#### 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.