Celebratio Mathematica

Ab­ra­ham Ad­ri­an Al­bert was an out­stand­ing fig­ure in the world of twen­ti­eth-cen­tury al­gebra, and at the same time a states­man and lead­er in the Amer­ic­an math­em­at­ic­al com­munity. He was born in Chica­go on Novem­ber 9, 1905, the son of im­mig­rant par­ents. His fath­er, Eli­as Al­bert, had come to the United States from Eng­land and had es­tab­lished him­self as a re­tail mer­chant. His moth­er, Fan­nie Fradkin Al­bert, had come from Rus­sia. Ad­ri­an Al­bert was the second of three chil­dren, the oth­ers be­ing a boy and a girl; in ad­di­tion, he had a half-broth­er and a half-sis­ter on his moth­er’s side.

Al­bert at­ten­ded ele­ment­ary schools in Chica­go from 1911 to 1914. From 1914 to 1916 the fam­ily lived in Iron Moun­tain, Michigan, where he con­tin­ued his school­ing. Back in Chica­go, he at­ten­ded Theodore Herzl Ele­ment­ary School, gradu­at­ing in 1919, and the John Mar­shall High School, gradu­at­ing in 1922. In the fall of 1922 he entered the Uni­versity of Chica­go, the in­sti­tu­tion with which he was to be as­so­ci­ated for vir­tu­ally the rest of his life. He was awar­ded the Bach­el­or of Sci­ence, Mas­ter of Sci­ence, and Doc­tor of Philo­sophy in three suc­cess­ive years: 1926, 1927, and 1928.

On Decem­ber 18, 1927, while com­plet­ing his dis­ser­ta­tion, he mar­ried Frieda Dav­is. Theirs was a happy mar­riage, and she was a stal­wart help to him throughout his ca­reer. She re­mains act­ive in the Uni­versity of Chica­go com­munity and in the life of its De­part­ment of Math­em­at­ics. They had three chil­dren: Alan, Roy, and Nancy. Tra­gic­ally, Roy died in 1958 at the early age of twenty-three. There are five grand-chil­dren.

Le­onard Eu­gene Dick­son was at the time the dom­in­ant Amer­ic­an math­em­atician in the fields of al­gebra and num­ber the­ory. He had been on the Chica­go fac­ulty since al­most its earli­est days. He was a re­mark­ably en­er­get­ic and force­ful man (as I can per­son­ally testi­fy, hav­ing been a stu­dent in his num­ber the­ory course years later). His in­flu­ence on Al­bert was con­sid­er­able and set the course for much of his sub­sequent re­search.

Dick­son’s im­port­ant book, Al­geb­ras and Their Arith­met­ics [e1] (Chica­go: Univ. of Chica­go Press, 1923), had re­cently ap­peared in an ex­pan­ded Ger­man trans­la­tion [e2] (Zurich: Orell Füss­li, 1927). The sub­ject of al­geb­ras had ad­vanced to the cen­ter of the stage. It con­tin­ues to this day to play a vi­tal role in many branches of math­em­at­ics and in oth­er sci­ences as well.

An al­gebra is an ab­stract math­em­at­ic­al en­tity with ele­ments and op­er­a­tions ful­filling the fa­mil­i­ar laws of al­gebra, with one im­port­ant qual­i­fic­a­tion — the com­mut­at­ive law of mul­ti­plic­a­tion is waived. (More care­fully, I should have said that this is an as­so­ci­at­ive al­gebra; non-as­so­ci­at­ive al­geb­ras will play an im­port­ant role later in this mem­oir.) Early in the twen­ti­eth cen­tury, fun­da­ment­al res­ults of J. H. M. Wed­der­burn had cla­ri­fied the nature of al­geb­ras up to the clas­si­fic­a­tion of the ul­ti­mate build­ing blocks, the di­vi­sion al­geb­ras. Ad­vances were now needed on two fronts. One wanted the­or­ems val­id over any field (every al­gebra has an un­der­ly­ing field of coef­fi­cient — a num­ber sys­tem of which the lead­ing ex­amples are the real num­bers, the ra­tion­al num­bers, and the in­tegers mod \( p \)). On the oth­er front, one sought to clas­si­fy di­vi­sion al­geb­ras over the field of ra­tion­al num­bers.

Al­bert at once be­came ex­traordin­ar­ily act­ive on both bat­tle­fields. His first ma­jor pub­lic­a­tion was an im­prove­ment of the second half of his Ph.D. thes­is [1]; it ap­peared in 1929 un­der the title “A de­term­in­a­tion of all nor­mal di­vi­sion al­geb­ras in six­teen units” [2]. The hall­marks of his math­em­at­ic­al per­son­al­ity were already vis­ible. Here was a tough prob­lem that had de­feated his pre­de­cessors; he at­tacked it with tenacity till it yiel­ded. One can ima­gine how de­lighted Dick­son must have been. This work won Al­bert a pres­ti­gi­ous postdoc­tor­al Na­tion­al Re­search Coun­cil Fel­low­ship, which he used in 1928 and 1929 at Prin­ceton and Chica­go.

I shall briefly ex­plain the nature of Al­bert’s ac­com­plish­ment. The di­men­sion of a di­vi­sion al­gebra over its cen­ter is ne­ces­sar­ily a square, say \( n^2 \). The case \( n = 2 \) is easy. A good deal harder is the case \( n = 3 \), handled by Wed­der­burn. Now Al­bert cracked the still harder case, \( n = 4 \). One in­dic­a­tion of the mag­nitude of the res­ult is the fact that at this writ­ing, nearly fifty years later, the next case \( (n = 5) \) re­mains mys­ter­i­ous.

In the hunt for ra­tion­al di­vi­sion al­geb­ras, Al­bert had stiff com­pet­i­tion. Three top Ger­man al­geb­ra­ists (Richard Brauer, Helmut Hasse, and Emmy No­eth­er) were after the same big game (just a little later the ad­vent of the Nazis brought two-thirds of this stel­lar team to the United States.) It was an un­equal battle, and Al­bert was nosed out in a photo fin­ish. In a joint pa­per [3] with Hasse pub­lished in 1932 the full his­tory of the mat­ter was set out, and one can see how close Al­bert came to win­ning.

Let me re­turn to 1928–1929, his first postdoc­tor­al year. At Prin­ceton Uni­versity a for­tu­nate con­tact took place. So­lomon Lef­schetz noted the pres­ence of this prom­ising young­ster, and en­cour­aged him to take a look at Riemann matrices. These are matrices that arise in the the­ory of com­plex man­i­folds; the main prob­lems con­cern­ing them had re­mained un­solved for more than half a cen­tury. The pro­ject was per­fect for Al­bert, for it con­nec­ted closely with the the­ory of al­geb­ras he was so suc­cess­fully de­vel­op­ing. A series of pa­pers en­sued, cul­min­at­ing in com­plete solu­tions of the out­stand­ing prob­lems con­cern­ing Riemann matrices. For this work he re­ceived the Amer­ic­an Math­em­at­ic­al So­ci­ety’s 1939 Cole prize in al­gebra.

From 1929 to 1931 he was an in­struct­or at Columbia Uni­versity. Then, the young couple, ac­com­pan­ied by a baby boy less than a year old, hap­pily re­turned to the Uni­versity of Chica­go. He rose stead­ily through the ranks: as­sist­ant pro­fess­or in 1931, as­so­ci­ate pro­fess­or in 1937, pro­fess­or in 1941, chair­man of the De­part­ment of Math­em­at­ics from 1958 to 1962, and dean of the Di­vi­sion of Phys­ic­al Sci­ences from 1962 to 1971. In 1960 he re­ceived a Dis­tin­guished Ser­vice Pro­fess­or­ship, the highest hon­or that the Uni­versity of Chica­go can con­fer on a fac­ulty mem­ber; ap­pro­pri­ately, it bore the name of E. H. Moore, chair­man of the De­part­ment from its first day un­til 1927.

The dec­ade of the 1930s saw a cre­at­ive out­burst. Ap­prox­im­ately sixty pa­pers flowed from his pen. They covered a wide range of top­ics in al­gebra and the the­ory of num­bers bey­ond those I have men­tioned. Some­how, he also found the time to write two im­port­ant books. Mod­ern High­er Al­gebra [5] (1937) was a widely used text­book — but it is more than a text­book. It re­mains in print to this day, and on cer­tain sub­jects it is an in­dis­pens­able ref­er­ence. Struc­ture of Al­geb­ras [6] (1939) was his defin­it­ive treat­ise on al­geb­ras and formed the basis for his 1939 Col­loqui­um Lec­tures to the Amer­ic­an Math­em­at­ic­al So­ci­ety. There have been later books on al­geb­ras, but none has re­placed Struc­ture of Al­geb­ras.

The aca­dem­ic year 1933–1934 was again spent in Prin­ceton, this time at the newly foun­ded In­sti­tute for Ad­vanced Study. Again, there were fruit­ful con­tacts with oth­er math­em­aticians. Al­bert has re­cor­ded that he found Her­mann Weyl’s lec­tures on Lie al­geb­ras stim­u­lat­ing. An­oth­er thing that happened was that Al­bert was in­tro­duced to Jordan al­geb­ras.

The phys­i­cist Pas­cu­al Jordan had sug­ges­ted that a cer­tain kind of al­gebra, in­spired by us­ing the op­er­a­tion \( xy + yx \) in an as­so­ci­at­ive al­gebra, might be use­ful in quantum mech­an­ics. He en­lis­ted von Neu­mann and Wign­er in the en­ter­prise, and in a joint pa­per they in­vest­ig­ated the struc­ture in ques­tion. But a cru­cial point was left un­re­solved; Al­bert sup­plied the miss­ing the­or­em. The pa­per ap­peared in 1934 and was en­titled “On a cer­tain al­gebra of quantum mech­an­ics” [4]. A seed had been planted that Al­bert was to har­vest a dec­ade later.

Let me jump ahead chro­no­lo­gic­ally to fin­ish the story of Jordan al­geb­ras. I can add a per­son­al re­col­lec­tion. I ar­rived in Chica­go in early Oc­to­ber 1945. Per­haps on my very first day, per­haps a few days later, I was in Al­bert’s of­fice dis­cuss­ing some routine mat­ter. His stu­dent Daniel Zel­in­sky entered. A tor­rent of words poured out, as Al­bert told him how he had just cracked the the­ory of spe­cial Jordan al­geb­ras. His en­thu­si­asm was de­light­ful and con­ta­gious. I got in­to the act and we had a spir­ited dis­cus­sion. It res­ul­ted in arous­ing in me an en­dur­ing in­terest in Jordan al­geb­ras.

About a year later, in 1946, his pa­per ap­peared. It was fol­lowed by “A struc­ture the­ory for Jordan al­geb­ras” [9] (1947) and “A the­ory of power-as­so­ci­at­ive com­mut­at­ive al­geb­ras” [11] (1950). These three pa­pers cre­ated a whole sub­ject; it was an achieve­ment com­par­able to his study of Riemann matrices.

World War II brought changes to the Chica­go cam­pus. The Man­hat­tan Pro­ject took over Eck­hart Hall, the math­em­at­ics build­ing (the self-sus­tain­ing chain re­ac­tion of Decem­ber 1942 took place a block away). Sci­ent­ists in all dis­cip­lines, in­clud­ing math­em­at­ics, answered the call to aid the war ef­fort against the Ax­is. A num­ber of math­em­aticians as­sembled in an Ap­plied Math­em­at­ics Group at North­west­ern Uni­versity, where Al­bert served as as­so­ci­ate dir­ect­or dur­ing 1944 and 1945. At that time, I was a mem­ber of a sim­il­ar group at Columbia, and our first sci­entif­ic in­ter­change took place. It con­cerned a math­em­at­ic­al ques­tion arising in aer­i­al pho­to­graphy; he gently guided me over the pit­falls I was en­coun­ter­ing.

Al­bert be­came in­ter­ested in cryp­to­graphy. On Novem­ber 22, 1941, he gave an in­vited ad­dress at a meet­ing of the Amer­ic­an Math­em­at­ic­al So­ci­ety in Man­hat­tan, Kan­sas, en­titled “Some math­em­at­ic­al as­pects of cryp­to­graphy.”1 After the war he con­tin­ued to be act­ive in the fields in which he had be­come an ex­pert.

In 1942 he pub­lished a pa­per en­titled “Non-as­so­ci­at­ive al­geb­ras” [8], [7]. The date of re­ceipt was Janu­ary 5, 1942, but he had already presen­ted it to the Amer­ic­an Math­em­at­ic­al So­ci­ety on Septem­ber 5, 1941, and he had lec­tured on the sub­ject at Prin­ceton and Har­vard dur­ing March of 1941. It seems fair to name one of these present­a­tions the birth date of the Amer­ic­an school of non-as­so­ci­at­ive al­geb­ras, which he single­han­dedly foun­ded. He was act­ive in it him­self for a quarter of a cen­tury and the school con­tin­ues to flour­ish.

Al­bert in­vest­ig­ated just about every as­pect of non-as­so­ci­at­ive al­geb­ras. At times a par­tic­u­lar line of at­tack failed to ful­fill the prom­ise it had shown; he would then ex­er­cise his sound in­stinct and good judg­ment by shift­ing the as­sault to a dif­fer­ent area. In fact, he re­peatedly dis­played an un­canny knack for se­lect­ing pro­jects which later turned out to be well con­ceived as the fol­low­ing three cases il­lus­trate.

1. In the 1942 pa­per [8] he in­tro­duced the new concept of iso­topy. Much later it was found to be ex­actly what was needed in study­ing col­lin­eations of pro­ject­ive planes.

2. In a se­quence of pa­pers that began in 1952 with “On non-as­so­ci­at­ive di­vi­sion al­geb­ras,” [13] he in­ven­ted and stud­ied twis­ted fields. At the time, one might have thought that this was merely an ad­di­tion to the list of known non-as­so­ci­at­ive di­vi­sion al­geb­ras, a list that was already large. Just a few days be­fore this para­graph was writ­ten, Giam­paolo Menichetti pub­lished a proof that every three-di­men­sion­al di­vi­sion al­gebra over a fi­nite field is either as­so­ci­at­ive or a twis­ted field, show­ing con­clus­ively that Al­bert had hit on a key concept.

3. In a pa­per that ap­peared in 1953, Er­win Klein­feld clas­si­fied all simple al­tern­at­ive rings. Vi­tal use was made of two of Al­bert’s pa­pers: “Ab­so­lute-val­ued al­geb­ra­ic al­geb­ras” [10] (1949) and “On simple al­tern­at­ive rings” [12] (1952). I re­mem­ber hear­ing Klein­feld ex­claim “It’s amaz­ing! He proved ex­actly the right things.”

The post­war years were busy ones for the Al­berts. Just the job to be done at the Uni­versity would have ab­sorbed all the en­er­gies of a less­er man. Mar­shall Har­vey Stone was lured from Har­vard in 1946 to as­sume the chair­man­ship of the Math­em­at­ics De­part­ment. Soon Eck­hart Hall was hum­ming, as such world-fam­ous math­em­aticians as Shi­ing-Shen Chern, Saun­ders Mac Lane, An­dré Weil, and Ant­oni Zyg­mund joined Al­bert and Stone to make Chica­go an ex­cit­ing cen­ter. Al­bert taught courses at all levels, dir­ec­ted his stream of Ph.D.s, main­tained his own pro­gram of re­search, and helped to guide the De­part­ment and the Uni­versity at large in mak­ing wise de­cisions. Even­tu­ally, in 1958, he ac­cep­ted the chal­lenge of the Chair­man­ship. The main stamp he left on the De­part­ment was a pro­ject dear to his heart: main­tain­ing a lively flow of vis­it­ors and re­search in­struct­ors, for whom he skill­fully got sup­port in the form of re­search grants. The Uni­versity co­oper­ated by mak­ing an apart­ment build­ing avail­able to house the vis­it­ors. Af­fec­tion­ately called “the com­pound,” the mod­est build­ing has been the birth­place of many a fine the­or­em. Es­pe­cially mem­or­able was the aca­dem­ic year 1960–1961 when Wal­ter Feit and John Thompson, vis­it­ing for the en­tire year, made their big break­through in fi­nite group the­ory by prov­ing that all groups of odd or­der are solv­able.

Early in his second three-year term as chair­man, Al­bert was asked to as­sume the de­mand­ing post of Dean of the Di­vi­sion of Phys­ic­al Sci­ences. He ac­cep­ted, and served for nine years. The new dean was able to keep his math­em­at­ics go­ing. In 1965 he re­turned to his first love: as­so­ci­at­ive di­vi­sion al­geb­ras. His re­tir­ing pres­id­en­tial ad­dress to the Amer­ic­an Math­em­at­ic­al So­ci­ety, “On as­so­ci­at­ive di­vi­sion al­gebra” [14] presen­ted the state of the art as of 1968.

Re­quests for his ser­vices from out­side the Uni­versity were wide­spread and fre­quent. A full tab­u­la­tion would be long in­deed. Here is a par­tial list: con­sult­ant, Rand Cor­por­a­tion; con­sult­ant, Na­tion­al Se­cur­ity Agency; trust­ee, In­sti­tute for Ad­vanced Study; trust­ee, In­sti­tute for De­fense Ana­lyses, 1969–1972, and dir­ect­or of its Com­mu­nic­a­tions Re­search Di­vi­sion, 1961–1962; chair­man, Di­vi­sion of Math­em­at­ics of the Na­tion­al Re­search Coun­cil, 1952–1955; chair­man, Math­em­at­ics Sec­tion of the Na­tion­al Academy of Sci­ences, 1958–1961; chair­man, Sur­vey of Train­ing and Re­search Po­ten­tial in the Math­em­at­ic­al Sci­ences, 1955–1957 (widely known as the “Al­bert Sur­vey”); pres­id­ent, Amer­ic­an Math­em­at­ic­al So­ci­ety, 1965–1966; par­ti­cipant and then dir­ect­or of Pro­ject SCAMP at the Uni­versity of Cali­for­nia at Los Angeles; dir­ect­or, Pro­ject ALP (nick­named “Ad­ri­an’s little pro­ject”); dir­ect­or, Sum­mer 1957 Math­em­at­ic­al Con­fer­ence at Bowdoin Col­lege, a pro­ject of the Air Force Cam­bridge Re­search Cen­ter; vice-pres­id­ent, In­ter­na­tion­al Math­em­at­ic­al Uni­on; and del­eg­ate IMU Mo­scow Sym­posi­um, 1971 hon­or­ing Vino­gradov’s eighti­eth birth­day (this was the last ma­jor meet­ing he at­ten­ded).

Al­bert’s elec­tion to the Na­tion­al Academy of Sci­ences came in 1943, when he was thirty-sev­en. Oth­er hon­ors fol­lowed. Hon­or­ary de­grees were awar­ded by Notre Dame in 1965, by Ye­shiva Uni­versity in 1968, and by the Uni­versity of Illinois Chica­go Circle Cam­pus in 1971. He was elec­ted to mem­ber­ship in the Brazili­an Academy of Sci­ences (1952) and the Ar­gen­tine Academy of Sci­ences (1963).

In the fall of 1971, he was wel­comed back to the third floor of Eck­hart Hall (the dean’s of­fice was on the first floor). He re­sumed the role of a fac­ulty mem­ber with a zest that sug­ges­ted that it was 1931 all over again. But as the aca­dem­ic year 1971–1972 wore on, his col­leagues and friends were saddened to see that his health was fail­ing. Death came on June 6, 1972. A pa­per pub­lished posthum­ously in 1972 was a fit­ting coda to a life un­selfishly de­voted to the wel­fare of math­em­at­ics and math­em­aticians.

In 1976 the De­part­ment of Math­em­at­ics in­aug­ur­ated an an­nu­al event en­titled the Ad­ri­an Al­bert Me­mori­al Lec­tures. The first lec­turer was his long-time col­league Pro­fess­or Nath­an Jac­ob­son of Yale Uni­versity.