Celebratio Mathematica

S. R. Srinivasa Varadhan

Probability and statistics


by Linda Kirby

Srinivasa Varadhan had planned to go in­to in­dustry after gradu­ation and as­sumed that the field of stat­ist­ics would be a bet­ter pre­par­a­tion. However, after meet­ing sev­er­al math­em­at­ic­al prob­ab­il­ists, he real­ized that math­em­at­ics was far more in­ter­est­ing and pro­duct­ive, and switched his sub­ject areas, even­tu­ally win­ning the Abel Prize in Math­em­at­ics in 2007.

Srinivasa Varadhan — or Raghu to his friends — was born on 2 Janu­ary 1940, in Chen­nai (formerly called Madras), In­dia. His fath­er, Ranga Iy­engar, was a sci­ence teach­er who be­came the Prin­cip­al of the Board High School in Pon­neri, a small town about 30 km from Chen­nai.

After his gradu­ation from high school in 1955, Varadhan was ac­cep­ted at Pres­id­ency Col­lege at the Uni­versity of Madras and re­ceived both his Bach­el­or’s (1959) and Mas­ter’s (1960) de­grees in stat­ist­ics. He con­tin­ued his stud­ies at The In­di­an In­sti­tute in Kolk­ata study­ing stat­ist­ic­al con­trol, which left him “totally un­sat­is­fied”. The math­em­aticians Varada­ra­jan, Parthas­arathy and Ranga Rao con­vinced him that math­em­at­ics would be more pro­duct­ive, and they began to work on a prob­lem con­cern­ing prob­ab­il­ity dis­tri­bu­tions on groups. In 1963, Varadhan got his Ph.D. in math­em­at­ics from the In­di­an In­sti­tute, with a thes­is on “Con­vo­lu­tion Prop­er­ties on Dis­tri­bu­tion of To­po­lo­gic­al Groups” [e1]. Per­haps pre-planned, the well-known Rus­si­an math­em­atician Kolmogorov was one of the ex­am­iners of Varadhan’s thes­is. Kolmogorov was so im­pressed with Varadhan that he wrote, “This is not the work of a stu­dent, but of a ma­ture mas­ter”.

In 1963, Varadhan came from In­dia as a postdoc­tor­al fel­low to the Cour­ant In­sti­tute in New York and nev­er left, spend­ing his en­tire pro­fes­sion­al life there, in­clud­ing serving two terms as its dir­ect­or (1980–1984; 1992–1994). He is cur­rently its Frank J.Gould Pro­fess­or of Sci­ence. It is at Cour­ant that he met Daniel Stroock, his fu­ture col­lab­or­at­or.

In 1964, he mar­ried Vasun­dra, who was born in 1947 in Chen­nai but had spent most of the first twelve years of her life in New York. Vasu is a pro­fess­or in me­dia stud­ies at Gal­lat­in School of In­di­vidu­al­ized Study at New York Uni­versity. They had two chil­dren, Go­pal (born in 1969) and his young­er broth­er Ashok. In Au­gust 2001, Go­pal joined Can­tor Fitzger­ald as the Man­aging Dir­ect­or of its in­terest-rate de­riv­at­ives busi­ness, and died in the north tower of the World Trade Cen­ter on Septem­ber 11, 2001.

Varadhan en­joys trav­el­ing and the arts — movies and both clas­sic­al In­di­an and west­ern mu­sic. He par­tic­u­larly likes read­ing Tamil lit­er­at­ure. “Not many people in the world are fa­mil­i­ar with Tamil as a lan­guage. It is a lan­guage which is 2,000-years old, al­most as old as Sanskrit. It is per­haps the only lan­guage which today is not very dif­fer­ent from the way it was 2,000 years ago. So, I can take a book of po­etry writ­ten 2,000 years ago, and I will still be able to read it.” [source]

In 2010, Varadhan re­ceived the Abel Prize in Math­em­at­ics by the Nor­we­gi­an Academy of Sci­ence and Let­ters “for his fun­da­ment­al con­tri­bu­tions to prob­ab­il­ity the­ory and in par­tic­u­lar for cre­at­ing a uni­fied the­ory of large de­vi­ations.”

Prob­ab­il­ity the­ory is the math­em­at­ic­al tool for ana­lyz­ing situ­ations gov­erned by chance. The law of large num­bers, dis­covered by Jac­ob Bernoulli in the eight­eenth cen­tury, shows that the av­er­age out­come of a long se­quence of coin tosses is usu­ally close to the ex­pec­ted value. Yet the un­ex­pec­ted hap­pens, and the ques­tion is, how? The the­ory of large de­vi­ations stud­ies the oc­cur­rence of rare events. This sub­ject has con­crete ap­plic­a­tions to fields as di­verse as phys­ics, bio­logy, eco­nom­ics, stat­ist­ics, com­puter sci­ence, and en­gin­eer­ing.

The law of large num­bers states that the prob­ab­il­ity of a de­vi­ation bey­ond a giv­en level goes to zero. However, for prac­tic­al ap­plic­a­tions, it is cru­cial to know how fast it van­ishes. For ex­ample, what cap­it­al re­serves are needed to keep the prob­ab­il­ity of de­fault of an in­sur­ance com­pany be­low ac­cept­able levels? In ana­lyz­ing such ac­tu­ar­ial “ru­in prob­lems,” Har­ald Cramér dis­covered in 1937 that stand­ard ap­prox­im­a­tions based on the Cent­ral Lim­it The­or­em (as visu­al­ized by the bell curve) are ac­tu­ally mis­lead­ing. He then found the first pre­cise es­tim­ates of large de­vi­ations for a se­quence of in­de­pend­ent ran­dom vari­ables. It took 30 years be­fore Varadhan dis­covered the un­der­ly­ing gen­er­al prin­ciples and began to demon­strate their tre­mend­ous scope, far bey­ond the clas­sic­al set­ting of in­de­pend­ent tri­als. [Abel Prize web­site]

Upon Varadhan’s re­ceipt of the Abel prize, the Pres­id­ent of New York Uni­versity, John Sex­ton, said:

We are so happy and proud of Raghu. Not only is he an out­stand­ing schol­ar, he is also a kind and won­der­ful col­league, a de­voted teach­er, and an ex­em­plary “Uni­versity cit­izen,” serving with ded­ic­a­tion and pro­fes­sion­al­ism as dir­ect­or of the Cour­ant In­sti­tute and on such bod­ies as the Uni­versity Sen­ate. This dis­tinc­tion is a well-de­served hon­our for a fac­ulty mem­ber whose mod­esty and dis­cre­tion are al­most as great as his schol­arly con­tri­bu­tions. In the time that Raghu has been at NYU, our Uni­versity has changed a great deal, but it is the per­sist­ent pres­ence of schol­ars such as he that has en­abled us to build NYU in­to what it is today and to con­tin­ue to at­tract top schol­ars and re­search­ers to our midst.

In 1996, Varadhan and Daniel Stroock re­ceived the Amer­ic­an Math­em­at­ic­al So­ci­ety’s Leroy Steele Prize for fun­da­ment­al con­tri­bu­tions to re­search. Their cita­tion [e6] reads,

To Daniel Stroock and Srinivasa Varadhan for their four pa­pers “Dif­fu­sion Pro­cesses with con­tinu­ous coef­fi­cients, I and II” (1969) [e3], [e2], “On the sup­port of dif­fu­sion pro­cesses with ap­plic­a­tions to the strong max­im­um prin­ciple” (1970) [e4], “Mul­ti­di­men­sion­al dif­fu­sion pro­cesses” (1979) [e5], in which they in­tro­duced the new concept of a mar­tin­gale solu­tion to a stochast­ic dif­fer­en­tial equa­tion, en­abling them to prove ex­ist­ence, unique­ness, and oth­er im­port­ant prop­er­ties of solu­tions to equa­tions which could not be treated be­fore by purely ana­lyt­ic meth­ods; their for­mu­la­tion has been widely used to prove con­ver­gence of vari­ous pro­cesses to dif­fu­sions.

In ad­di­tion, Varadhan is a mem­ber of the Na­tion­al Academy of Sci­ences, the Amer­ic­an Academy of Arts and Sci­ences, the Roy­al So­ci­ety of Lon­don, and the Third World Academy of Sci­ences.

He also has hon­or­ary de­grees from Uni­versité Pierre et Mar­ie Curie in Par­is (2003) and from the In­di­an Stat­ist­ic­al In­sti­tute in Kolk­ata, In­dia (2004).

In ad­di­tion, Varadhan re­ceived the fol­low­ing prizes: The Birk­hoff Prize in 1994 from the Amer­ic­an Math­em­at­ic­al So­ci­ety; The Mar­garet and Her­man Sokol Award of New York Uni­versity’s Fac­ulty of Arts and Sci­ence in 1995; an Al­fred P. Sloan fel­low­ship, and a Gug­gen­heim Fel­low­ship. He is a fel­low of the In­sti­tute of Math­em­at­ic­al Stat­ist­ics of the In­di­an Academy of Sci­ences.

In 2008, the gov­ern­ment of In­dia awar­ded him the Padma Bhush­an, one of the coun­try’s highest ci­vil­ian hon­ors.

Daniel Stroock de­scribes [e7] his co-au­thor and col­lab­or­at­or as fol­lows:

I am not go­ing to claim that Varadhan does not en­joy his suc­cess; he does. Nor am I about to say that he is some sort of saint; we could nev­er have be­come friends if he were. Non­ethe­less, what dis­tin­guishes Varadhan from nearly all the oth­er gif­ted people whom I have met is the re­mark­able com­mand he ex­er­cises over his own gift. In par­tic­u­lar, he has learned how to pre­vent his un­usu­al in­tel­lec­tu­al powers from pois­on­ing his re­la­tions with less­er in­tel­lects. For ex­ample, Varadhan can tol­er­ate be­ing wrong, at least oc­ca­sion­ally. In ad­di­tion, he is not one of the many math­em­at­ic­al princes who es­pouse the no­tion that all their ob­lig­a­tions to hu­man­ity can be met through their con­tri­bu­tions to math­em­at­ic­al re­search.