Paul Roesel Garabedian (1927–2010) · 2014. 2. 10. · Paul Roesel Garabedian (1927–2010) Jerry L. Kazdan Paul Garabedian, 2000. Background Mathematics was in Paul’s genes:

Paul Roesel Garabedian(1927–2010)Jerry L. Kazdan

Paul Garabedian, 2000.

BackgroundMathematics was in Paul’s genes: both his fatherand his two uncles were mathematicians. Paul andhis sister, Caroline Coulon, were home schooled.At age sixteen he was not admitted to Harvardbecause Paul’s father’s Ph.D. advisor, G. D. Birkhoff,was concerned that he might be immature. Instead,Paul went to Brown—and excelled. He simply was aprodigy. In the enlightening oral history conductedby Philip Davis in 2004 [2], Paul remarked (withinsight), “I was a child prodigy. I am still a childprodigy, but there are very few people who knowthat, perhaps yourself and a few others.” PhilipDavis laughed.

After graduating from Brown in 1946 be begangraduate school at Harvard. Two years later atthe age of twenty he received his Ph.D., working

Jerry L. Kazdan is professor of mathematics at the Uni-versity of Pennsylvania. His email address is [email protected].

DOI: http://dx.doi.org/10.1090/noti1088

with Lars Ahlfors. Paul also benefited from manyconversations with Max Schiffer, who was visitingHarvard’s applied mathematics department. Thiswas the beginning of Paul’s important mathematicalinteraction with Schiffer—and a lifetime of workwhere complex analysis played a major role eitherfor itself or as a key ingredient in applications.

In 1948, after receiving his Ph.D., Paul was aNational Research Council Fellow at Stanford. Thefollowing year he became an assistant professorat the University of California, Berkeley. One yearlater he simply resigned from the university beforeit became mandatory to sign the loyalty oathrequired of University of California faculty.

Stanford immediately offered Paul a position. Heflourished there: promoted to associate professorin 1952 and to professor in 1956. In 1959 Paulcame to NYU’s Courant Institute. He was appointedthe director of the Division of Computational FluidDynamics at the Courant Institute in 1978. Paulremained a member of the NYU faculty for fifty-oneyears. Paul also often spoke fondly of his ten yearsat Stanford, maintained many friends there, andreturned often to visit and give lectures.

The obituary published by the Courant Institute[1] is a good source for more information.

ResearchOne early influential result with Schiffer (1950)was where they used Hilbert space ideas withthe Bergman kernel function to prove existencetheorems for elliptic partial differential equations.

Another problem he considered deeply was theBieberbach Conjecture for the family of holomor-phic functions f (z) that are 1-1 maps of the unitdisk to the complex plane normalized by f (0) = 0and f ′(0) = 1. Then f (z) = z + a2z2 + a3z3 + · · · .In 1916 Bieberbach proved that |a2| ≤ 2 withequality essentially only for the Koebe function

f (z) := z(1− z)2 = z + 2z

2 + 3z3 + · · · .

Bieberbach conjectured that |an| ≤ n, with equalityonly for this function.

244 Notices of the AMS Volume 61, Number 3

In 1985, after more than sixty years, the con-jecture was proved true by de Branges. But theroad was not straight. In 1923 Löwner proved that|a3| ≤ 3, but it took until 1955 for Garabedianand Schiffer to prove that |a4| ≤ 4. The proof byGarabedian and Schiffer that |a4| ≤ 4 involvedtedious numerical calculations. Although Paul’sdesk was always notoriously empty except for onesmall pad, this work was a real exception. Thesecalculations on the Bieberbach Conjecture for a4were not done in his head.

Subsequently Charzynski and Schiffer (1960)found a much simpler, more conceptual, proofthat |a4| ≤ 4, one that was also now shorter thanLöwner’s proof that |a3| ≤ 3 but did not imply that|a3| ≤ 3. Since one anticipates that the difficulty ofthe proof should increase with n, this violation ledPaul to question the conjecture itself. Some relatedconjectures had already been found to be naive.

As a thesis topic for me as a graduate studentat the Courant Institute, Paul suggested that Itry to find a counterexample. So I tried. With EvaSwenson’s superb help on the computer, I sought—but did not find. This lack of monotonicity in thedifficulty of proof persisted. Peterson and Ozawa(1968, 1969, respectively) proved that |a6| ≤ 6before Pederson and Schiffer (1972) proved that|a5| ≤ 5. The eventual proof by de Branges usesLöwner’s approach. Crooked paths make life moreinteresting. I personally value this experience ofstruggling with a conjecture whose resolutionis uncertain. It taught me to be dubious aboutconjectures—and diffuse evidence—until they arereally proved.

In the early 1950s, Paul, in joint work withDonald Spencer on the ∂-Neumann problem, wasattempting to understand some problems involvingfunctions of several complex variables. This workwas seminal. See Denny Hill’s note below for hispersonal insight on the collaboration.

In 1952 Paul’s article with Lewy and Schiffer onRiabouchinsky flow was one of his earliest workson fluid flow.

Beginning in the mid 1950s, Paul’s research focuswas decisively influenced by specific questionson transonic fluid flow that were asked by DavidYoung at Ramo-Wooldridge Corporation. Paul’scontributions involved analytic continuation to thecomplex plane and were viewed with suspicion—until people calmed down and realized that eventhough they were mystified, the results seemed tobe both correct and better than anything their ownresearch could yield. This work was influencedby earlier results of Hans Lewy. Paul’s beautifulPartial Differential Equations text is a wonderfulintroduction to these seminal ideas. This workwas also the beginning of his pioneering use ofcomputers to solve basic problems in science

and engineering. It is surprising that althoughPaul didn’t write computer code himself, hewas superb as a leader to produce fundamentalresults. The article below by Antony Jamesongives an introduction to the problems and Paul’sprofound contributions, particularly to the designof a transonic airfoil, now used in commercialairplanes. This is a wonderful example of veryapplied mathematics at its best.

In his next period, Paul focused on magneticfusion, where one studies fluid flow coupled withan electromagnetic field. One goal is the importantpractical application of designing a nuclear “powerplant” based on fusion, not fission. The valuablearticles by Geoffrey McFadden [3], [4] discuss boththis and Paul’s work on transonic airfoils.

While most of Paul’s work after 1970 was onapplied mathematics, he published a few veryshort gems on topics in pure mathematics, suchas a simple proof of a variant of the Lewyexample of a linear unsolvable PDE. These alwaysinvolve complex analysis and often some functionalanalysis.

Paul had twenty-seven Ph.D. students and manypostdocs.

Honors

Fairchild Distinguished Scholar, Caltech (1975)NASA Public Service Group Achievement Award

(1976)Boris Pregel Award, New York Academy of Sciences

(1980)Birkhoff Prize of the AMS and SIAM (1983)Theodore von Kármán Prize, SIAM (1989)National Academy of Sciences Award in Applied

Mathematics and Numerical Analysis (1998)American Physical Fellow (2004)SIAM Fellow in the inaugural class (2009)

He was a member of both the National Academy ofSciences and the American Academy of Arts andSciences.

Paul is survived by his wife, Lynnel, his daughters,Emily and Cathy, and two grandchildren, as well ashis sister, Caroline Coulon. He was an active father,who played an important role in his daughters’lives. Once when Paul and I happened to meet inJapan, he described proudly several trips he tookto Kyoto accompanied by his younger daughter,Cathy, and how she helped “take care” of him. Forme, these stories were a model. I regret that I neverhad an opportunity to have a similar experiencewith either of my own daughters.

References[1] Courant Institute’s obituary for Paul Garabedian,

http://cims.nyu.edu/webapps/content/special/Garabedian_Obituary.

March 2014 Notices of the AMS 245

http://cims.nyu.edu/webapps/content/special/Garabedian_Obituaryhttp://cims.nyu.edu/webapps/content/special/Garabedian_Obituary

[2] Phillip Davis, Oral history: An interview withPaul Garabedian, http://history.siam.org/oralhistories/garabedian.htm.

[3] Geoffrey B. McFadden, Garabedian obituary in SIAM,http://siam.org/pdf/news/1809.pdf.

[4] , Garabedian obituary in Stellarator, http://www.ornl.gov/sci/fed/stelnews/pdf/sn126.pdf.

Peter D. LaxPaul Garabedian was an outstanding pure math-ematician as well as one of the most originalapplied mathematicians. His strength was that heused very sophisticated pure mathematics to solveapplied mathematics problems. Sometimes thishad the effect that the engineering community didnot comprehend his methods and therefore wasreluctant to use his results (more about this later).

Paul’s father was a mathematician, a Ph.D.student of G. D. Birkhoff at Harvard; his motheralso had an advanced university degree. Paul andhis sister were educated at home. When Paul turnedsixteen, he was already excelling in mathematics;his father took him to Birkhoff as a prospectiveHarvard student. Birkhoff was concerned that homeschooling did not prepare Paul for college life. Ashe put it, “Suicides give a school a bad name.” Hesuggested that Paul go to boarding school for ayear, and then he would be admitted to Harvard.His father took Paul to Brown University; there,too, the admission people were worried about thelack of social skills and suggested that Paul go toboarding school for a year. “If he goes to boardingschool, he goes to Harvard,” said his father, soBrown admitted him right away. His roommatewas the fifteen-year-old Al Novikoff. According toAl, Paul taught him a lot of mathematics, and hetaught Paul about women.

Paul was a brilliant undergraduate; he finishedin 1946 and was admitted to Harvard as a graduatestudent. He earned his Ph.D. in 1948 under thedirection of Lars Ahlfors; his dissertation was onthe Szegő kernel function.

His first academic position was at Berkeley.He arrived at a time of anticommunist hysteria.The trustees of the university demanded that thefaculty sign a loyalty oath. Paul was among thefaculty who refused (Hans Lewy was another). Hewas immediately hired by Szegő at Stanford, wherehe spent nine fruitful years. He collaborated withhis colleagues Don Spencer and Max Schiffer andmade his first venture into applied mathematics.An outstanding problem in the 1950s was whethera ballistic missile would burn up upon reenteringthe earth’s atmosphere. Since at that time ballisticmissiles existed only on the drawing board, the

Peter D. Lax is emeritus professor of mathematics at NewYork University-Courant Institute. His email address is [email protected].

Paul with Peter Lax (1968).

decision had to be made by theoreticians. Paul gavea brilliant mathematical formulation of the problemwhich could then be solved on the relatively slowcomputers available at the time. His calculationshowed that ballistic missiles would not burn upupon reentry.

While at Stanford, Paul married Gladys Rappa-port, a graduate student in statistics. She wrotethe computer code for the reentry problem; mirac-ulously, the code worked immediately. A second,more sophisticated, program contained some bugs;Paul gave her hell.

In 1957 Paul was named scientific attachéin London; the duties of the attaché were totravel around Europe and report on new scientificdevelopments. During his visit to Italy, Paul learnedthat De Giorgi had solved a Hilbert problem aboutthe regularity of solutions of nonlinear ellipticequations (at about the same time John Nash alsosolved this problem by an entirely different method).The paper by De Giorgi was published in a veryobscure journal; the international mathematicalcommunity learned about it from Paul’s report.

In 1959 Jim Stoker, Courant’s successor asdirector of the institute at NYU, visited Stanford.He had had his eye on Garabedian for some time;so then and there he made an offer to Paul tocome to NYU. After an afternoon of negotiations,well lubricated by martinis, Paul accepted. It was ahappy outcome for both parties; Paul enjoyed andwas stimulated by the atmosphere of the CourantInstitute, and he liked to live in New York City.

Paul’s father was of Armenian descent; hismother was not, and Paul resembled his mother.


http://history.siam.org/oralhistories/garabedian.htmhttp://history.siam.org/oralhistories/garabedian.htmhttp://www.ornl.gov/sci/fed/stelnews/pdf/sn126.pdfhttp://www.ornl.gov/sci/fed/stelnews/pdf/sn126.pdf

Paul with his sister, Caroline, and niece, Aline(France, 1962).

When interviewed for admission to Harvard gradu-ate school, Paul noticed that the interviewer wasmaking notes; left alone momentarily, he took apeek at the notes; they said, “Wrong name, rightappearance.”

Paul was a hero to the Armenian mathematicalcommunity. He was once invited to a mathematicalcongress in Yerevan, the capital of Soviet Armenia.He accepted gladly. When he stepped off the plane,the reception committee was shocked to beholdthe blue-eyed, blond guest; “That’s not Garabedian;it is a CIA agent.”

While in New York his first marriage ended indivorce. His second marriage, to Lynnel, was veryhappy. They had two adored daughters.

I would like to give now the broad outlines ofsome of Paul’s research. It started in function theoryand potential theory. At Stanford he began hiscollaboration with Max Schiffer on the Bieberbachconjecture; very likely they were introduced tothis problem by their colleague Charles Loewner,who did the deepest work on this problem back in1923. Their approach was to use the calculus ofvariations. The formulas involved in this researchwere formidable. Ultimately, the complete solution,by de Branges, used the Loewner representation ofschlicht functions.

There is a natural connection of analyticfunctions to fluid dynamics; two-dimensionalincompressible, irrotational flows are describedby analytic functions. Paul’s interest in fluid me-chanics was much broader. He tackled many ofthe classical problems in fluid dynamics, such asthe flow around a rising bubble, the shape of anelectrified droplet, the vertical entry of a wedgeinto water, and other problems of flows with a freeboundary.

In the 1970s aircraft companies were designingplanes that could fly near the speed of sound. That

meant that over part of the wing the airflow wouldbe supersonic. Cathleen Morawetz had shown thatin general such transonic flows contain shocks;shocks increase the drag of the airplane andtherefore they should be avoided if possible.

One of Paul’s most influential works, with DavidKorn, was the design of airfoils that carry shock-less transonic flows. This required solving partialdifferential equations that are partly elliptic, partlyhyperbolic. They accomplished this by introducingcomplex coordinates. This was so sophisticateda mathematical idea that the aerodynamic com-munity was unable to comprehend it; thereforethey ignored it. Finally a mathematically mindedaerodynamicist in Canada tested the airfoil in awind tunnel and found the flow to be indeed freeof shocks.

After that, aerodynamicists pounced on theGarabedian-Korn design. One of the leading Frenchaircraft companies invited Paul to be a consultant.Paul accepted the offer, under the condition thatthey fly him to France on the Concord.

In 1964 Paul published a text on the theoryof partial differential equations, designed as atext for a graduate course. It treated all the usualtheoretical subjects, as well as numerical methodsfor solving partial differential equations, includingthe use of complex coordinates. The book becameextremely popular and still is today.

Starting about thirty years ago Paul turned hisattention to the mathematical and engineeringproblems of nuclear fusion. The physics subject,called magneto-hydrodynamics (MHD), is aboutflows of high temperature plasmas that typicallycontain shock-like discontinuities. Paul publishedabout sixty papers on the subject, more than a thirdof his total publication of 167 papers (the last oneappeared in 2010). He believed in the practicalityof generating energy by nuclear fusion. Until thelast months of his life, very weak physically butrazor sharp mentally, he continued to work onproblems of MHD. If fusion energy ever becomes areality, Paul’s work will have played an importantpart in its success.

With much outstanding mathematics one canimagine the work having been done by a numberof mathematicians. Not so with most of Paul’saccomplishments; his outlook was unique. He willbe remembered for a long time.

Albert B. J. NovikoffHere goes (stream of consciousness): we were botharound sixteen years old, too young to be at (Brown)University, but it was war time, and compromises

Albert B. J. Novikoff is professor of mathematics at NewYork University, Courant Institute. His email address [email protected].


were the order of the day. I came first, fromBronx Science, and Paul arrived a year later, freshfrom home schooling in a professor’s household. Iwas, comparatively, the sophisticate, but we bothwore long underwear in the winter, scandalizingour respective roommates and condemning us toroom together. Items of forbidden winter clothingremain in my mind, stigmata of being under ageand of controlling mothers, specifically, mittensand galoshes. He was already tall, but like a giantboy Pinocchio, all kneecaps and shins. I had notyet started the viola, and he, miraculously, alreadyplayed Chopin ballades and (on the violin) thegreat Bach Chaconne. I had no one from BronxScience to compare him to.

Mathematically, he had a comparable back-ground. His father still marveled at his own experi-ence at Harvard, under the great G. D. Birkhoff, andwhat it was like to be in class and watch the greatman thinking his way before your very eyes. Hisfather, by the way, played me an organ symphonyby Widor, whose existence had until then escapedme.

Music, mathematics, and the intellectual lifewere his heritage, but apparently he had been leftout of “kidhood,” and my great (ha!) contributionwas to introduce him to the humbler parts ofhuman destiny. He was quick to enjoy the companyof the graduate students we both frequented andremained tremendously grateful to me over thedecades, when I did no more than turn a key in ametaphorical door.

As his roommate (and hopeless competitor inmath classes) I can report that he was a “cleandesk” type from the get-go; only the prescribedtexts sullied the empty plane of his desk. What heworked on was his own, in every sense. I think hetook pleasure in being obstinately independent,unbowing to written authority.

By the time he was taking senior (and also grad-uate) courses, the graduate students all recognizedthat he was exceptional. I never told him, but once agroup of us were remarking on this when the seniormember of our circle, a future physicist, remarkedwryly, “He’s going to be annoyingly useful someday.”

Now the public in general has (or had) littleunderstanding or interest in what mathematiciansactually do. Regarding this, Fritz John noted someyears ago that the mathematician’s sole recognitionis “the grudging appreciation of a few friends”.Clearly, Paul started collecting his quota of suchrecognition from an early age.

As I learned from an early visit to his parentsin Wheaton, MA, where his father was not onlyprofessor of mathematics but school organist anddirector of the (girls) choir, a lot was pretty muchset in stone.

Paul with David Gilbarg (1950s).

A few years later I was at Stanford and heardSchiffer boast of the recent discoveries of hisstudent, Paul, at Harvard. I had not ever been partyto such generous pride. I remember him lookingback at Harvard and commenting that he had been“pretty good” at ideal theory and wondering justwhat units he was using.

Other associations are Saint-Venant’s problem(whatever that was, involving René de Possel), theBieberbach Conjecture, et al.

He mellowed enormously with a happy marriageand adoring daughters and took to making jokesat his own expense. Just after moving to his newapartment, not so long ago, he had occasion todecide which reprints he wanted to save, and said,“I seem to have written each paper five times.”

With more emotion than I care to admit,Al Novikoff

C. Denson HillIn late August of 1961 I arrived at Idlewild [nowKennedy] Airport, having $300 in my pocket andno idea how to get to NYU, where I was supposed tobecome a new graduate student. So I squanderedpart of it by taking a taxi into Manhattan andthen asked the taxi driver if he knew of a cheaphotel near Washington Square Park. After a nightof squishing cockroaches, I went to look for the

C. Denson Hill is professor of mathematics at StonyBrook University (SUNY). His email address is [email protected].


famous Courant Institute. I was dismayed to findout that it was located in an old hat factory, andI was shocked at how rude people were on thestreet and how much of a slum the neighborhoodappeared to be. I had expected something quitedifferent. But they had offered me a fellowship,and I had no other place to go, so I went inside tolook for somebody to talk to. Almost immediatelyI found a kind secretary, who took me to “tea,”saying I might find some faculty member there.At Rice University I had already developed aninterest in PDE, applied mathematics, and hadsome experience with numerical analysis. It wasbecause of those interests that I had been advisedto go to Courant, but nobody had suggested whomI should see there or exactly what I should do uponarrival.

At tea, there was only one guy, who I thoughtwas another graduate student. But the secretaryintroduced him to me as “Professor Garabedian.”He looked much too young to be a professor.Over tea he politely asked me what my interestswere, which courses I had taken, etc. Then heannounced that he was writing a book on PDE andthat he needed somebody to read and correct themanuscript, and he asked me if I would like to helphim on his project. Of course I agreed. That is how,within twenty minutes of entering the building, Ibecame a graduate student of Paul Garabedian.

So for my first two and a half years as a graduatestudent, I had the side job of going over variousdrafts of Garabedian’s book [1]. At that time Paulhad another student, Jerry Kazdan. He was a fewyears ahead of me, and Jerry had made detailedcriticisms of the first draft of the book. My jobwas to go over various subsequent drafts. I woundup making extensive revisions to Chapter 10 onintegral equations and Chapter 15 on free boundaryproblems. Needless to say, I learned an enormousamount, not only from the manuscript itself, butalso from the sometimes heated discussions withPaul about my suggestions for changes. I wasalways pressing him to make precise definitionsand state precise theorems, and Paul was alwaysresisting. He kept telling me that he wanted thebook to be read by engineers and other appliedscientists and that he was trying to provide insight,not a list of theorems. His point was that if hetried to prove the sharpest versions of varioustheorems, it would clutter up the exposition andobscure the elegance of many arguments. Probablyat that time I was too much enamored of my recentaffair with Bourbaki. But luckily I had also readGoursat, so I did in fact get his point. After morethan fifty years, I see clearly now that he was right.

What impressed me the most was Garabedian’samazing ability to step out into imaginary direc-tions and use several complex variables to gain

insight into various questions about PDE’s. Inthis aspect he was very much in the spirit ofJ. Hadamard and H. Lewy. So I read Hadamard’sbook on the Cauchy problem, and a number ofLewy’s papers. (This set me up for my later workwith Aldo Andreotti.) But eventually when I wentto Paul and asked him to suggest a thesis problemwhich had something to do with several complexvariables (a subject I knew nothing about at thetime), he said, “Well, I used to work in SCV, but Igave it up because I did not understand it.” LaterI found out that Garabedian was the man whoinvented the ∂-Neumann problem.

The story, as told to me by Paul, went like this:For some period he was working with D. Spencer.It was a friendly collaboration in which each wasquite polite and respectful to the other. But theproblem was that he never understood anythingSpencer said, and he was not sure if Spencerunderstood him either. So it wound up with eachof them writing his own paper and putting theother guy’s name on it [2], [3]. The ∂-Neumannproblem was formulated in the paper [2] writtenby Garabedian. As is well known, Spencer laterpushed his student J. J. Kohn to solve the problem,which was difficult due to the noncoercive natureof the Neumann boundary condition for ∂.

Another story Paul told me was how he managedto get his Ph.D. at Harvard, in only two years, bysolving his thesis problem over the weekend.Ahlfors had returned to Harvard as a full professorin 1946, and Garabedian was his first Ph.D. studentthere. Paul walked into Ahlfors’s office on aThursday, or maybe a Friday, and asked him fora thesis problem to work on. Ahlfors promptlysuggested a problem involving the Szegő kernel. ButGarabedian did not understand exactly what Larswas getting at. Fortunately another mathematician,Menahem Schiffer, had also arrived at Harvard asa research lecturer in 1946, but he was over inapplied mathematics. Paul had discovered thatSchiffer was easy to talk to, so he went over to theapplied mathematics department to ask Schifferjust what it was that Ahlfors wanted him to do.Schiffer was much more clear than Ahlfors hadbeen and was able to explain to Paul what theproblem actually was. Knowing now just what thequestion was, Garabedian went home, worked hardover the weekend, and went back to Ahlfors’s officeon Monday and presented the solution. Ahlforssaid, “You have a thesis.” Later Garabedian andSchiffer became colleagues and collaborators atStanford.

References[1] Paul R. Garabedian, Partial Differential Equations,

John Wiley and Sons, New York, 1964.


Alex Chorin, Paul, and Lynnel at the NationalAcademy of Sciences award celebration.

[2] Paul R. Garabedian and D. C. Spencer, Complexboundary value problems, Trans. Amer. Math. Soc. 73(1952), 223–242.

[3] , A complex tensor calculus for Kählermanifolds, Acta Math. 89 (1953).

Antony Jameson

Paul Garabedian’s Contributions to Tran-sonic Airfoil and Wing DesignThis note on Paul Garabedian’s work on transonicairfoil and wing design is written from the perspec-tive of aeronautical engineering as well as appliedmathematics. Paul’s contributions in this area hada profound and lasting impact on the way peopleset about designing wings in the aircraft industry.Transonic flow is of great relevance to aircraftdesign because it is the most efficient regime forlong-range transport aircraft. Transonic flow isalso of great mathematical interest. Outside theboundary layer and wake the flow is well repre-sented by the transonic potential flow equationwhich is of mixed type, elliptic in the subsoniczone and hyperbolic in the supersonic zone. Thisequation proved quite intractable to analyticalmethods of solution. In order to reduce the dragone must look for shapes that minimize the shockstrength or even produce shock-free flow. Thiswas the problem that Paul chose to tackle. It hadbeen established, however, by Cathleen Morawetzthat shock-free solutions are isolated points andshocks will appear with small perturbations of theshape or the flight condition. So the problem ofdesigning a shock-free shape is not well posed.

Paul elected to pursue an inverse approach.Following earlier work by Lighthill, he used thehodograph transformation in which the velocity

Antony Jameson is Thomas V. Jones Professor of Engi-neering, Department of Aeronautics and Astronautics,Stanford University. His email address is [email protected].

components u and v are treated as the independentvariables and the coordinates x and y becomethe dependent variables. While this results in alinear equation of mixed type, it remains hard tofind solutions in the hodograph plane which corre-spond to physically realizable shapes. Nieuwlandhad previously generated a family of hodographsolutions which resulted in airfoils that were notpractically useful. Paul applied the method ofcomplex characteristics which he had successfullyused to solve the supersonic blunt body prob-lem in earlier work to solve the equations in thehodograph plane. He was able to find boundaryconditions and integration paths that resultedin usable shock-free airfoils for a range of Machnumbers and lift coefficients. Working with hisassistant Frances Bauer and his doctoral studentDavid Korn, he published the first results in thebook Supercritical Wing Sections.

In this period he made contact with Richard Whit-comb at NASA Langley, who had experimentallydeveloped a supercritical airfoil with a flat-toppedshape and heavy rear camber which produced acomparatively weak shock at its design condition.Paul’s shock-free 78-06-10 airfoil had a similarthough smoother shape. This influenced Whit-comb’s thinking, and he decided to fund furtherstudies of supercritical airfoils at the CourantInstitute. Paul also made contact with R. T. Jonesat NASA Ames and obtained additional fundingfrom Jones to pursue studies of yawed wings.

In 1970, as a staff engineer at Grumman, I wasasked to look into the state of the art in supercriticalwing technology. I soon found out that Paul’s groupwas at the cutting edge. Eventually I joined thegroup as a senior research scientist in 1972. Paulwas now working on a second book, SupercriticalWing Sections II , which presented an improvedseries of shock-free airfoils, a transonic analysismethod (Program H) which included a boundarylayer correction, some results of experimentaltests, and some preliminary results for yawedwings. My principal assignment was to write thethree-dimensional analysis code for yawed wings(Program J, or Flo17) which subsequently evolvedinto a widely used code for calculating transonicflow over swept wings (Flo22). Program H andFlo22 are still in use today for preliminary designwork at Boeing.

The concept of a yawed flying wing for super-sonic cruise was the subject of intensive studiesat NASA fifteen years later, but no viable designemerged. In the meanwhile Paul continued hisstudies of supercritical wing design, issuing athird book, Supercritical Wing Sections III , in 1977.With Geoffrey McFadden he also developed a three-dimensional inverse design method. By 1980 hisinterest had switched to magnetic containment


of plasma for fusion reactors, and this remainedthe main focus of his research for the rest of hiscareer.

In the period I worked for him, Paul was awonderful mentor. He exposed me to broad areasof mathematics in which my knowledge was quitedeficient. He would do this in a very subtle way bycasually asking if I was not aware of this or that, forexample, the Bateman variational principle. Then Iwould be forced to go and find out what he wastalking about. He had an extraordinary youthfulappearance—at age forty-four one might easilyhave taken him to be twenty-eight.

To the best of my knowledge none of the airfoilslisted in either of the two books was directly usedin an actual aircraft, but they had a profoundand lasting impact on the aircraft industry byshowing for the first time that practically usefulsupercritical airfoils which are shock free orproduce very weak shocks could be designed. Thispermanently changed the way engineers thinkabout transonic wing design.

The 75-06-12 “Garabedian-Korn” airfoil hasbeen widely used as a benchmark to validatenew numerical methods for computational fluiddynamics (CFD). Due to three-dimensional effects,particularly near the fuselage, a satisfactory sweptwing cannot be designed with a fixed wing sectionfrom root to tip. In a numerical experiment I havesubstituted the Garabedian-Korn section into arepresentative modern transonic wing design, theNASA Common Research Model (CRM), which isthe test shape for the latest AIAA Drag PredictionWorkshops. The initial design produced a verystrong shock wave across the entire span. However,using an optimization method based on techniquesdrawn from control theory for partial differentialequations, the wing can be redesigned to producean essentially shock-free flow. This demonstratesthat the Garabedian-Korn section could still beused as the starting point for a competitive wing.This calculation took four hours using a quad-coreworkstation which is about 5,000 times fasterthan the Control Data 6600 computers at theCourant Institute in the early 1970s and hasabout 8,000 times the memory. Evidently such acalculation would not have been feasible in thatera. Nevertheless, the outcome after forty yearsis that all modern transonic commercial aircraft,including business jets as well as airliners, havewing sections which strongly resemble the sectionsdesigned by Paul Garabedian.

Eva V. SwensonPaul Garabedian was my Ph.D. thesis supervisorduring 1961–1965. I am indebted to him for takingme under his wing, for patiently coaching andsupporting me through the Ph.D. process, and formaintaining an interest in my career and my lifeever after. He provided me with a steady compasspoint that I knew I could turn to anytime I felt Ineeded to.

As a graduate student and research assistant, Ifelt fortunate to be assigned the office adjoininghis which was quite small, enough for only oneperson. It allowed me to concentrate on work,and it provided enough space for the voluminouscomputer printouts that I was generating at thattime. By contrast, his own office was a huge corneroffice minimally furnished with desk, chair, plant,and a couple of file cabinets. The desk itselfheld very few items; most notably he always hada 4′′ × 6′′ notepad and a pen. His office spokevolumes of his clear and uncluttered thinking. WithPaul’s steady guidance and gentle encouragement,I pursued various explorations until one day Paulexcitedly declared, “Eva, I think you have yourresult!” He showed me that producing a Ph.D. thesiscan be challenging and satisfying, and he enabledme to experience that fantastically rich worldwhere pure and applied mathematics intersect.

He wasn’t all work, though. Paul knew verywell how to live in balance. I remember well thegatherings on Friday evenings with Paul and Lynneland his graduate students and their respectivegirl/boyfriends of the day. On those occasions, Ifelt that he was just one of us, having a good time.We would go to the public pool for a swim, thento his apartment for martinis. Then we would goto the Old Mill Restaurant for cheap but excellentsteaks. One summer, he invited us all out to hisplace on Fire Island. I remember a scene wherewe were all sunbathing on the beach, except thatPaul was off to one side with his 4′′ × 6′′ notepadand pen, unobtrusively continuing to work on thetheorem of the day.

In retrospect, I realize how extremely fortunateI was to have Paul as my mentor. He taught methrough lectures, discussions, and, especially, byexample. I learned to stay focused: to look for theessential and to trim the superfluous. I learnedthat if an idea can’t be contained in a 4′′ × 6′′notepad, it is not yet “ready for prime time.”

Eva V. Swenson’s email address is [email protected].


Paul with daughters Cathy and Emily, 1989.

Ruth Bers ShapiroThe following is drawn from a talk that Ruth BersShapiro gave at the memorial for Paul on Decem-ber 4, 2010. Ruth Bers Shapiro’s email address [email protected].

Remembering PaulI am honored that Lynnel asked me to speak todayabout Paul as a friend and as a husband and father.Paul, Lynnel, and I often spoke of the many waysour lives intersected. Here is a brief chronology. Ifirst met Paul in the early 1950s on a visit to PaloAlto. Paul had been a student of my father, LipmanBers, at Brown University in the early 1940s. Beforeour meeting, my father described Paul with greataffection as a highly gifted mathematician. He hadneglected to tell me, however, that Paul was alsohandsome, had an irrepressible twinkle in his eye,and loved having a good time. He was by far thecoolest mathematician I had ever met.

In the fall of 1957, Lynnel and I met in ourfreshman year at the University of Michigan, becameroommates, and have remained best friends sincethen. After graduating from college and whileworking towards a master’s degree in Englishliterature at NYU, Lynnel took a job at the CourantInstitute. I was not surprised that when she metPaul, she found him charming, adventuresome—before their meeting he had sailed from SanFrancisco to Hawaii on a four-man sailboat—andromantic.

They were from the start and remained through-out their marriage closely matched on how theysaw the world. In politics, vociferous liberals. Inreligion, confirmed atheists but fully tolerant ofothers’ beliefs. In friendships, unswerving in theirloyalty and devotion to their friends. It is hard tospeak of one without the other, but I do want tosay a few words about Paul’s essential values.

Although he could be exacting and discriminat-ing about mathematical elegance, scientific rigor,

or musical performance—perhaps even elitist orsnobbish—in terms of political and social justice,Paul was consummately egalitarian. He could nottolerate discrimination or injustice of any kind,and he felt that every human being deserved a fairbreak, an opportunity to have a good life. It waspainful to him that in our wealthy nation, peoplewere left stranded. He believed in a visceral waythat the privileged and fortunate needed to care forthe poor and disenfranchised. He was outspokenand courageous.

Paul was a wonderful friend. If there was aproblem and he could help, he never turned away,even when it was uncomfortable and might puthim in a difficult position. He remained grateful topeople who befriended him or helped him in largeor small ways.

Before becoming a parent, Paul expressed theworry that his preoccupation with mathematicswould make it difficult for him to be a goodfather. Lynnel reassured him. In fact, Paul was apassionate, empathic, and supportive parent. It isLynnel’s view that he fell in love with his daughtersand was transformed by the love he felt for them.I would second that.

I am sure everyone here knows that Paul could bemeticulous—he worked on a cleared desk with onlythe sheet of paper on which he was writing. Whenit came to Emily and Cathy, only the constructiveaspects of order remained. He wanted them to feelfree, valued their spontaneity, and was charmedby their developing personalities. When they wereyoung, Paul was known to refer to his daughtersas his two best theorems.

Paul loved life. He fought against the ravages ofcancer, refusing even an Advil because he did notwant anything to interfere with the clarity of histhought—stubborn in ways, but passionate andintensely alive. It is not surprising that his lastpaper, written with his devoted student GeoffreyMcFadden, was published just four months beforehe died.

I will miss him as long as I live: his humor,warmth, compassion, and his intensity.

Cathy Garabedian

Memories of My DadBy the time I was born in 1975, my father was forty-eight years old and was already a well-establishedmathematician. Earlier that year, before I was evenborn, he had been elected to the National Academyof Sciences. This honor was lost to me, as myearliest memory of my father is being carried tobed on his shoulder, watching our cozy living room

Cathy Garabedian’s email address is [email protected].


grow smaller as I clung to the neck of his six-footframe. There I am sure he kissed me good nightand told me he loved me before tucking me intobed, as he did year after year as I grew up.

As I grew older I slowly became aware that myfather had an important role outside our homeas someone other than “my Daddy.” I rememberlearning when I was about eight years old that hewas writing a book about his work, so I asked himif he would dedicate it to me. When the book cameout, I didn’t even understand the title, but I wasextremely proud that it was dedicated to me, mysister, Emily, and the coauthor’s daughter. I stillhave a copy of Magnetohydrodynamic Equilibriumand Stability of Stellorators on my shelf with mydad’s handwritten dedication and a smiley facealong with the printed inscription. It never matteredto me what the book was about; I was proud of myfather just for being my dad. Looking back at mylife, however, I realize that my father’s work had agreat influence on me.

My father had great confidence in his ownintellectual abilities and taught me and my sisterto have the same confidence in ours. When I wasin high school, he would help me with my calculushomework. He would read the problem and thenthink it through and start explaining how we wouldsolve it, sketching out a plot and carefully writingdown a couple of formulas. “See, it’s easy,” hewould say, “You can do this.” His words cameback many times over the years: while studyingfor organic chemistry tests in college, trying tounderstand the Hodgkin and Huxley model of theaction potential in graduate school, and teachingmyself how to program in Matlab to analyze datafor my graduate thesis. “It’s easy,” I’d think tomyself. I knew that my dad believed in me. Hisconfidence in me is a strength I carry in myself.He believed in his daughters so strongly that wecouldn’t help but believe in ourselves.

In fact, as a father of two daughters, he wasfervent in his support of equal opportunity for all.He firmly told us that women were as capable asmen in math and expressed his respect for thewomen he had advised as graduate students. Thisrespect extended to his racially and internationallydiverse group of students and colleagues. Hisegalitarian belief system informed his social andpolitical views, such as his belief in a strong publiceducation system for all and assistance programsfor the disadvantaged. I believe that part of hischoice and comfort with living in New York Citywas based in his appreciation of the multitudes ofdifferent people who live together in the city.

My father’s work as a mathematician took myfamily around the world, giving us exposure tocultures beyond what we could see even in NewYork. I spent my sixth birthday in Hong Kong,

on the way to Kyoto, Japan. My dad had becomeenamored with Japan the year before while he wasthere for a conference, and he and my motherdecided to take the family there for his sabbatical.During our year in Japan, Dad started teachinghimself Japanese, proudly practicing phrases fromhis language books and pointing out familiar Kanjisigns to us in the subways.

Some of my dad’s travels that could have kepthim away from his family instead became preciousmemories of time I shared alone with him. When Iwas in middle school, he took me back to Japanwhile he visited colleagues there and gave talksat Kyoto University. My sister was in high school,too busy with classes to join us, and my momnow worked full time. For three years these tripswere our special father-daughter time. On theplane the airline attendants would dote on us,thinking it sweet that a single father was lookingafter his little girl. During my dad’s meetings Iwould do my school assignments, preparing forthe presentations on “My trip to Japan” that Iwould give to my class when I returned.

Aside from math my dad’s passion was music.He would sit down almost every evening at ourgrand piano and play pieces by Chopin, Beethoven,or Mendelssohn. For a man whose work involved thelogic and precision of mathematical calculations,his love of the Romantic composers revealed amore emotional side. He would fill the room withstrong, intense passages followed by soft, beautifulmelodies. My parents encouraged me to take pianolessons from the age of seven so I could share thisinterest. Later my dad brought out his old violinso we could play duets; we practiced Beethoven’sSpring Sonata together, culminating in a concertfor my piano teacher that earned us both stickersas a token of our accomplishment. But my dadloved nothing more than hearing me sing. Inthe mornings he’d brush my hair before school,and as he wove it into two braids, I would singhim all the songs I could think of: SomewhereOver the Rainbow , Tomorrow (from “Annie”), andtunes from “The Sound of Music”. When I startedperforming in musicals, he would come to everyshow, making sure to get there early to get a goodseat and beaming when I came out to take my bow.There was no more proud, dedicated parent.

My dad looked forward to spending the summerat his sanctuary, our house on Fire Island. Therehe would take walks along the beach each day,enjoying the hard sand and crashing waves at lowtide. He would disappear for hours by the ocean. Iimagine him mulling over some research problemas he walked mile after mile in the sun, finallyreturning to our family as the sun got low in thesky. He would wave to us as the outline of hisfigure got closer and closer on the beach, and he


Paul escorting daughter Emily at her wedding,2003.

would finally sit down in the sand with us to enjoythe last warm breeze of a summer day. Sometimeshe would sit on the porch of our cottage and watchthe birds, finding joy in the new hatchlings learningto fly in the spring. My father’s sometimes quicktemper was balanced by an unbelievable soft spotfor the beauty and innocence of nature.

My father was successful at mathematics ata young age and found fulfillment in his workthrough the very end of his life. He continued tofeel passion about his research as he raised hisfamily, running to the office after the presentswere opened on Christmas morning to check hiscomputer run, and quietly scribbling ideas inhis notebook when he was on vacation with us.However, I never for a moment thought that hisfamily was anything but his top, most importantpriority. I have never seen as much pride in hiseyes as when he walked my sister down the aisleat her wedding, helped my little nephew put onhis shoes, and held my giggling niece in his armsbefore she could even say “Grampa”. My familyhas lost a loving father and husband and we willmiss him, but we will never forget the wonderfulmemories he gave to us, and we will hold themwith us always.

Emily Garabedian

The following is drawn from a talk Emily Garabediangave at the memorial for Paul on December4, 2010. Emily Garabedian’s email address [email protected].

Memories of My FatherI am Paul’s older daughter, Emily. Many of you knowme. Some of you watched me grow up. Most of thisday commemorates Paul’s academic achievementsand mathematical genius. But I am glad we willspend some time remembering his personal life,celebrating him for the father, husband, brother,grandfather, and friend that he was. I would liketo share with you some of my memories of Paul asa father.

Growing up, I remember my father as anintellectual, an academic. He did math, and heplayed the piano. I never thought of him as anathlete. However, in preparing this eulogy, I realizemany of my childhood memories are of sportsactivities my father did with me, including teachingme how to play tennis. He took me to ParagonSports and bought me a racquet, and he wouldreserve a court on the roof of the Coles SportsCenter. I’m afraid I never excelled at tennis. Fora while, to my father’s dismay, I insisted that hereserve a basketball court instead. He didn’t playbasketball, but he got me a basketball and tookme to Coles and practiced with me. I expected myfather to help me with my math homework and togive me confidence in my intellectual abilities. But Irealize that he also tried to give me the confidenceI needed to try out for the basketball team. WhenI didn’t actually make the team, he was there toconsole me.

My father brought me to the pool at Coles, andwe would swim together. Sometimes we wouldmeet people he knew from NYU, and I could hearthe pride in his voice when he introduced me, andI knew he thought I was one of the most importantpeople in the world.

My father tried to teach me how to ride atwo-wheeled bicycle. He tried to explain to me, atthe age of five or six, the physics of keeping thedevice upright. I learned about gyroscopes andangular momentum, but I wasn’t able to ride thebike. In the end it was our doorman who intervened.He would run behind me as I pedaled, holding onto the back of the bike just like my father did, onlywhen I had achieved some speed, the doormanwould let go. This was something my father wasafraid to do. Although he knew I would have tofall to learn how to ride, he couldn’t bear to feelresponsible for it.

Paul enjoyed sailing and had a boat, a FlyingJunior, that he kept on Fire Island where my familyspent the summers. I recall him strapping a lifejacket on me over my protests that I was a biggirl and I knew how to swim even in water overmy head. Paul encouraged me to learn how to sailmyself and taught me how lift helps pull a boatforward when it is going upwind. When I provedto be much better at sailing than I was at tennis or


Paul, a lifelong pianist.

basketball, he sold his beloved Flying Junior andbought me a Sunfish, the style of boat that wassailed by the other teenage kids.

In the winter my father would take me iceskating, first at Sky Rink, an indoor rink on thetop floor of a skyscraper, and later at the WollmanRink in Central Park. He knew a few tricks, and wewould do figure eights and little leaps he calledbunny hops. I wanted him to teach me how to twirllike the figure skaters we would watch together.He wasn’t able to do that trick, but he explained tome again about angular momentum and pointedout how the skaters would spin faster when theypulled their arms in close to their bodies. I lovedholding hands with my father as we skated aroundand around the rink. I continued to hold his handwhile we skated, many years after I no longerneeded it for balance.

My daughter is turning four years old soon,and her big gift is going to be her first two-wheelbicycle. I’m sad that my father won’t be aroundto see her learn to ride it. I was sad when I tookmy children ice skating last week. I told both mychildren that their grandfather could have taughtthem how to skate. I am trying to help my childrenretain their memories of their grandpa, talking tothem about the kind of person he was and thethings he liked to do with them, like helping themput on their shoes.

And so we keep my father’s memory alivethrough our recollections of him, his dedication tohis family and to his work which will be carried on.

The creators of MathJobs.Org welcome you to:

MathPrograms.Org

Announcing...

Service is FREE to applicants.

Institutions pay annually for one program or for multiple programs.

Receive, read, rate, and respond to electronic

applications for your mathematical sciences

programs, such as undergraduate summer research programs and travel grant competitions.

Customize your settings and control the application form; also set secure access for the admissions committee.

Enter program announcements for public display.

Download data to personal computers for use in word processing and spreadsheets or as a full permanent storage fi le.

http://MathPrograms.Orghttp://MathJobs.Org

Documents

Paul Roesel Garabedian (1927–2010) · 2014. 2. 10. · Paul Roesel Garabedian (1927–2010) Jerry L. Kazdan Paul Garabedian, 2000. Background Mathematics was in Paul’s genes: