One of the first collaborations Xinyi Yuan and Wei Zhang ever undertook was a trip to the Social Security office. It was the fall of 2004 and the two of them were promising young graduate students in mathematics at Columbia University. They were also friends from their college years at Peking University in Beijing. Yuan had come to Columbia a year earlier than Zhang, and now he was helping his friend get a Social Security number. The trip did not go well.

“We went there, and we were told that some document of Wei’s was missing and that he couldn’t do it at that time,” Yuan recalled.

That failed attempt was one of the few unsuccessful team efforts the two have undertaken since coming to the U.S. Zhang, who is now a professor at Columbia, and Yuan, now an assistant professor at the University of California, Berkeley, are members of an unofficial quartet of Chinese mathematicians who have been friends since their undergraduate days at Peking University in the early 2000s and now hold positions in some of the best mathematics departments in the world.

That a number of elite mathematicians would come out of the same class at a top university is unusual, but not unprecedented. The most recent example is Manjul Bhargava, Kiran Kedlaya and Lenny Ng, freshman classmates at Harvard University who went on to become distinguished mathematicians. They remain good friends and all traveled to Seoul in 2014 when Bhargava won the Fields Medal.

What’s unusual about the group formed by Zhang, Yuan and their two friends is the degree to which they continue to collaborate and the extraordinary amount of successes that they’ve had.

“They are not only good, they work in almost the same areas, and because they learned together, they influenced each other, and even as mature mathematicians they’re collaborative,” said Shou-Wu Zhang, a mathematician at Princeton University who knows all four and was influential in recruiting Zhang and Yuan to study in the U.S.

In addition to Zhang and Yuan, the other members of the group are Zhiwei Yun, an associate professor at Stanford University, and Xinwen Zhu, an associate professor at the California Institute of Technology. Yun and Zhu work in the field of algebraic geometry, while Zhang and Yuan work in number theory. This split in fields provides them with complementary perspectives on what is probably the single biggest project in mathematics, the Langlands program, which has been described by the Berkeley mathematician Edward Frenkel (who was Zhu’s graduate adviser) as “a kind of grand unified theory of mathematics.” The program, first envisioned by the mathematician Robert Langlands in the late 1960s, seeks to draw connections between number theory and geometry, so as to use tools from one field to make discoveries in the other.

One obstacle to pursuing the Langlands program is that it’s difficult for a single mathematician to know both fields deeply enough to see all the connections between the two. Yet mathematicians from different fields may have trouble communicating with one another. The best collaborations involve mathematicians who have deep knowledge of different fields, but who also know just enough in common to talk to each other.

That is the case with these four mathematicians. They are all individually talented, and each has pursued his own research interests over the years. But they are also close friends with a shared background and a similar approach to mathematics. This has allowed them to prompt each other, teach each other, and make discoveries together that they might not otherwise have made so easily. These include several smaller papers they’ve written in tandem and now, most recently, their biggest collaborative discovery yet — a forthcoming result by Zhang and Yun that’s already being hailed as one of the most exciting breakthroughs in an important area of number theory in the last 30 years.

**The Early Years**

Before their mathematical abilities drew them together, the four grew up in different parts of China. Zhu is from Chengdu, a provincial capital in the southwest. Yun grew up in a town outside Shanghai called Changzhou. At first he was more interested in calligraphy than math. Then, when he was in third grade, a teacher, recognizing Yun’s potential, explained to him that the repeating decimal 0.9999… is exactly equal to one. Yun puzzled over this unexpected fact for months. After that, he was hooked.

Yuan started out in the least auspicious circumstances of the four. He was born in a village close to Wuhan, a poor area with few resources for cultivating mathematical genius. But his teachers quickly noticed his talent.

“My math teachers liked me very much in first and second grade, and I could tell they were surprised by my ability,” he said. “Mainly that I got very high scores, usually perfect scores on exams.” Later, he enrolled in the prestigious Huanggang High School.

In China, as in other countries, there are structures in place that make it likely that top mathematical talents will eventually meet. Zhu and Zhang, who grew up 300 miles away from Chengdu, first met at a summer math camp after 10th grade. Yun and Yuan were both members of the Chinese national Math Olympiad team, a status that reflected their particular technical skill and prowess at solving problems.

In August 2000, the four were among 200 students in the entering class at Peking University. Many of their classmates were good at math, but most aspired to careers in practical fields like finance or computer science. By their junior year, their class had divided up according to interests, and Yuan, Zhang, Yun and Zhu found themselves placed together in a small group focused on pure math.

At that point the four became friends in the typical college way. They’d watch movies, go hiking, and play soccer and basketball together. Yuan, whom they all describe as the most athletic of the group, usually won. During this period, in class and in discussions they organized among themselves, the four also encountered for the first time some of the mathematical concepts, such as automorphic forms, that would later form the focus of their careers. And as they made their way into the world of higher mathematics, they realized they were all fascinated by the same kind of mathematical research.

“By the end of college it was pretty clear to me that the four of us shared a similar taste in mathematics,” Yun said. “That taste is structure-based mathematics. Instead of doing computation, all of us are interested in the big picture and finding interesting examples demonstrating general principles.”

Yuan was the first of the group to take this perspective to the United States. In 2003 he went to Columbia to work with Shou-Wu Zhang. He was drawn abroad by the feeling that in China, he wouldn’t be able to realize his potential as a mathematician.

“I somehow thought that the professors [with whom I interacted] at Peking University were not good enough, were not top mathematicians,” he said. “I wanted to come to the United States earlier just to see these great mathematicians.”

Yuan’s experience as a graduate student surpassed his expectations. It wasn’t just that he suddenly found himself attending conferences and colloquia with the brightest mathematicians in the world. It was also that as he observed these mathematicians up close, he gained a new appreciation for the immense potential in the discipline he’d chosen to pursue.

“In China, mathematicians were not that happy, like somehow they didn’t seem to enjoy math. They gave off the impression that math was hard and you needed to be cautious to choose math as your lifetime career,” he said. “Columbia was totally different. One important thing I saw there was happiness in math, motivation, optimism. These are the parts I didn’t see in China.”

A year later, Yuan’s friends followed him to graduate school in the United States: Zhu went to Berkeley, Yun to Princeton, and Zhang to Columbia. Zhang remembers that soon after arriving in the United States, he realized he’d miscalculated when he would receive his first stipend check and was going to run out of cash. Yuan, who’d had a year to figure out the intricacies of direct deposit, gave him some money to get by.

Even more crucially, Yuan helped Zhang get his bearings in the math department at Columbia. “He gave me more direct access in understanding what professors here studied,” Zhang said. Zhang was particularly attracted to Shou-Wu Zhang’s research. Shou-Wu Zhang, who later left Columbia for Princeton, worked simultaneously in number theory and arithmetic algebraic geometry. Wei Zhang was impressed by what he describes as Shou-Wu Zhang’s ability to “expose ideas directly without hiding them behind a lot of technical ideas.”

Eventually Wei Zhang decided to focus his dissertation research on L-functions, a central topic in modern number theory and one of the most interesting. In particular, he was interested in generalizing the Gross-Zagier formula, which applies to a certain subset of L-functions, to a much broader range of L-functions. This work, which presaged his most recent discovery with Yun, was closely related to Shou-Wu Zhang’s own research, but not confined by it. The freedom to chart one’s own mathematical path, even as a graduate student, is something Wei Zhang likely would not have found had he stayed in China.

“In the Chinese way, you 100 percent follow your teacher and do the problem that’s left from the teacher’s research area,” Shou-Wu Zhang said. “The American way is, you take teacher’s advice with some modification.”

At the same time that Wei Zhang was exploring L-functions, Yuan was finding his own way in number theory, and Yun and Zhu were establishing their research programs in algebraic geometry. During and after graduate school, the four stayed in regular contact. Their paths often crossed in the country’s math centers — in Cambridge, where Yun was a postdoctoral fellow at the Massachusetts Institute of Technology and Zhu was at Harvard, and at Princeton, where Yuan and Yun overlapped during the 2008-2009 academic year.

During that year at Princeton, Yuan and Yun met regularly and began to develop their collaborative style. In informal conversations, Yuan explained the intricacies of number theory to his geometer friend. They spoke in Mandarin and conversed easily; Yuan had a good understanding of what Yun knew and didn’t know, and Yun was able to ask questions, even simple ones, without fear of looking naive. “Because he was able to explain many things to me,” Yun said, “I didn’t find it very difficult, while before I found number theory too difficult for me.”

These conversations, along with the work of the 2010 Fields medalist Ngô Bảo Châu, helped Yun understand that many of the techniques he knew from algebraic geometry could be used to attack problems in number theory. This was the goal of the Langlands program, and it had been made apparent to Yun in a very direct way. Now all he needed was a question to address.

**The Breakthrough**

In December 2014, Zhang flew from New York to the West Coast, where he saw Yun and Yuan. The reason for the trip was a 60th-birthday conference at the Mathematical Sciences Research Institute in Berkeley for the Columbia mathematician Michael Harris, but Zhang also arrived with an idea he wanted to share with his friends. That idea had grown out of a conversation he’d had with Yun back in 2011. At that time, Yun had been thinking about work Zhang had done even earlier on a problem in the Langlands program known as the arithmetic fundamental lemma. Yun thought that some of those ideas could be combined with techniques from algebraic geometry, but he told Zhang he wasn’t sure if it was possible.

“I had some geometric idea which could be true, but I couldn’t make it precise because I was lacking some vision in number theory,” Yun said. “I told Wei, Do you think this thing could be true? He wasn’t sure.”

They left the conversation there for several years. Then in 2014, Zhang realized that Yun’s intuition was correct, and he began to see what it would take to prove it. The problem at hand involved L-functions, which Zhang had studied in graduate school. L-functions have what’s known as a Taylor expansion, in which they can be expressed as a sum of increasing powers. In 1986 Benedict Gross and Don Zagier were able to calculate the first term in the series.

Although L-functions were initially purely objects of number theory, they can also have a geometric interpretation, and powerful techniques from algebraic geometry can be used to study them. Yun had guessed that every term in the Taylor expansion should have a geometric interpretation; Zhang was able to precisely define what such an interpretation would look like. Whereas Gross and Zagier (and the French mathematician Jean-Loup Waldspurger) had been able to obtain exact formulas for the first and second term in the expansion, the new work would show how to obtain a geometric formula for every term.

Zhang explained his thinking to Yun and Yuan at Yuan’s house. As he listened, Yun remembers thinking that Zhang’s ideas fit together so well, they had to be true.

“He had the vision for this sort of global picture that made what I had vaguely in my mind very precise,” Yun said. “I think I was really kind of astonished when he laid out the whole thing. It was so beautiful.”

After that night, it took Zhang and Yun about nine months to prove their ideas. By September of this year, they had an early draft of a paper and began to give informal talks on their efforts. By the end of November, they had a completed draft. Shou-Wu Zhang, who has seen the work, estimates they completed the work at least a year faster than Wei Zhang could have managed on his own — assuming, that is, that the approach would have even occurred to him.

The result still has to go through peer review, but it is already generating excitement in the math world. Among other implications, it opens a whole new window onto the famed Birch and Swinnerton-Dyer conjecture, which is one of the seven Millennium Prize Problems that carry a $1 million award for whoever solves them first.

But the effects of Zhang and Yun’s latest work go beyond math. Zhang and Yun met as teenagers, grew up with Zhu and Yuan across two continents, and came of age together as mathematicians. Now the benefits of the friendship are spilling over into the rest of the mathematical world.

“These four people have different styles and methodologies to attack problems, so combined together, it’s simply great,” says Shou-Wu Zhang.

*Editor’s note: Benedict Gross is a member of Quanta Magazine’s advisory board.*

*Update on December 10, 2015: This article has been updated to include links to the new work.*

*This article was reprinted on Wired.com.*

It reminds me of the golden age of Bourbaki and some description of how Serre, by knowing exactly what Grothendieck knew could find the right way to explain him in a few sentences a difficult paper he had read.

Wonderful story, thank you.

http://arxiv.org/abs/1512.02683

While I'm positive that stories about math at the level of sophistication described here require some technical underpinning which the writer and editors may potentially lack and which therefore make it both more difficult to be aware of them (to know to report on them in the first place) but also to objectively lay out the topic, from a journalistic perspective, this particular story, which references prior work by Benedict Gross, should have at least mentioned, as a caveat, the relationship between Quanta as a publisher and Gross's position as an advisory board member to it.

While Gross's bona fides are certainly not in question given the topic or his distinguished career, what, if any, was his involvement with the piece? Did he suggest it? Did he provide background? (I notice he wasn't directly quoted, which makes me wonder even further). Was he paid? There's certainly the possibility that something potentially not quite kosher may be going on here (even if it is just free publicity for him and his past work, particularly given that the final result discussed here hasn't been published or independently verified), and not providing at least a mention of the relationship from an editorial standpoint is a glaring error.

Aside from this small journalistic oversight, I otherwise laud Quanta for consistently providing interesting, entertaining, and timely coverage of the world of math and science in such a fantastic fashion. Helping to humanize and even idolize mathematicians and scientists should be a more common effort in our society.

Given the somewhat broad nature of the readership and range of backgrounds, for those who may not have the high level of background for reading the related papers (once they're publicly available) yet, many may be interested in knowing that Dr. Gross has an excellent and free online series of lectures on basic abstract algebra available (http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra) which may go a reasonable part of the way for helping the not-so-technical readers to better understand some of these areas of mathematics and how they interrelate.

@Chris Aldrich, thank you for your comment, and for your kind words about Quanta. As you note, Benedict Gross is a member of our advisory board, as are a number of other distinguished scientists and journalists. We agree it is important to make that relationship as transparent as possible. In addition to the advisory board listing in the right-hand column of the article (and on the homepage), we have appended a note to the bottom of the article, to make our relationship with Prof. Gross even more apparent.

Our advisory board members provide suggestions and feedback to help ensure that we’re covering the most important work in mathematics, physics, computer science and basic biology, and that our reporting is as accurate and precise as possible. Quanta's editors evaluate all advisory board suggestions, independently assessing the importance of the research, and its timeliness and suitability for coverage in Quanta Magazine. As with all of our articles, we verify the importance of the work with independent scientists who are active in the same subfield of science as the research in question. All articles are fact-checked with independent experts; board members do not review articles prior to publication. Suggestions from our valued board members are just that — only a fraction of our advisors' recommendations ever become Quanta articles. These, in turn, represent only a small fraction of the articles that Quanta publishes. The role of our board is not to drive Quanta's coverage, but to help reveal our blind spots.

Deputy editor Michael Moyer, who edited Kevin Hartnett's article, describes how the story came together: "This spring, Professor Gross alerted us to a potentially interesting story in mathematics—that of four friends from the same class in Peking University who went on to immigrate to the U.S. and do terrific work. He also mentioned that they were working on an important proof in number theory. We began looking into this story in the spring, and found that while all the ideas were in place, and that the researchers had begun to give lectures on their work, the proof was not yet complete. We also verified the importance of the work with independent researchers involved in the Langlands program.

"As we detail in the story, the new work expands on the three-decade-old work of Gross and Zagier, who demonstrated how to calculate the first and second terms of the Taylor expansion of an L-function. (The new result provides a geometrical way of calculating all terms.) The Gross-Zagier formula was an important milestone in number theory; Gross and Zagier (along with Dorian Goldfeld) went on to receive the Cole Prize of the American Mathematical Society for their work. That the new achievement of Zhang and Yun moves the research program substantially further—and does it for the first time in three decades—we took to be an important verification that the new research was worth covering in Quanta Magazine. We held off on reporting and publishing the piece until the proof was complete, and other researchers had examined the work. (The paper is now available on the arxiv.)"

I hope these details can help alleviate your concerns regarding the relationship between Quanta's editorial staff and its advisory board. We consider our board a source of invaluable expert advice in our effort to produce a reliably accurate math and science publication founded on the bedrock of journalistic independence. Thank you so much for reading Quanta and for your thoughtful comments.

Thomas Lin

Editor in Chief

I am extremely sad to note that this story dwells on trivial biographical details and completely omits the explanation of the concepts and mathematics involved. Quanta is literally the best magazine on the planet about higher mathematics and it is your onus to make that math accessible. If the writer does not understand it himself, he should not even be writing the article. I understand that the ideas are difficult, but it is not impossible to at least sketch the basic metaphors underlying them.

And this is not the only article in which I have seen this grave error, there was another about the Kadison-Singer problem which was highly vague. I was surprised because that particular writer is one of the most consistently lucid ones at Quanta.

I hope my objection is taken seriously.

I couldn't agree more with @fadesingh.

The thing I loved about quanta is how the writers tried to make very difficult subject accessible to non specialist but still not dumbing down below a level of university graduates.

It was often possible to grasp the intricacies, subtleties and beauty involved in the research cited.

This article is just plain trivial from beginning to end. It does not even try to explain anything, or give a picture of what is discussed. To compensate for that total lack of scientific explanation all the focus is on personal stories. Please leave this level of explanation for television; even off the shelf science magazine do a better job, but I came to expect a lot lot more from quanta.

I must say that I have grown more and more disappointed with the latest offerings, this one being absolutely the worst.

I was so glad to have found a science magazine at this level, is it over already?

Please Quanta, bring back who ever was in charge of this wonderful place.

I will dearly miss you.

A side point to this article, but Manjul, Kiran, and Lenny were not freshman classmates at Harvard. Manjul and Kiran were sophomores when Lenny was a freshman.

I agree with fadesingh. While I enjoyed the biographies, I would have also preferred more mathematical detail.

I agree with @fadesingh that it would be much more worthwhile to explain the math better in articles for public consumption. I can recall only a few times when this was accomplished, using simple examples and illustrations, but these examples were almost miraculous in their ability to stimulate my interest. I'd like something between a mere display of personalities and a deep journal publication, particularly when intelligent illustrations are employed. I often read that math is too obscure and difficult for the average reader. Too bad, but "average" readers should never be a target audience for math- or science-oriented journalism, they don't care enough to want to explore new thinking. The abysmal state of math understanding in schools is holding the U.S. back in a big way; even teachers often don't like or understand anything beyond arithmetic and simple algebra, which is why they spend so much time being boring. Similarly, the authors of most "pseudo-math" articles seem not to understand the material, so how can they explain it engagingly? At a minimum, they should get the researchers cited to do more entry-level explaining, and publishers should cultivate more talented writers.

Arxiv Can only save the paper, the important is peer review.

I am not much into Mathematics as a subject,so forgive my ignorance.But didn't Godel proved that it was impossible to unify all of mathematics under one theory?