years. me. This was the first joint work that I had un- dertaken. I am forever in his debt for all the things Although Lou and I did not work together on any mathematical and otherwise that I learned from research projects after this year in Berkeley, we him during this year. We ended up publishing a talked about mathematics and other topics fre- lengthy jointly authored AMS Memoir [3] on our quently and remained good friends. Lou continued work. We made good progress on the problem but his work on harmonic analysis on solvmanifolds did not solve it, and it was a few years later that and nilmanifolds with many interesting papers. The Lou and Bert Kostant made the real breakthrough focus of his work then shifted to analysis of dis- and obtained a result for solvable Lie groups fully crete Fourier transforms, where he developed new as elegant, complete, and penetrating as the Kir- fast algorithms of interest in computer science. He illov result. During his stay in Berkeley Lou also at- applied harmonic analysis to problems in radar tracted a Berkeley graduate student, Jon Brezin, to (analysis of the radar ambiguity function) and then work with him. Jon followed Lou to New York the to general mathematical problems in various as- next year, completing his dissertation and making pects of signal processing. Our scientific paths significant contributions to the representation the- crossed briefly once more just recently in our com- ory of solvable Lie groups. Jon was Lou’s first Ph.D. mon interest in the phase reconstruction problem student at City University. in crystallography, a problem in finite Fourier During the 1963–64 year when Lou was in Berke- analysis. ley, he also introduced me to nilmanifolds and I have many fond memories of working with solvmanifolds (these are manifolds of the form Lou—his insight and intuition were marvelous, G/H where G is a nilpotent or solvable Lie group and it was a joy to work with him. I learned a lot H a closed subgroup). The most interesting case about mathematics and how to do mathematics and is when G/H is compact. Lou had written a num- about many other things. Lou was certainly what ber of important and path-breaking papers on I would call “streetwise” (interestingly, a descrip- solvmanifolds and nilmanifolds, including an ex- tion that I see Tom Kailath uses as well to describe tension of the Bieberbach theorem on space groups Lou), and I recall his recounting stories from his where n-dimensional Euclidean space is replaced youth about outsmarting other kids and cleverly by a connected and simply connected nilpotent Lie avoiding getting into fights, especially with kids big- group. Lou, with Leon Green and Frank Hahn, had ger than he was. I am forever very much in his debt done some beautiful work [2] on analyzing when and will miss him.

MARCH 1998 NOTICES OF THE AMS 391 general case we faced coadjoint orbits which were Bertram Kostant no longer necessarily simply connected (more than I met Lou Auslander for the first time when we were one quantum line bundle per orbit). Worse, there graduate students at Chicago in the early 1950s. could be no invariant real polarizations. Still worse It was an exciting time and place for mathematics. was the problem of how to characterize type 1-ness. If I remember correctly, Lou’s primary interest at R. F. Streater visited MIT in the mid-1960s and that time was . He, like so complained to me that the 4-dimensional solvable many others (including me), was highly influenced Lie group he was studying (the oscillator group) had by Chern. After Chicago I remember a productive no invariant real polarizations, so how was he to conversation with Lou in the 1950s on holonomy find its unitary dual? I suggested using an invari- groups. The conversation certainly influenced a ant positive complex polarization. It worked. This paper I was writing on that subject. However, the was an encouragement for us. main interaction I had with Lou Auslander began My collaboration with Lou was fun. Each week after a lecture he gave, I believe, at MIT in the mid- alternately either I would go to CUNY or he would 1960s. After a come to MIT. It was also mutually educational. He long, heated dis- taught me Mackey theory, and I taught him the ins Ph.D. Students of Louis Auslander cussion we both and outs of geometric quantization. Together we Clifford Perry (1965) came away with were able to translate successfully the relevant as- pects of the Mackey machine to the language of John Scheuneman (1966) a common excit- ing agenda: find symplectic geometry. The final result was nicer Jonathan Brezin (1967) the unitary dual than we had hoped for. Not only was the unitary Robert Johnson (1969) of a type 1 sim- dual neatly determined, but the condition for type Richard Tolimieri (1969) ply connected 1-ness turned out to be expressible geometrically Carol Heinz-Jacobowitz (1972) solvable Lie in an elegant way. The paper [4] was published in Harvey Braverman (1973) group. The back- Inventiones in 1971. It is not the best-written paper. ground for such A much nicer version, which gets to the heart of Evelyn Mayer Roman (1973) an enterprise is the matter in a clearer way than we had seen it, ap- Sharon Goodman (1978) as follows. Lou pears in a Bourbaki seminar (1973–74) by Michèle Ephraim Feig (1980) had written an Vergne [9]. Bharti Temkin (1983) important paper I have fond memories of Lou. He brought to the Michael Vulis (1983) with Cal Moore table excitement, brilliance, and marvelous in- Myoung Shenefelt (1986) on unitary repre- sights. After our paper was written Lou liked to sentations of tease me by telling a story in gatherings about our James Seguel (1987) solvable Lie lunch times together. When I visited New York, he Michael Cook (1989) groups. As a con- would take me out for an elaborate meal. When he Frank Bernard Geshwind (1993) sequence he was came to Boston, I would take him to the sandwich Jeffrey Litwin (1995) heavily armed machines in the basement of MIT. (I had no idea Irina Gladkova (1998) with Mackey the- at the time how painful this was to him.) ory [8]. From My collaboration with Lou Auslander was one Mackey to Moore of the best collaborative experiences I have ever to Auslander. On the other hand, I had seen a cou- had. I am very grateful to him. ple of years earlier that what the work of Kirillov, Borel-Weil, Harish-Chandra, and Gelfand had in common was a quantization of a symplectic struc- ture on coadjoint orbits. Among other things I in- Anna Tsao troduced the concept of polarization of a sym- I met Lou Auslander while a member of the tech- plectic manifold. This term with this meaning, now nical staff at AT&T Bell Laboratories in 1987. It was well known, appeared for the first time in my sub- a pivotal moment in my life, as Lou became a men- sequent joint paper with Lou. A very strong moti- tor, colleague, and above all a close friend who had vation for me in my collaboration with Lou was to a major impact on both my professional and per- show that what is now called geometric quantiza- sonal development. At Bell Labs I was assigned the tion could break new ground in representation task of designing a parallel algorithm for adaptive theory. Besides Lou’s earlier paper with Moore, beamforming. A critical part of the processing was there were preceding us other steps beyond Kir- solving a symmetric eigenvalue problem. Lou had illov’s nilpotent result. Bernat found the unitary some intriguing ideas about computing eigenval- dual of any exponential solvable Lie group. In the ues of large dense matrices by thinking of the so-

Bertram Kostant is professor emeritus at the Massachu- Anna Tsao is program manager, Applied and Computa- setts Institute of Technology. His e-mail address is tional Mathematics Program DARPA/Defense Sciences [email protected]. Office. Her e-mail address is [email protected].

392 NOTICES OF THE AMS VOLUME 45, NUMBER 3 lution as a direct sum decomposition of certain Auslander with commutative algebras. Since I was trained as a granddaughters pure complex analyst, I was apprehensive about Michelle and plunging headlong into areas in which I had a less- Danielle, top than-comprehensive background. Lou would exhort photo, and me not to be concerned because (paraphrased) grandson Taran, “Mathematics is like a river. You just jump in some bottom. place, and the current will help take you where you need to go.” Our subsequent collaboration [6] on eigenproblems after I joined the Supercomputing Research Center led to considerable activity in nu- merical linear algebra. Shortly thereafter Lou decided to join DARPA as the program manager for the Applied and Com- putational Mathematics Program. To Lou the sig- nificant personal sacrifice of putting his research aside for two years was more than offset by the opportunity to make a difference to both the De- partment of Defense mission and to mathematics in general. And make a difference he did, in pro- found ways. Several programs that Lou started led to spectacular mathematical and scientific achieve- ments that are receiving wide attention in the sci- entific and Department of Defense communities. Two examples are wavelets and the Fast Multipole and engineering Method for solving the Helmholtz equation. The im- not only resulted pact on important DoD problems resulting from the in substantial significant funding in these areas by Lou and his funding of math- DARPA successors has been highlighted in briefings ematical re- to members of Congress and the Pentagon nu- search at DARPA merous times, including detailed briefings to offi- but influenced cials such as the then Secretary of Defense William other program Perry. managers to in- Lou’s success at DARPA was a direct result of corporate more his ability to inspire others with his vision of math- mathematical di- ematics as a cohesive, integrated, and powerful dis- rections into cipline that can open exciting new vistas to be their own pro- conquered when combined with science and en- grams. Lou’s ex- gineering. He was considered to be a singularly ef- tended vision of fective mathematician and program manager at the role of a DARPA who actively sought out and created new mathematics mathematical opportunities. His quick intellect program man- and congenial manner enabled him to engage a ager lives on as a wide range of program managers from competing guiding principle offices and disciplines in meaningful discussions of DARPA’s Ap- of their technical programs which led to joint for- plied and Com- mulation of the underlying mathematical chal- putational Math- lenges. His success in this is legend at DARPA and ematics Program required great energy, objectivity, creativity, and and is one which I, as a current DARPA program knowledge of the breadth of mathematics. Often manager, try to emulate. after first learning an entirely new field, he would Lou’s research interests and collaborations in re- then identify the appropriate mathematical ex- cent years were extremely diverse. Lou’s outlook pertise needed to meet these challenges and seek is exemplified by the fresh, original approaches that out experts in those areas to team with the appli- characterized his research, in collaboration with cation scientists. On other occasions he would several others, on the Fourier transforms. Lou was draw on his considerable scientific insights and convinced of the significant role that core math- abilities to identify application areas to which cur- ematics should play in the design of computa- rent areas of mathematical activity could make an tional algorithms. To elaborate, he once stated impact. Lou’s tireless, personal dedication to that the almost endless capability that mathematics demonstrating the power of mathematics in science provides to reformulate problems and solutions

MARCH 1998 NOTICES OF THE AMS 393 should be exploited in seeking new algorithms. IBM Research, whose director, Shmuel Winograd, He observed that mathematicians are often sur- had charged me with putting together a summer prised that mathematically equivalent formula- program on the Mathematics of Signal Processing tions can have radically differing computational for the Institute for Mathematics and its Applica- properties. This duality of pure math and compu- tions in Minneapolis. Seeing Lou in action led me tational applications is beautifully presented in to invite him to be on the organizing committee, Lou’s AMS Bulletin article [5] with R. Tolimieri on of which he soon became a dominant member, ex- the Fourier transform, which additionally enticed pressing himself quite freely and caustically, which many young mathematicians into harmonic analy- will not surprise many of you. However, Lou had sis. He was involved in a long-term effort to in- many good ideas, several of which we adopted. vestigate how the algebraic structure of Fast Fourier A couple of years later he landed himself at Transform (FFT) algorithms could be exploited to DARPA as program manager for Mathematics. automatically implement optimized FFT algo- [How this came about is a story in itself — briefly rithms. The mathematical culmination of Lou’s in- (and perhaps not completely accurately), a few terest in this subject was a group representation years earlier Lou had taken it upon himself to theoretic result obtained jointly with J. R. Johnson complain directly to the director of DARPA that the and R. W. Johnson shortly before Lou’s death, in value of mathematics was not adequately recog- which “all” possible additive FFT algorithms are nized by DARPA. His complaint worked, but the characterized mathematically [7]. Although the program that was launched was floundering after full implications of this result will take some time a while, so the director apparently insisted that Lou to realize, it has already led to multidimensional come and fix the problems.] However, by the time FFTs that have significantly improved computa- Lou arrived at DARPA, there was no money left in tional properties. the Mathematics budget; moreover the program Lou left a rich legacy of professional contribu- had found itself in the Materials (later Defense) Sci- tions to mathematics. In addition, he gave those ence Division. Undaunted, Lou began to adapt to of us who had the opportunity to know him as a this new environment. Exposure to various dis- friend and colleague may wonderful memories of cussions and seminars convinced him that though his warmth, generosity, and wisdom. there was a lot of interest in the manufacture of various new types of materials, the tools of con- trol, optimization, signal processing, and simula- Thomas Kailath tion were not much used by the materials scien- Coordinator’s Note: This segment is a lightly edited tists working on these problems. Lou called me up version of an electronic message sent by Thomas to discuss the issues and then to suggest that we Kailath to “Friends in the Stanford Manufacturing take a look at filling this gap. While I agreed that Group” shortly after Lou Auslander’s death. our tools could help, as a theoretical engineer who Not all of you may have heard that Lou Aus- had never worked with real equipment, I was re- lander passed away last week. This sad news, which luctant to get involved. “I only write papers,” I I received while traveling, was especially poignant said, “while here it looks as if you really want because Lou was going to be visiting the Bay Area something done.” However, Lou persisted, with later this month. the promise (possible only by DARPA managers, Lou, as some of you know from direct experi- or perhaps usually only dared to be made by them) ence, was a very unusual man. He won a fine rep- of freedom to explore whatever we found to be utation in several different fields of mathematics best, backed up by reasonably generous funding. before turning in the last dozen years or so to the He added that it was OK if we failed. Put that way, study of multidimensional Fourier transforms and it was an offer we couldn’t refuse. to the design of signal sets for communications and As you know, thanks to our environment of radar. These areas bring difficult mathematical great students and cooperative colleagues, we were challenges, but they also got Lou very interested able to rise to Lou’s challenge and to vindicate his in applications. There are mathematicians who vision of the importance of control and signal pro- have difficulty coping with the informal way in cessing in manufacturing. Lou was as pleased as which engineers often handle mathematical dis- we were when one of our efforts won the 1994 Out- cussions, but not Lou. I remember first meeting him standing Paper Prize of the IEEE Transactions on at some signal processing workshop in the mid- Semiconductor Manufacturing, a journal we had 1980s. He seemed to enjoy the meeting and had hardly been aware of a few years ago. Lou’s suc- discussions with a variety of people. Lou was then cessor at DARPA, Jim Crowley, built on Lou’s ini- a consultant for the mathematics department of tiative by winning approval for a much larger MURI (Multidisciplinary University Research Initiative) Thomas Kailath is Hitachi America Professor of Engi- program in modeling, simulation, and control of neering at Stanford University. His e-mail address is materials manufacturing processes, now under [email protected]. way with six university teams. Some technology

394 NOTICES OF THE AMS VOLUME 45, NUMBER 3 transfer has already taken place, and I have no doubt that even in the relatively conservative semi- conductor manufacturing industry we shall see increasing use of these tools. However, theory in manufacturing was only one of Lou’s initiatives at DARPA. Other major ones were on wavelets (with both classified and un- classified applications), advanced radars (one of his personal research areas), matrix-oriented com- puting architectures, and several others far be- yond my ken. Our project, and the related ones in industry, gave me the pleasure of many discussions with Lou. He was always available as a source of advice on a variety of issues, including some very prickly ones. He attributed a lot of his attitudes and per- sonality to his grandmother and mother, early im- migrants from Eastern Europe whose intelligence, determination, and common sense made them successful despite their lack of formal education. In fact, one of Lou’s epitaphs might well be “He was a ‘streetwise’ mathematician”. But that is only one of them. He was too rich a personality, and his range of contributions is too wide to be captured in a single phrase. We shall really miss him, a thought that might bring a chuckle to his spirit, wherever and what- ever it is now—perhaps, in Shelley’s words, “an un- bodied joy whose race is just begun.”

References [1] L. AUSLANDER, On curvature in Finsler geometry, Trans. Amer. Math. Soc. 79 (1955), 378–388. [2] L. AUSLANDER, L. GREEN, and F. HAHN, Flows on homo- geneous spaces, Ann. Math. Stud., No. 53, Princeton Univ. Press, Princeton, NJ, 1963. [3] L. AUSLANDER and C. C. MOORE, Unitary representations of solvable Lie groups, Mem. Amer. Math. Soc., No. 62, 1966. [4] L. AUSLANDER and B. KOSTANT, Polarization and unitary representations of solvable Lie groups, Invent. Math. 14 (1971), 255–354. [5] L. AUSLANDER and R. TOLIMIERI, Is computing with the finite Fourier transform pure or applied mathemat- ics?, Bull. Amer. Math. Soc. (N.S.) 1 (1979), 847–897. MR 81c:42020. [6] L. AUSLANDER and A. TSAO, On parallelizable eigen- solvers, Adv. Appl. Math. 13 (1992), 253–261. MR: 93- F:65098. [7] L. AUSLANDER, J. R. JOHNSON, and R. W. JOHNSON, Multi- dimensional Cooley-Tukey algorithms revisited, Adv. Appl. Math. 17 (1996), 477–519. MR 97k:65301. [8] G. W. MACKEY, The theory of unitary group repre- sentations. [Based on notes by James M. G. Fell and David B. Lowdenslager of lectures given at the Uni- versity of Chicago, 1955.] Chicago Lectures in Math., Univ. of Chicago Press, Chicago, 1976. [9] M. VERGNE, Représentations unitaires des groupes de Lie résolubles, Séminaire Bourbaki, Vol. 1973/74, 26ème année, Exp. 447, pp. 205–226.

MARCH 1998 NOTICES OF THE AMS 395