The Paper's reporter Shao Wen
"The beauty of mathematics is an abstract beauty, it is possible to narrate a truth in a very simple way, and many special examples can be summed up in a few sentences after reading through, and its beauty lies in simplicity and consistency." On the afternoon of August 21, Mo Yiming, the new winner of the Future Science Prize for Mathematics and Computer Science, told the surging news (www.thepaper.cn).
In April 2008, Mo Yiming answered the reporter's question about the beauty of mathematics to mention "conciseness", and now 14 years later, for Mo Yiming, the answer seems to be the same as yesterday. Mo Yiming explained at the time that the so-called conciseness is to see things that are already known from another perspective.
From Mo Yiming's experience, the most prominent source of this "other perspective" is the philosophy of language, who has different degrees of mastery and specialization in English, German, French, Italian, Spanish, Portuguese, Japanese and even Indian Sanskrit.
In Mo Yiming's view, if a person understands more widely, there are more natural sources of inspiration. For example, he said, "There is a concept in linguistics called 'reconstruction', and the concept of reconstruction is related to mathematics, such as VMRT (very small rational tangent cluster), can reconstruct this space by the geometric properties of a point. In addition to linguistics, the concept of reconstruction may also appear in other disciplines, such as chemistry is decomposition, decomposition after reconstruction, this view is also commonly used in biology. ”
In addition to the understanding of concepts, this "reconstruction" is also implicitly reflected in Mo Yiming's methodology on how to do "wonderful mathematics". In a sense, the study of pure mathematical geometry, number theory, algebra involves a lot of creative components, even the idea of the problem itself needs to be discussed, mathematics has its technical aspects, but also related to literature and art. To do mathematics well, there is no formula that can tell you the answer, to be able to integrate the key concepts in different fields to find new directions, to be able to create a new research theme, this kind of creativity has something to do with being a film director. ”
According to the award speech of the Future Science Prize, "the creation of the the theory of the Very Small Rational Tangent Cluster (VMRT) and its use to solve a series of conjectures in the field of algebraic geometry, as well as the proof of the Ax-Schanuel conjecture on the Shimura cluster", is the reason for Mo's award.
"Professor Mok's work sets an example of successful collaboration between complex geometry, algebraic geometry, and number theory." Xia Zhihong, a member of the Science Committee of the Future Science Prize and a Pancoe Chair Professor at Northwestern University, commented.
Xia Zhihong further explained that the objects of study of complex geometry and algebraic geometry are geometric shapes with complex and algebraic structures, respectively. The purpose of complex geometry research is to understand the properties of these geometric shapes, as well as the mapping of maintaining complex structures between them. Yiming Mo and his collaborators created and developed the theory of the Very Small Rational Tangent Vector Family (VMRT), which solves a series of pending mathematical guesses by studying the analytical mapping between manifolds with a set of algebraic clusters with special structures.
The significance and achievements of this work may not be easy for most people to understand. But the logic and way of thinking about doing a good job can often be transferred and borrowed in different fields.
In the interview, Mo Yiming recalled the beginning of his mathematical journey, talked about how to deal with setbacks in the research process, and patiently and meticulously provided his own experience to young people who love mathematics. At the same time, Mo Yiming also answered the questions of what kind of environment mathematics needs, and how the level of Chinese mathematics compares with the forefront of the world.
Mo Yiming
"Mathematics is not equal to numbers, logical reasoning is only a necessity"
Mo Yiming believes that many times mathematics requires absolute truth, but how to discover this absolute truth is often based on the perspective of the researcher, and the perspective of different people with different training backgrounds is different.
As far as Mo Yiming is concerned, the training of personal mathematics and the cultivation of perception actually come from interest in other different fields. "I like to understand what is being said in other fields", in Mo Yiming's view, mathematics is not equal to numbers, and logical reasoning is only a necessary condition for mathematics, but not a sufficient condition. Mathematical researchers need to have the ability to touch bypass and the ability to be intuitive. How to achieve this ability?
"It's about training, by improving your understanding of different areas." Most scholars have many other hobbies and interests, but they only do it because they do it well. Mo Yiming once said.
It is easy to think of mo Yiming's first journey to mathematics.
Mok was born in 1956 in Hong Kong, China. In elementary school, Mo Yiming, like other classmates, trained himself to solve math problems quickly in order to enter middle school. But Mo Yiming will think deeper, "I have been thinking about this question for a long time, why is this faster?" What is its mechanism other than speed, for example, when learning the speed algorithm, the teacher will tell you the method, so why does this method hold? Mo Yiming recalled to the surging news reporter (www.thepaper.cn).
In Mo Yiming's retrospective, the first important person to appear is his father. "My father was more interested in mathematics, he was in World War II at that time, and he didn't have the opportunity to learn deep mathematics. But he was very interested in mathematics and often told me small stories. At that time, I knew a lot of stories, and when I was very young, I wanted to do something that contributed to China and chinese civilization, which was my dream when I was very young. Mo Yiming said.
The second was his middle school teacher. In Mo Yiming's account, the highly skilled teacher compiled a textbook that taught the old mathematics in the language of the new mathematics, and introduced the basic concepts of the new mathematics such as linear algebra earlier, including the probability theory that could not be learned until college in the old mathematics. In addition, Mo Yiming also benefited greatly from the old math textbook of his sister who was one level higher than himself.
In 1975, Mo Graduated from High School and received a scholarship from the University of Chicago to continue his studies in the United States.
"When I left Hong Kong, my mathematical roots were already relatively solid, and the scope was relatively wide and deep, so I could do research early in college." Mo Yiming told the surging news (www.thepaper.cn).
Mo Yiming is not the stereotype of a mathematician who is obsessed with the world of abstract numbers all day, and he has a considerable curiosity about many things. He once said that mathematics is the foundation of the natural sciences (such as mathematical biology), and linguistics is the key to opening the doors of the humanities (such as literature, especially poetry and philosophy).
"When I learn something, from the beginning there are more components of self-study, and I will first see what the goal of the discipline is, and then go back along the tool to try to solve the problem." I'm bold in doing math, hoping to find some of its core problems, try to deal with them in ways that may not be the most traditional and classic, and sometimes find new directions. Mo Yiming said.
For beginners who have firm ideas in the field of mathematics, Mo Yiming's advice is to have a long-term plan, that is, to stabilize their roots. Because even if you only care about a certain problem, researchers may use knowledge in other fields in the research problem, so Mo Yiming believes that it is necessary to enrich the basic knowledge and have a certain depth.
Speaking of this, Mo Yiming mentioned an experience when he was a doctor. At that time, Mo Yiming was able to graduate, but received advice from his mentor. The tutor told Mo Yiming that what he had learned now was not enough, and later Mo Yiming postponed graduation by a year and spent a year learning new things. In this year's time, Mo Yiming's future research plans have been outlined.
In 1980, Mo received his Ph.D. from Stanford University under the teacher mathematician Xiao Yintang. In the same year, Mo Yiming entered the United States to teach at Princeton University, and later served as a professor at Columbia University and the University of Paris in France.
Mo Yiming said, "Of course, over time, the plan will be modified, but I first determined what I want to study. My research is problem-oriented, and this problem is sometimes raised in a certain field, sometimes it is not, sometimes even in a certain field, but it can be reorganized into a problem in other fields. The study of that subject matter must have a certain importance, there are historical elements in it. ”
Regarding the widely discussed "publication of papers", Mo Yiming believes that "publishing papers is of course good, and it is natural to have any discoveries to write them down." But publishing a paper depends on the quality of the paper, your work should go beyond myself and beyond the contributions of others in this regard, and even sometimes beyond this era, you can set high goals, which is also the advice my supervisor gave me, to set high goals, even if you can't achieve them, at least on this road is wonderful. ”
Synergies – "Mathematics Needs discussion"
During the interview, Mo Yiming mentioned the importance of partners several times, and he believed that mathematics needed a lot of discussion. In terms of personal experience, he considers himself relatively fortunate to have found a good direction.
Yiming Mo has long been committed to the research of multivariate function theory, complex differential geometry and algebraic geometry, and has won the Sloan Award, the Young Researcher Award of the President of the United States, the 1998 Croucher Award, the second prize of the National Natural Science Award in 2007, the Bergman Award of the American Mathematical Society in 2009, and the Chen Provincial Award of the World Congress of Chinese Mathematicians in 2022.
"I have very capable mentors, mathematical collaborators who work with me, and I have a good environment to discuss with other mathematicians. For example, I originally did complex differential geometry and complex analysis, and I know a lot of people in this field, why can I work in algebraic geometry and number theory in the future? It's because I've met some cross-cutting researchers who, in the middle of the discussion, found that they have a lot of interesting things that I can try. Mo Yiming said.
Together with his collaborators, Mo developed a geometric theory using very small rational curve clusters (VMRT) to study Finow manifolds, which proved the rigidity theorem of irreducible Ermitt symmetrical spaces under Keller's deformation in 1998, and answered the Lasasfeld problem in the field of algebraic geometry in 1999 and 2004.
Yau Chengtong, the first Chinese winner of the Fields Medal, also mentioned the "synergy effect" in his autobiography "My Geometric Life":
Synergy refers to the interaction of two or more things, the effect will be more than the sum of the effects of each independent thing, or the independent things can not produce any results at all. Synergies abound in nature, for example, two hydrogen atoms and one oxygen atom combine to form hydrogen monoxide, or water, which makes up 71% of the Earth's surface. Water is amazing, it can sustain life, but by separating hydrogen from oxygen, these magical effects disappear without a trace. The same is true of bees and ants. Individual bees or ants can do only a limited number of tasks if they join forces. The same is true of neurons. One neuron can do nothing, but when a hundred billion neurons are connected by a hundred trillion synapses, they form a brain, and its function is beyond the imitation of today's technology.
"Looking back at history, mathematical breakthroughs have come from individual or group cooperation, and important problems have never relied on setting up committees, splitting up the work, and then distributing it as if it were homework. Even so, I still believe that it is still desirable to bring together elites from different fields but related fields for the exchange of ideas. They will have ample space and resources to conduct research according to their interests, regardless of time constraints. Looking back on my work, I also benefit from a similar environment. Yau Chengtong wrote.
As for how to deal with the setbacks in the research, Mo Yiming believes that when the method is not general, it is necessary to sum up the experience, even if there is no way to solve the problem. "I can say what level I have reached to solve this problem, and then I will look for a way out." The mind should be very quiet, and irritability is useless. ”
In Mo Yiming's view, the difference between basic mathematics and other disciplines lies in theory. Mathematics pursues truth, and the rigor of the theory is very high, even if there is a little bit of failure to look at the whole problem. Often when it seems that it can be successful, it will find that there are problems, so Mo Yiming believes that to do research, you must have the endurance, be able to accept the difficulties in the middle, and have the ability to accept failure.
Since about 2010, Mo yiming has been conducting research work spanning the fields of algebraic geometry, complex differential geometry, and number theory. Mo Yiming and his collaborators worked to develop a theory of differential geometry on submanifolds on single straight-grained projective manifolds filled with straight lines, and used complex differential geometry methods to solve a series of geometric problems from number theory in quotient space in bounded symmetry domains.
So from a more macroscopic perspective, what kind of environment does mathematics need? How does the level of Chinese mathematics compare to the highest level in the world?
"Compared with today in 1980, China's progress in the field of mathematics is very obvious, it can be said that it is a very excellent result, in some fields of mathematics has reached the world level, but at the same time, it will take time as a whole." Because mathematics is a historically heavy discipline, there are often some problems that have not been solved in a hundred years, it has a mathematical culture, which needs to be accumulated, and there is no need to rush in this regard. Mo Yiming said.
What conditions can make China's mathematical level improve faster? Mo Yiming believes that the environment of research and the environment of communication are important. For example, the Institute of Mathematics holds seminars or mathematical conferences, which is a more formal method of communication. "We should find more opportunities to share research inspiration, share our results and share our ideas with outstanding talents in all aspects of the world, and we may find new paths in the process of mutual communication, and I personally benefit from this opportunity."
Mo Yiming continued, "At present, the return of talents has begun to have important achievements in mathematics in China, and talents are also emerging, and I am full of confidence in the development of Chinese mathematics. ”
At the end of the interview session, the host asked Mo Yiming if there was anything else he wanted to talk to everyone about. Perhaps many reporters at the scene thought that he was tired and would not take the initiative to open the topic, but he paused slightly, and then said the following two paragraphs seriously:
I hope to tell young people that as long as they are interested in mathematics, as long as they have a certain confidence in their own abilities, there are still many opportunities to develop possibilities on the road of mathematics. Sometimes you don't know which aspect of mathematics will perform well in the beginning, and it doesn't matter, because you can only know after more discussion, so look for such opportunities.
Mathematics is a big world, and the direction of future development is a lot, pure mathematics, applied mathematics, and applied mathematics in the natural sciences. For example, there is a great application in biology, in physics, from the traditional 18th and 19th centuries mathematics and physics coexisted, and I feel that I will slowly return to the tradition in the future. The development of mathematics has a lot of contact with other disciplines, and there are many contacts in all aspects of mathematics, so let our younger generation discover it and then carry it forward and make Chinese contributions to mathematics.
Editor-in-Charge: Wu Yuewei Photo Editor: Zhang Tongze
Proofreader: Shi Gong