The news that Ngô Báo Châu, a 38-year-old Vietnamese mathematician, was awarded the Fields Medal, an International Medal for outstanding discoveries in Mathematics, at the 26th International Congress of Mathematicians (held on August 19, 2010 in Hyderabad, India) was quite unexpected to many who were unfamiliar with the mathematics scene in Vietnam. In fact, Châu's proof of the Fundamental Lemma in the theory of automorphic forms through the introduction of new algebro-geometric methods had also been listed as one of the top ten scientific discoveries of 2009 by Time magazine.
People now look at mathematics in Vietnam in a different light because of Ngô Báo Châu's outstanding achievement. In addition, Châu's experience and development in the field of mathematics piqued the curiosity and attracted attention of people from all over the world.
When interviewed by Science Times, Ngô Báo Châu said, "I have only proved the Fundamental Lemma, but not everything in the program. I think it may take my entire lifetime to prove everything in the program."
Ngô Báo Châu was born into a Vietnamese family of scholars in 1972. At the age of 15, he was admitted into a class in the Vietnam National University High School specialising in mathematics. Châu received gold medals at both the 29th and 30th International Mathematical Olympiad (IMO) in 1988 and 1989 respectively. He finished his undergraduate study in France and began to research on the Langlands Program as a postgraduate. He then proved the Fundamental Lemma of the Langlands Program in 2008.
The Langlands Program was proposed by Robert Langlands, a Canadian-American mathematician. He developed an ambitious and revolutionary theory that connected two branches of mathematics called number theory and group theory in 1979. In a dazzling set of conjectures and insights, the theory captured deep symmetries associated with equations that involve whole numbers, laying out what is now known as the Langlands Program. Langlands knew that the task of proving the assumptions that underlie his theory would be the work of generations. However, he was convinced that one stepping stone that needed confirmation — dubbed the "fundamental lemma" — would be reasonably straightforward. He, his collaborators and his students were able to prove special cases of the fundamental theorem. Yet proving the general case proved more difficult than Langlands had anticipated — so difficult, in fact, that it took 30 years to finally achieve. Ngô Báo Châu finally proved the lemma through his novel method in 2008.
Ngô Báo Châu's proof of the Fundamental Lemma had been listed as one of the top ten Scientific Discoveries of 2009 by Time magazine. Châu became a full professor of University of Chicago on September 1, 2010.
Recently, Châu made an academic visit to Beijing at the invitation of Professor Shing-Tung Yau, Director of the Mathematical Science Centre of Tsinghua University and Professor of Harvard University. During this period, Châu accepted the interview by Science Times to talk about how he embarked on his academic journey into mathematics.
Showing Mathematical Talent in Vietnam
"I began to really love mathematics after participating in Mathematical Olympiad. After graduating from high school, I decided to pursue mathematics for a career."
Ngô Báo Châu was born in June 1972 in Hanoi, Vietnam. His father, Professor Ngô Huy Cân, was a professor of physics at the Vietnam National Institute of Mechanics. His mother, Trân Luu Vân Hiên, was an associate professor at National Traditional Medicine Hospital in Hanoi. Châu was the only child in the family.
As his father obtained his PhD in applied mathematics in the Soviet Union and worked there for a long time, Châu spent his childhood with his mother's family. By the time his father returned to Vietnam, Châu had already started attending primary school.
Ngô Báo Châu's father had a great influence on him. "I studied in an experimental primary school (Giang Vo experimental primary), which used special teaching methods such as encouragement of independent reading and freedom of expression. However, my father didn't like it and he sent me to a school for gifted students with mathematical talents. From then on, because of my parents, I did a lot of mathematical exercises and began to love mathematics."
Ngô Báo Châu studied in the special class in Trung Vuong lower secondary school where students were specially selected through admission tests. After graduating from junior high school in 1987, Châu was admitted into a class in the Gifted High School of Hanoi National University of Natural Sciences which was specialised in mathematics and aimed at gifted students. Châu participated in the 29th and 30th IMO during his two years in the High School.
According to Châu, "The school for the gifted had a good system of organising mathematical competitions. All the participants had been selected through municipal, provincial and national examinations. We passed many tests, just like sports competitions. Youths like sports but I didn't like mathematical competitions as there were too many competitions, each with a long preparation process, and the competitions themselves were nerve wrecking. In 1988, I won the 29th IMO gold medal with full marks but after that, I was no longer interested in competitions. I participated in the 30th IMO at the request of school. I also won a gold medal but I didn't really enjoy it that time."
In Vietnam, it is a special honour to win the gold medal of International Mathematical Olympiad. Ngô Báo Châu was received by a general. "He congratulated me. I was very happy because it was a kind of recognition. However, I couldn't remember whether there had been prizes." In addition, the medallists of IMO could receive scholarships to study in universities in the Soviet Union or Eastern European countries.
Châu was offered a scholarship by the Hungarian government. After graduating from high school in 1989, he prepared to study in Hungary because of his love for combinatorial mathematics.
"I had learned Hungarian for one year. However, in the aftermath of the fall of communism in Eastern Europe, the new Hungarian government stopped providing scholarships to students from Vietnam. I lost the chance due to this unexpected event."
Beginning to Research on Mathematics in France
"It was by pure coincidence that I decided to research on the Langlands Program. I wanted to do something and it was a good decision at that great time."
It was at that time that a professor from France visited the Institute of Mechanics where Châu's father worked. After knowing that Ngô Báo Châu had won IMO gold medals, the professor tried to help him secure a scholarship from the French government.
"Thanks to this scholarship as it enabled me to go to Paris."
Having studied in France before, Châu's grandfather began to teach him French. "The French education system is different from other countries". After spending two years in high school, I went to École Normale Supérieure for undergraduate study. Michel Broue, my instructor, suggested that I follow Prof. Gérard Laumon, from Université Paris Sud 11. So I started undertaking doctoral research in university."
Studying in France at the high school level had a considerable impact on Châu. "High schools in France have two-year college preparatory study, which is quite different from Vietnam, where high schools focus on examination preparation."
The doctoral training program in France is very different from the United States. During the time when Châu started his doctoral research, the Langlands Program was a well-known project among the French mathematicians. Mathematician Roger Godement, known as the father of automorphic forms in France, introduced the Langlands Program and automorphic forms to France. These had great influence on French mathematicians, including Professor Laumon.
"Almost all the mathematicians were studying automorphic forms. Many people joined this field. Most of them were associated to Godement. They were very strong in this area and as a result this field became very active… Almost every mathematics student was trying to solve this problem. Upon Professor Laumon's suggestion, I began to research on the Langlands Program in 1993."
In 1997, Ngô Báo Châu obtained his PhD at the age of 25 from Université Paris Sud 11. "I solved a problem very similar to the Fundamental Lemma and I began to understand that the key was the geometric model of the trace formula."
Ngô Báo Châu became a member of the National Centre for Scientific Research (CNRS) at Université Paris Nord in 1998. It was his first job. At that time, his goal was to prove the Fundamental Lemma of the Langlands Program.
Châu said, "The training system in France is very different from that in America. In the United States, you must work as a postdoctor for two to three years after getting the doctorate degree. There is great pressure to publish papers before applying for a job and even after getting a job, the pressure stays. In France however, I didn't have this pressure. I didn't need to produce papers. All I needed to do was to do research in mathematics."
During the first seven years after getting his PhD, Châu worked as a researcher, not a professor. "I worked with Professor Laumon at first. When I went back to the problem of the Fundamental Lemma, I tried different methods and got new ideas."
Ngô Báo Châu was very happy in France because he could concentrate on mathematics. "CNRS is similar to the Chinese Academy of Sciences (CAS). Researchers are affiliated to CNRS. That is to say, CNRS pay them salaries; although the researchers work with professors in related universities but they don't have to teach. I don't know whether this arrangement is good or bad, but the time after I got my doctorate degree was really a golden era for me. It is a lifelong position when one becomes a researcher of CNRS. I don't have the pressure to apply for funds, to publish papers, to worry about tenure or to teach. All I need to do is to stay there and spend more time on research in mathematics rather than doing other things."
According to the statistics of MathSciNet of American Mathematical Society, Ngô Báo Châu has published 15 papers so far. He said, "I am not interested in publishing papers of low quality. I have written few papers, but all of them are good. My colleague told me, ‘Don't waste time to write lousy papers. One good paper is better than 100 papers of bad quality.’ It is not my way but the standard in France."
How do colleagues evaluate his work if he has no papers? "I accept evaluation every year. I only need to report what I have done during the year. The French National Research Council evaluates me every five years. I report what I have achieved and what I intend to do. If they feel that it is not bad and give me a good evaluation, the CNRS will continue to support me."
It was a turning point in 2003. "At that time, I was extremely clear about every problem related to geometry. Things became simpler and clearer. I believed that I got a new idea, but it was just the beginning." In that summer, Gérard Laumon went to Vietnam for a trip at Châu's invitation. Laumon became interested in Châu's idea and they succeeded in proving the Fundamental Lemma for unitary groups together. They received the Clay Research Award in 2004.
In 2005 Châu received the title of professor at Université Paris Sud 11 and at the age of 33, also became the youngest full professor ever in Vietnam.
Becoming a Distinguished Guest of the Institute for Advanced Study, Princeton
"The most exciting moment was the time when I found the solution. I was so excited. I became almost exhausted after that. I didn't know all the details even though I had found the method. I wrote out all the steps which were more than 200 pages. This was a long process, filled with hardship and pressure. It took me two months to correct a very serious error."
The Langlands Program has fascinated Ngô Báo Châu. He spent nearly 17 years working on it.
Châu indicated, "Every mathematician knows the importance of the Langlands Program. You will understand mathematics and geometry in a new way if you know the Langlands Program. Andrew Wiles proved Fermat's Last Theorem by using the ideas of the Langlands Program. You can see how beautiful and powerful the program is. It is really exciting."
After proving the Fundamental Lemma for unitary groups, Laumon decided to quit, but Ngô Báo Châu persisted. "Unitary groups were inapplicable to common forms. As such, I spent a long time on the problem. In 2006, I almost believed it was impossible to obtain the proof."
Things started to change at this time. Ngô Báo Châu was invited to visit the Institute for Advanced Study (IAS) at Princeton in 2006. It was his first visit there. "It was around December 2006, during a conversation with Mark Goresky of the IAS that provided me the missing piece to solve this jigsaw puzzle. I realised that I found the right way and I believed that I could crack the problem in its full form. It took me more than one year to complete the proof."
Châu had planned to stay for three months at Princeton, but the IAS hoped that he would stay longer: five years. However, he still returned to France. "This is because I belonged in CNRS. I went back to Princeton in 2007 and remained there."
In June 2007, Châu completed the first draft of his paper, consisting of 200 pages. Then he gave a talk on his proof in a workshop held in France. "Some were suspicious of its validity, but most people were convinced by my proof." After returning to Princeton, Ngô Báo Châu continued delivering reports during many conferences.
"During that five months, I relentlessly continued to give lectures, explained my ideas and corrected my own mistakes. In May 2008, I sent my paper to Publications Mathématiques de I'IHÉS in France. It was a long time of scrutiny. Only a few people could check the details, but I didn't know who the reviewers were."
By the end of 2009, almost everyone in this field believed that Ngô Báo Châu had proved this conjecture. The Fundamental Lemma was listed as one of the top ten scientific discoveries of 2009 by the Time magazine:
Over the past few years, Ngô Báo Châu, a Vietnamese Mathematician working at Université Paris-Sud and the Institute for Advanced Study (IAS) of Princeton, formulated an ingenious proof of the fundamental lemma. When it was checked this year and confirmed to be correct, mathematicians around the globe breathed a sigh of relief. Mathematicians’ work in this area in the last three decades was predicated on the principle that the fundamental lemma was indeed accurate and would one day be proved. "It's as if people were working on the far side of the river waiting for someone to throw this bridge across," says Peter Sarnak, a number theorist at IAS. "And now all of a sudden everyone's work on the other side of the river has been proven."
In January 2010, Ngô Báo Châu's paper "The Fundamental Lemma for Lie Algebras" was accepted and published by Publications Mathématiques de lI'IHÉS in France.
Châu said, "I heard of the selections made by the Time magazine, but I had no idea how they knew about it."
The Invitation from the University of Chicago
"He is one of the greatest mathematicians of this age. He is very intelligent. I really hope this young man will do more great things."
- Robert Fefferman, Dean of Physical Sciences Division, professor in Mathematics Department, University of Chicago.
Of course, there was another man who was extremely excited about the proof. It was Robert Langlands, who once left this field but is back now.
Châu said, "Langlands must have thought that it was easy to prove this theorem when he came out with it. He worked on it for ten years with his students. Therefore he named it 'Fundamental Lemma'. However, he encountered more and more geometry problems which were not clear to him at that time. He left the field of automorphic forms and began to research on mathematical physics. He was very happy when he first saw me proving the Fundamental Lemma with new methods in Paris. He returned to working on automorphic forms. Maybe I encouraged him, but I don't know the exact reason for his return."
In 2010, Ngô Báo Châu published a paper collaboratively with Robert Langlands.
In January 2010, Ngô Báo Châu joined the Mathematics Department of the University of Chicago as a full professor.
Constantine, Chairman of the Mathematics Department, University of Chicago, has this to say about Ngô Báo Châu. "He proved a fundamental theory, a conjecture called the Fundamental Lemma. It was named so because it was the key to the door for the progress of the Langlands Program... Châu's proof had dramatically opened this door."
Why choose to go to the University of Chicago? Châu answered, "The University of Chicago gave me very good conditions. If I wanted to, I could teach. If I just wanted to do research, I could stop teaching. I had a tenured professorship. I could do what I wanted to do. If you were a professor in France, you would have to teach, which is a heavy responsibility. However, in Chicago, they didn't require me to teach and they supported me. And at the University of Chicago there are many people working on mathematics that I like, such as Robert Kottwitz, Alexander Beilinson, Kazuya Kato, Vladimir Drinfeld, and Spencer Bloch and others. So there are more colleagues with whom I can discuss mathematics with."
Robert Kottwitz, a mathematician from the University of Chicago, had developed the method to solve the Fundamental Lemma of the Langlands Program in a joint work with Mark Goresky and Robert MacPherson from IAS, Princeton University. Châu said, "Other than my supervisor Prof. Laumon, Prof. Kottwitz also had a great impact on me. I used to go to Chicago to visit him. He always generally told me many of his thoughts and opinions. He didn't compete with me and instead, he helped me clarify many problems."
Another important reason Ngô Báo Châu chose to move to Chicago was for the benefits of his children. "There are excellent experimental primary schools and high schools in Chicago."
At the beginning of 2010, Ngô Báo Châu became a French citizen. "When I was in France, it was not necessary for me to obtain French citizenship because I had the permit for permanent residence. However, after I had decided to go to America, it became difficult for me to return to France. I had left but did not resign as I had still kept my French University position. I hope that I could return to France regularly to meet and talk to my friends and colleagues, but it is not easy to obtain a French visa in America."
"It is Pleasant to Work on Mathematics."
"It is very important to take part in good seminars, and it is necessary to keep on conversing with others. When I attended the seminar in the first year, I couldn't understand a single word, but I persisted on listening."
It was not easy for Ngô Báo Châu to become a mathematician from a Mathematical Olympiad medallist. Not all of the winners of Mathematical Olympiad grow up to be mathematicians, but in Vietnam, almost all of the mathematicians are medallists of the Mathematical Olympiad.
Looking back on his mathematical journey, Châu said, "Participating in the Mathematical Olympiad is different from researching on mathematics. To participate, one needs to master various skills which will help to solve complicated and high level problems in a limited time. The danger in this is that one may not respect the natural conciseness and beauty of mathematics. Whether or not one can become a mathematician ultimately depends on the person himself and his ability to appreciate mathematics. Such a transformation is not direct or obvious. In my opinion, it is necessary to be a ‘connoisseur’ of mathematics in order to become a good mathematician."
How can one develop one's taste in mathematics? "To do so requires a lot of time spent on mathematics, to study and learn more about it."
Châu suggested this to the students who are beginning to work on mathematics, "In France, students have to take part in many fundamental courses and fruitful discussions. You can develop a good taste from the involvement in discussions during the time as an undergraduate. You can learn how the mathematicians put forward questions, why they are interested in them, how they discuss them and how to prove them from the talks of good mathematicians. I had fortunately participated in many discussions and projects, and had learned a lot from these involvements. I came out with the proof for the problem when I was a postgraduate. If I had not joined the discussions, I would not have been able to come out with the problem by myself and hence would not have been able to work on the project."
Ngô Báo Châu and his wife have three children. His wife does not go to work but stays home full time. Usually Châu does research in his office. "When working, I don't speak to anyone. I will talk to the children at home when I am not stressed. After they go to bed, I begin to work. I don't sleep much."
When talking about mathematics, he said, "It is pleasant to work on mathematics when you want to research on it. You will feel it in its most natural form; mathematics is the most beautiful language that describes the world. It is very simple, hence it is also the most practical language. It is exact and succinct."
Since Châu has learned English by himself, he spoke to the journalist in English. "I have read lots of books and papers. Reading has always been my favourite hobby for relaxation."
When speaking about the future, he said, "I have only proved the Fundamental Lemma, not everything in the program. Our next goal is the Langlands Program, of which the Fundamental Lemma is just the foundation akin to a small mountain. After peaking this mountain, we can see the Langlands Program in its full glory. There is a big mountain in front of us, but the issue now is how to scale it. One good thing is that Langlands has returned and he will direct us in a new direction to solve the problems. I think it may take my entire lifetime to prove everything in the program."
The original article was published on Science Times, November 18, 2010. Acknowledgement also goes to Dr Chaozhong Wu (Tshinghua University, China) for the help during preparation of this article.