Celebratio Mathematica

Shiing-Shen Chern

Interview with Shiing Shen Chern

by Allyn Jackson

Shi­ing Shen Chern is one of the greatest liv­ing geo­met­ers. He was born on Oc­to­ber 28, 1911, in Jia Xin, China. His fath­er had a de­gree in law and worked for the gov­ern­ment. When Chern was a young­ster, China was just start­ing to es­tab­lish West­ern-style col­leges and uni­versit­ies. He entered Nankai Uni­versity at the age of fif­teen and was drawn to phys­ics but, find­ing him­self clumsy with ex­per­i­ment­al work, even­tu­ally settled on math­em­at­ics. In 1930 he entered the gradu­ate school of Tsinghua Uni­versity, where there were a num­ber of Chinese math­em­aticians who had ob­tained Ph.D.s in the West. Among these was Guangy­uan Sun (Dan Sun), who had been a stu­dent of E. P. Lane at the Uni­versity of Chica­go. Some twenty years later, Chern be­came Lane’s suc­cessor at Chica­go. In 1932 Wil­helm Blasch­ke, a math­em­atician from the Uni­versity of Ham­burg, vis­ited Tsinghua, and his lec­tures had a great in­flu­ence on Chern.

No­tices: After your stud­ies in China, you de­cided to get a doc­tor­ate in the West.

Chern: I was giv­en a fel­low­ship to come to the West by Tsinghua Uni­versity in 1934, after one year of as­sist­ant­ship and three years in the gradu­ate school. I de­cided Europe was a bet­ter place than the United States. The nor­mal thing to do was to come to the United States, but I was not in­ter­ested in Prin­ceton or Har­vard.

No­tices: Why not?

Chern: Not so good. I wanted to be a geo­met­er. The United States did not have the type of geo­metry I wanted to work on, so I went to Europe. At that time, I think I had the ad­vant­age that al­though I was a be­gin­ning stu­dent, I had some ideas about what I wanted, about the math­em­at­ic­al situ­ation in the world, who are the good math­em­aticians, where are the best cen­ters. My eval­u­ation could have been wrong, but I had my ideas. And I de­cided to go to Ham­burg. In fact, it was a very good choice. At the end of the nine­teenth cen­tury the cen­ter of sci­ence was Ger­many, in­clud­ing math­em­at­ics. And the cen­ter of math­em­at­ics in Ger­many was Göttin­gen, with Ber­lin and Mu­nich not far be­hind. And Par­is, of course, was al­ways a cen­ter.

I gradu­ated from Tsinghua gradu­ate school in 1934. In 1933 Hitler took power in Ger­many, and there was great move­ment in Ger­man uni­versit­ies. The Jew­ish pro­fess­ors were re­moved, and so on, and Göttin­gen col­lapsed. And Ham­burg be­came a very good place. Ham­burg was a new uni­versity foun­ded after the First World War. It was not so dis­tin­guished, but the math de­part­ment was ex­cel­lent. So I went there at the right time.

It was in Ham­burg that Chern first came in­to con­tact with the work of Elie Cartan, which had a pro­found in­flu­ence on Chern’s ap­proach to math­em­at­ics. At that time, Erich Kähler, a Privat­dozent at Ham­burg, was one of the main pro­ponents of Cartan’s ideas. Kähler had just writ­ten a book, the main the­or­em of which is now known as the Cartan–Kähler the­or­em, and he or­gan­ized a sem­in­ar in Ham­burg. On the first day of the sem­in­ar all of the full pro­fess­ors — Blasch­ke, Emil Artin, and Erich Hecke — at­ten­ded.

Chern: [The sem­in­ar] looked like a kind of cel­eb­ra­tion. The classroom was filled, and the book had just come out. Kähler came in with a pile of the books and gave every­body a copy. But the sub­ject was dif­fi­cult, so after a num­ber of times, people didn’t come any­more. I think I was es­sen­tially the only one who stayed till the end. I think I stayed till the end be­cause I fol­lowed the sub­ject. Not only that, I was writ­ing a thes­is ap­ply­ing the meth­ods to an­oth­er prob­lem, so the sem­in­ar was of great im­port­ance to me. I even came to see Herr Kähler after the sem­in­ar. A lot of times we had lunch to­geth­er. There was a res­taur­ant near the in­sti­tute, and we had lunch to­geth­er and talked about all kinds of things. My Ger­man was very lim­ited, and Herr Kähler did not speak Eng­lish at that time. Any­way, we got along. So, as a res­ult, I fin­ished my thes­is very quickly.

Every­body knew that Elie Cartan was the greatest dif­fer­en­tial geo­met­er. But his writ­ings were very dif­fi­cult. One reas­on is that he uses the so-called ex­ter­i­or dif­fer­en­tial cal­cu­lus. And in our sub­ject of dif­fer­en­tial geo­metry, where you talk about man­i­folds, one dif­fi­culty is that the geo­metry is de­scribed by co­ordin­ates, but the co­ordin­ates do not have mean­ing. They are al­lowed to un­der­go trans­form­a­tion. And in or­der to handle this kind of situ­ation, an im­port­ant tool is the so-called tensor ana­lys­is, or Ricci cal­cu­lus, which was new to math­em­aticians. In math­em­at­ics you have a func­tion, you write down the func­tion, you cal­cu­late, or you add, or you mul­tiply, or you can dif­fer­en­ti­ate. You have something very con­crete. In geo­metry the geo­met­ric situ­ation is de­scribed by num­bers, but you can change your num­bers ar­bit­rar­ily. So to handle this, you need the Ricci cal­cu­lus.

Chern had a three-year fel­low­ship, but fin­ished his de­gree after only two years. For the third year, Blasch­ke ar­ranged for Chern to go to Par­is to work with Cartan. Chern did not un­der­stand much French, and Cartan spoke only French. On their first meet­ing, Cartan gave Chern two prob­lems to do. After some time they happened to meet on the stairs at the In­sti­tut Henri Poin­caré, and Chern told Cartan he had been un­able to do the prob­lems. Cartan asked Chern to come to his of­fice to dis­cuss them. Chern there­after came reg­u­larly to Cartan’s of­fice hours, which of­ten at­trac­ted a large num­ber of vis­it­ors who wanted to meet with the fam­ous math­em­atician. After a few months, Cartan in­vited Chern to meet with him at his home.

Chern: Usu­ally the day after [meet­ing with Cartan] I would get a let­ter from him. He would say, “After you left, I thought more about your ques­tions…” — he had some res­ults, and some more ques­tions, and so on. He knew all these pa­pers on simple Lie groups, Lie al­geb­ras, all by heart. When you saw him on the street, when a cer­tain is­sue would come up, he would pull out some old en­vel­ope and write something and give you the an­swer. And some­times it took me hours or even days to get the same an­swer. I saw him about once every two weeks, and clearly I had to work very hard. This las­ted for a year, till 1937, and then I went back to China.

When he re­turned to China, Chern be­came a pro­fess­or of math­em­at­ics at Tsinghua, but the Sino-Ja­pan­ese War severely lim­ited his con­tact with math­em­aticians out­side China. He wrote to Cartan about his situ­ation, and Cartan sent a box of his re­prints, in­clud­ing some old pa­pers. Chern spent a great deal of time read­ing and think­ing about them. Des­pite his isol­a­tion Chern con­tin­ued to pub­lish, and his pa­pers at­trac­ted in­ter­na­tion­al at­ten­tion. In 1943 he re­ceived an in­vit­a­tion from Os­wald Veblen to come to the In­sti­tute for Ad­vanced Study in Prin­ceton. Be­cause of the war it took Chern a week to reach the United States by mil­it­ary air­craft. Dur­ing his two years at the In­sti­tute, Chern com­pleted his proof of the gen­er­al­ized Gauss–Bon­net The­or­em, which ex­presses the Euler char­ac­ter­ist­ic of a closed Rieman­ni­an man­i­fold of ar­bit­rary di­men­sion as a cer­tain in­teg­ral of curvature terms over the man­i­fold. The the­or­em’s mar­riage of the loc­al geo­metry to glob­al to­po­lo­gic­al in­vari­ants rep­res­ents a deep theme in much of Chern’s work.

No­tices: What among your math­em­at­ic­al works do you con­sider the most im­port­ant?

Chern: I think the dif­fer­en­tial geo­metry of fiber spaces. You see, math­em­at­ics goes in two dif­fer­ent dir­ec­tions. One is the gen­er­al the­ory. For in­stance, every­body has to study point set to­po­logy, every­body has to study some al­gebra, so they get a gen­er­al found­a­tion, a gen­er­al the­ory that cov­ers al­most all math­em­at­ics. And then there are cer­tain top­ics which are spe­cial, but they play such an im­port­ant role in ap­plic­a­tion of math­em­at­ics that you have to know them very well, such as the gen­er­al lin­ear group, or even the unit­ary group. They come out every­where, wheth­er you do phys­ics or do num­ber the­ory. So there is the gen­er­al the­ory, which con­tains cer­tain beau­ti­ful things. And the fiber space is one of these. You have a space whose fibers are very simple, are clas­sic­al spaces, but they are put to­geth­er in a cer­tain way. And that is a really fun­da­ment­al concept. Now, in fiber spaces the no­tion of a con­nec­tion be­comes im­port­ant, and that’s where my work comes in. Usu­ally the best math­em­at­ic­al work com­bines some the­ory with some very spe­cial prob­lems. The spe­cial prob­lems call for de­vel­op­ment in the gen­er­al the­ory. And I used this idea to give the first proof of the Gauss–Bon­net for­mula.

The Gauss–Bon­net for­mula is one of the im­port­ant, fun­da­ment­al for­mu­las, not only in dif­fer­en­tial geo­metry, but in the whole of math­em­at­ics. Be­fore I came to Prin­ceton [in 1943] I had thought about it, so the de­vel­op­ment in Prin­ceton was in a sense very nat­ur­al. I came to Prin­ceton and I met An­dré Weil. He had just pub­lished his pa­per with Al­lendo­er­fer. 1 Weil and I be­came good friends, so we nat­ur­ally dis­cussed the Gauss–Bon­net for­mula. And then I got my proof. I think this is one of my best works, be­cause it solved an im­port­ant, a fun­da­ment­al, clas­sic­al prob­lem, and the ideas were very new. And to carry out the ideas you need some tech­nic­al in­genu­ity. It’s not trivi­al. It’s not something where once you have the ideas you can carry it out. It’s subtle. So I think it’s a very good piece of work.

No­tices: One of your oth­er most im­port­ant works was the de­vel­op­ment of char­ac­ter­ist­ic classes.

Chern: The char­ac­ter­ist­ic classes — they are not that im­press­ive. Char­ac­ter­ist­ic classes are very im­port­ant, be­cause these are the fun­da­ment­al in­vari­ants of fiber spaces. Fiber spaces are very im­port­ant; there­fore, the char­ac­ter­ist­ic classes come up. But it did not take me that much thought. They come up of­ten, even the char­ac­ter­ist­ic class \( c_1 \) comes up, be­cause in elec­tri­city and mag­net­ism you need the no­tion of com­plex line bundles. And the com­plex line bundles lead to \( c_1 \), which comes up in Dir­ac’s pa­per on quantum elec­tro­dynam­ics. Of course, Dir­ac did not call it \( c_1 \). When \( c_1 \) is not zero, that’s re­lated to the so-called mono­pole. So char­ac­ter­ist­ic classes are im­port­ant in the sense that they come up nat­ur­ally in con­crete prob­lems, fun­da­ment­al prob­lems.

No­tices: When you first de­veloped the the­ory of Chern classes in the 1940s, were you aware of Pontry­agin’s work and the fact that the Pontry­agin classes of a real bundle could be re­covered from the Chern classes of its com­plexi­fic­a­tion?

Chern: My main idea is that you should do to­po­logy or glob­al geo­metry in the com­plex case. The com­plex case has more struc­ture and is in many ways sim­pler than the real case. So I in­tro­duced the com­plex Chern classes. I read the Pontry­agin pa­pers, but the real case is much more com­plic­ated. I didn’t see his full pa­pers, but I think he made some kind of an­nounce­ment in Dok­lady in Eng­lish. I learned from Hirzebruch the re­la­tions between Chern classes and Pontry­agin classes.

Chern classes can be ex­pressed in terms of the curvature, in terms of the loc­al in­vari­ant. I was mainly in­ter­ested in the re­la­tions between loc­al prop­er­ties and glob­al prop­er­ties. When you study spaces, what you can meas­ure are the loc­al prop­er­ties. It’s very re­mark­able that some loc­al prop­er­ties are re­lated to the glob­al prop­er­ties. The simplest case of the Gauss–Bon­net for­mula is that the sum of angles of a tri­angle is 180 de­grees. It shows up already in very simple cases.

No­tices: You are seen as one of the main ex­po­nents of glob­al dif­fer­en­tial geo­metry. Like Cartan you have worked with dif­fer­en­tial forms and con­nec­tions and so on. But the Ger­man school, of which Wil­helm P. A. Klin­gen­berg is one of the ex­po­nents, does glob­al geo­metry in a dif­fer­ent way. They don’t like to use dif­fer­en­tial forms, they ar­gue with geodesics and com­par­is­on the­or­ems, and so on. How do you see this dif­fer­ence?

Chern: There is no es­sen­tial dif­fer­ence. It’s a his­tor­ic­al de­vel­op­ment. In or­der to do, say, geo­metry on man­i­folds, the stand­ard tech­nique was Ricci cal­cu­lus. The fun­da­ment­al prob­lem was the form prob­lem, which was solved by Lipschitz and Chris­tof­fel, par­tic­u­larly Chris­tof­fel. And the Chris­tof­fel idea went to Ricci, and Ricci has his book on the Ricci cal­cu­lus. So all these people, in­clud­ing Her­mann Weyl, learned math­em­at­ics through the Ricci cal­cu­lus. Tensor ana­lys­is had such an im­port­ant role, so every­body learned it. That’s how every­body star­ted in dif­fer­en­tial geo­metry, with tensor ana­lys­is. But some­how in cer­tain as­pects, the dif­fer­en­tial forms should come in. I usu­ally like to say that vec­tor fields is like a man, and dif­fer­en­tial forms is like a wo­man. So­ci­ety must have two sexes. If you only have one, it’s not enough.

After spend­ing 1943–45 at the In­sti­tute in Prin­ceton, Chern re­turned to China for two years, where he helped to build the In­sti­tute of Math­em­at­ics at the Aca­demia Sin­ica. In 1949 he be­came a pro­fess­or of math­em­at­ics at the Uni­versity of Chica­go and in 1960 moved to the Uni­versity of Cali­for­nia, Berke­ley. After his re­tire­ment in 1979 he con­tin­ued to be act­ive, and in par­tic­u­lar helped launch the Math­em­at­ic­al Sci­ences Re­search In­sti­tute in Berke­ley, serving as its first dir­ect­or from 1981 to 1984. Chern has had forty-one Ph.D. stu­dents. This num­ber does not count the many stu­dents he has had con­tact with on his fre­quent vis­its to China. Be­cause of the Cul­tur­al Re­volu­tion in China, the coun­try lost many tal­en­ted math­em­aticians and the tra­di­tion of math­em­at­ic­al re­search al­most died out. Chern did much to re­gen­er­ate this tra­di­tion. In par­tic­u­lar, he was in­stru­ment­al in start­ing the Nankai In­sti­tute for Math­em­at­ics in Tianjin, China, in 1985.

No­tices: How of­ten do you get back to China?

Chern: In re­cent years I go back every year, usu­ally stay­ing one month or longer. I star­ted the in­sti­tute at Nankai, and the most im­port­ant thing is to get some good young people who will stay in China. In this re­spect we have been suc­cess­ful. Our new fac­ulty in­cludes Yim­in Long (dy­nam­ic­al sys­tems), Wil­li­am Chen (dis­crete math­em­at­ics), Weiping Zhang (in­dex the­ory), and Fuquan Fong (dif­fer­en­tial to­po­logy). There are oth­er very good young people. I think the main ob­struc­tion to the pro­gress of math­em­at­ics in China is the low pay. By the way, the In­ter­na­tion­al Math­em­at­ic­al Uni­on has chosen Beijing for the next In­ter­na­tion­al Con­gress of Math­em­aticians.

No­tices: Do you think that will be a big boost for math­em­at­ics in China?

Chern: Oh, yes. But I think what I am wor­ried about is that there will be too many math­em­aticians in China.

No­tices: It’s a large coun­try; maybe they need a lot of math­em­aticians.

Chern: I think they don’t need too many math­em­aticians. China is a large coun­try, so nat­ur­ally it has a lot of tal­ent, par­tic­u­larly in the smal­ler places. For in­stance, there’s the In­ter­na­tion­al Math­em­at­ic­al Olympi­ad for the high school stu­dents, and China gen­er­ally does very well. In or­der to achieve well in com­pet­i­tions like this, the stu­dents need train­ing, and as a res­ult oth­er top­ics could be ig­nored. Now the par­ents in China want their chil­dren to know more Eng­lish, go in­to busi­ness, and make more money. And these ex­ams don’t give money. One year I think they just did less of this train­ing, and China im­me­di­ately dropped. What do you do for a coun­try with 1.2 bil­lion people? It means that the stand­ard of liv­ing can­not be very high, if you have any so­cial justice.

In 1934, when Chern chose to go to Ger­many for gradu­ate study, geo­metry was a peri­pher­al sub­ject in the United States. By the time of his re­tire­ment in 1979, geo­metry coun­ted as one of the most vi­brant spe­cial­ties on the U.S. math­em­at­ic­al scene. Much of the cred­it for this trans­form­a­tion goes to Chern. Still, he is mod­est about his achieve­ments.

Chern: I don’t think I have big views. I only have small prob­lems. In math­em­at­ics a lot of con­cepts and new ideas come in, and you just ask some ques­tions, you try to get some simple an­swer, and you want to give some proofs.

No­tices: That’s how you get your ideas, just ob­serving things?

Chern: Yes, in most cases you don’t have an idea. And in even more cases your ideas don’t work.

No­tices: You’re de­scrib­ing your­self as a prob­lem solv­er, rather than some­body who builds a the­ory.

Chern: I think the dif­fer­ence is small. Every good math­em­atician has to be a prob­lem solv­er. If you are not a prob­lem solv­er, you only have vague ideas, how can you make a good con­tri­bu­tion? You solve some prob­lems, you use some con­cepts, and the mer­it of math­em­at­ic­al con­tri­bu­tions, you prob­ably have to wait. You can only see it in the fu­ture.

It is very dif­fi­cult to eval­u­ate a math­em­atician or a part of math­em­at­ics. Like the concept of dif­fer­en­ti­ab­il­ity. Some time ago, twenty or thirty years ago, a lot of people just didn’t like dif­fer­en­ti­ab­il­ity. I heard a lot of people who told me per­son­ally, “I’m not in­ter­ested in any math­em­at­ics with a no­tion of dif­fer­en­ti­ab­il­ity.” These are the people who tried to make math­em­at­ics simple. If you ex­clude no­tions in­volving dif­fer­en­ti­ab­il­ity you could ex­clude a lot of math­em­at­ics. But it’s not enough. New­ton and Leib­n­iz, they should play a role. But it’s in­ter­est­ing be­cause there are ideas in math­em­at­ics which are con­tro­ver­sial.

No­tices: Can you give some ex­amples of con­tro­ver­sial ideas in math­em­at­ics?

Chern: One thing is that some of the pa­pers nowadays are too long. Like the clas­si­fic­a­tion of fi­nite simple groups. Who is go­ing to read 1,000 pages of proof? Or even the proof of the 4-col­or prob­lem. I think people have to make math­em­at­ics in­ter­est­ing. I think math­em­at­ics won’t die soon. It will be around for some time, be­cause there are a lot of beau­ti­ful things to be done. And math­em­at­ics is in­di­vidu­al. I don’t be­lieve you can do math­em­at­ics with a group. Ba­sic­ally, it’s in­di­vidu­al. So it’s easy to carry on. Math­em­at­ics does not need much equip­ment. It’s not like oth­er sci­ences. They need more ma­ter­i­al sup­port than math­em­at­ics. So our sub­ject will last for some time. Now, hu­man civil­iz­a­tion, I don’t know how long it will last. That’s a very much big­ger prob­lem. But math­em­at­ics it­self, we will get along for some time.

At the age of eighty-six Chern con­tin­ues to do math­em­at­ics. In re­cent years he has been es­pe­cially in­ter­ested in Finsler geo­metry, which he dis­cussed in a No­tices art­icle two years ago (“Finsler geo­metry is just Rieman­ni­an geo­metry without the quad­rat­ic re­stric­tion”, Septem­ber 1996, pages 959–963).

Chern: Finsler geo­metry is much broad­er than Rieman­ni­an geo­metry and can be treated in an el­eg­ant way. It will be the sub­ject of the ba­sic course on dif­fer­en­tial geo­metry with­in the next ten years in many uni­versit­ies.

I have no dif­fi­culty in math­em­at­ics, so when I do math­em­at­ics, I en­joy it. And there­fore I’m al­ways do­ing math­em­at­ics, be­cause the oth­er things I can­not do. Like now, I am re­tired for many years, and people ask me if I still do math­em­at­ics. And I think my an­swer is, it’s the only thing I can do. There is noth­ing else I can do. And this has been true throughout my life.