Celebratio Mathematica

Alberto Pedro Calderón

Mathematician Alberto Calderón, 1920–1998

from the University of Chicago News Office

Al­berto Calderón, a Uni­versity of Chica­go math­em­atician widely re­garded as one of the most in­flu­en­tial math­em­aticians of the 20th cen­tury, died Thursday in North­west­ern Me­mori­al Hos­pit­al after a brief ill­ness. He was 77 and lived in Chica­go.

Calderón, Uni­versity Pro­fess­or Emer­it­us of Math­em­at­ics, is best known for his con­tri­bu­tions to math­em­at­ic­al ana­lys­is, the large branch of math­em­at­ics that in­cludes cal­cu­lus, in­fin­ite series and the ana­lys­is of func­tions. To­geth­er with his ment­or, Ant­oni Zyg­mund, he foun­ded the Chica­go school of ana­lys­is, the most in­flu­en­tial school in that branch of math­em­at­ics in the 20th cen­tury.

“He was one of the most ori­gin­al and pro­found math­em­at­ic­al ana­lysts of the past 50 years,” said Fe­lix Browder, the Max Ma­son Dis­tin­guished Ser­vice Pro­fess­or Emer­it­us of Math­em­at­ics at the Uni­versity of Chica­go, and a former vice-pres­id­ent of Rut­gers Uni­versity. “Calderón was one of the cent­ral links between two ma­jor areas of math­em­at­ic­al ana­lys­is, namely Four­i­er ana­lys­is and par­tial dif­fer­en­tial equa­tions.He made out­stand­ing con­tri­bu­tions to both fields and laid much of the ground­work for oth­er people’s work in these areas.”

Browder ad­ded, “Calderón was also a man of really re­mark­able up­right char­ac­ter. He was uni­ver­sally re­spec­ted and ad­mired be­cause of his ex­treme prob­ity and gen­er­os­ity.”

Calderón’s many hon­ors in­clude the 1991 Na­tion­al Medal of Sci­ence; the 1989 Wolf Prize; the 1989 Steele Prize from the Amer­ic­an Math­em­at­ic­al So­ci­ety; Ar­gen­tina’s Con­sagra­cion Nacion­al Prize, awar­ded in 1989; and the 1979 Bôcher Me­mori­al Prize from the Amer­ic­an Math­em­at­ic­al So­ci­ety.

Calderón was born in Ar­gen­tina on Sept. 14, 1920. As a child, he was edu­cated in Ar­gen­tina and in Switzer­land, where he first be­came in­ter­ested in math­em­at­ics. He re­ceived a de­gree in civil en­gin­eer­ing at the Uni­versity of Buenos Aires in 1947, where in 1948 he first met world-renowned Uni­versity of Chica­go math­em­atician Ant­oni Zyg­mund, who would be­come his ment­or.

Ac­cord­ing to Robert Fef­fer­man, Louis Block Pro­fess­or and Chair­man of Math­em­at­ics at the Uni­versity of Chica­go, Zyg­mund gave a sem­in­ar in Buenos Aires in which he dis­cussed a proof of an im­port­ant clas­sic­al the­orm that was in­cluded in his cel­eb­rated book. Calderón ap­proached Zyg­mund after the lec­ture and asked him why the proof he had giv­en in the sem­in­ar was so much longer than the one he had out­lined in his book. Zyg­mund told him it was the same proof.

“It turned out that Calderón had a habit of read­ing math­em­at­ics and do­ing proofs on his own,” said Fef­fer­man, “and later check­ing with the book to make sure he’d done it cor­rectly. Only this time he didn’t check the book and it turned out he had come up with a much sim­pler and more el­eg­ant proof than the stand­ard one in Zyg­mund’s book.”

Zyg­mund re­cog­nized Calderón’s bril­liance as a math­em­atician, and brought him back to Chica­go in 1949 as his stu­dent. Calderón com­pleted a thes­is in re­mark­ably short or­der, re­ceiv­ing his Ph.D. in 1950.

In pi­on­eer­ing work with his ment­or, Calderón for­mu­lated a the­ory, now known as the Calderón–Zyg­mund the­ory, of what are called sin­gu­lar in­teg­rals. “These sin­gu­lar in­teg­rals have tre­mend­ous ap­plic­a­tions with­in pure math­em­at­ics, as well as very im­port­ant ap­plic­a­tions in in­dus­tri­al prob­lems,” said Fef­fer­man. “It is un­ques­tion­ably one of the most im­port­ant de­vel­op­ments in ana­lys­is in the 20th cen­tury.”

Sin­gu­lar in­teg­rals are math­em­at­ic­al ob­jects that look in­fin­ite, but when in­ter­preted prop­erly are fi­nite and well-be­haved. Calderón later showed how these sin­gu­lar in­teg­rals could be used to ob­tain es­tim­ates of solu­tions to equa­tions in geo­metry and to ana­lyze func­tions of com­plex vari­ables. He also showed how sin­gu­lar in­teg­rals could provide en­tirely new ways to study par­tial dif­fer­en­tial equa­tions, which are widely used to solve prob­lems in phys­ics and en­gin­eer­ing.

In or­der to in­ter­pret sig­nals or waves, such as those ob­tained when an im­age is rep­res­en­ted by a di­git­al sig­nal, the clas­sic meth­od of ana­lys­is was to con­vert the sig­nals in­to a series of sine or co­sine waves through Four­i­er trans­forms. But it was shown that the tools de­veloped in Calderón–Zyg­mund the­ory could be used to rep­res­ent these sig­nals more ef­fect­ively, break­ing down the com­plic­ated sig­nal in­to smal­ler build­ing blocks called wave­lets. Wave­lets are of­ten far more use­ful than Four­i­er trans­forms be­cause they re­veal a sig­nal’s loc­al­ized be­ha­vi­or much more ef­fect­ively.

When an im­age is trans­mit­ted over a com­mu­nic­a­tion chan­nel, for ex­ample, the rough de­tails can be sent first, then finer tex­tures are ad­ded at smal­ler scales.This is a real-life im­ple­ment­a­tion of the Calderón–Zyg­mund de­com­pos­i­tion.

After his gradu­ation from Chica­go, Calderón served as a vis­it­ing as­so­ci­ate pro­fess­or at Ohio State Uni­versity, a vis­it­ing mem­ber of the In­sti­tute for Ad­vanced Study and then an as­so­ci­ate pro­fess­or at M.I.T. He re­turned to Chica­go as Pro­fess­or of Math­em­at­ics in 1959, and aside from brief peri­ods in which he re­turned to M.I.T. and to Ar­gen­tina, has been at Chica­go ever since. He has also been an hon­or­ary pro­fess­or at the Uni­versity of Buenos Aires since 1975.

In ad­di­tion to his re­search, Calderón’s in­flu­ence as a teach­er ex­ten­ded world­wide. “Calderón was in­stru­ment­al in pro­mot­ing a re­new­al of math­em­at­ic­al spir­it in Spain,” said Miguel de Guzmán, a former stu­dent of Calderón’s who is a math­em­at­ics pro­fess­or at Uni­ver­sid­ad Com­plutense in Mad­rid. “He brought many Span­ish stu­dents to Chica­go who worked to­geth­er with him and Ant­oni Zyg­mund on Four­i­er ana­lys­is.”

Calderón was a mem­ber of the Na­tion­al Academy of Sci­ences; the Amer­ic­an Academy of Arts and Sci­ences; the Na­tion­al Academy of Ex­act, Phys­ic­al and Nat­ur­al Sci­ences in Ar­gen­tina; the Academie des Sci­ences in France; the Roy­al Academy of Sci­ences in Spain; the Lat­in Amer­ic­an Academy of Sci­ences in Venezuela; and the Third World Academy of Sci­ences in Italy.

Calderón is sur­vived by his wife, noted math­em­atician Al­ex­an­dra Bel­low (née Bag­dasar), re­cently re­tired from North­west­ern Uni­versity, whom he mar­ried in 1989; and two chil­dren from his first mar­riage, Mary Josephine, of St. Charles, Ill., and Pablo, of New York, N.Y. His first wife, Ma­bel (née Molinelli Wells), to whom he was mar­ried for 35 years, died in 1985.