Recently, Tsinghua University held a defense meeting for the 2023 Undergraduate Special Scholarship, and the competition was extremely fierce. I received "A Letter from a Tsinghua Student" sent by the Student Affairs Department of the Tsinghua Party Committee. The letter said: "I still remember when we were in high school, we heard about the 'Tsinghua University Special Scholarship', which many people called the 'Fighting of the Immortals.' While learning about the deeds of the immortals of our seniors, we also silently planted the seeds of 'I want to go to Tsinghua' in our hearts. Now that we are in the garden, the halo of the college entrance examination has faded, and we are still working tirelessly. Strive for your dreams. As a recipient of the special scholarship from Tsinghua University, you are our predecessors and role models. I look forward to chatting with you and hearing your stories. "This makes me very ashamed. Although I have been working hard for many years on topological geometry, my results have been mediocre and many core theoretical problems have not been solved for a long time. I feel ashamed of my elders. Looking back on my study career at Tsinghua University, I received help from many mentors and friends, and I have very good memories. Here I summarize some experiences and lessons, hoping to be of benefit to my fellow students.
Combined major in science and engineering
I think the most important thing during college is to find academic "true love" and thus determine a lifelong goal. When I enrolled in the Department of Computer Science and Technology in 1989, the senior professors in the Department of Computer Science were far-sighted and formulated the Computer Science Theory major. According to the later explanation of the class teacher Huang Liansheng, the seniors believed that computer science will develop rapidly, which will inevitably require the integration of modern mathematical theories and methods into computer science. The purpose of establishing this major is to cultivate talents with both basic mathematics and computer science to prepare for the future. I was very lucky to be selected by Teacher Huang to enter the theory class and study basic mathematics and computer science at the same time. Professor Chen Tianquan teaches us mathematical analysis, using the textbook written by Professor Zolich of Moscow State University. At the beginning of school, we are exposed to abstract modern concepts, such as the bud layer of functions; at the same time, we learn recursive algorithms in computer courses, such as solving the Eight Queens Problem. I came to Beijing from a remote area with relatively backward economy and culture. When I first came into contact with the concept of recursion, I couldn't understand it, but I quickly mastered its inner principles and became comfortable in using it. But I didn’t really understand the essence of sheaf theory until more than ten years later. To this day, the cohomology of sheaves has become a sharp tool in my academic research. The students have made rapid progress in computer programming skills, but they are still unable to master mathematical theory construction. The abstract concepts in basic mathematics are very difficult to understand. Zhuo Liqi used the proof of many classic theorems in history as homework, which was difficult, complicated and time-consuming. Frustrated, many students protested to the department. They believed that computer science students did not need to learn "useless" basic mathematics, but should strengthen "practical" computer skills. Some students even changed classes in anger after their protests failed. Decades later, some students who transferred to another class are now working on Wall Street, living a comfortable life, and becoming a financial elite. It seems that they do not understand the cohomology theory of the upper class, which does not hinder their job hunting. On the contrary, another classmate loves mathematics, is talented, and has a high level of understanding. He has been practicing with great concentration and has reached a certain level of internal mathematics. His undergraduate thesis was to use manifold theory for speech recognition. He later went to Carnegie Mellon University to pursue a doctoral degree and continued to use mathematics to study artificial intelligence. He is currently working on Wall Street. It seems that the secular world does not favor pure mathematics. At that time, I was also very confused. On the one hand, I was in awe of the obscurity and difficulty of modern mathematics. On the other hand, I also doubted the value of pragmatism and tended to believe that major decisions in life should be based on a long-term perspective.
However, with the deepening of my studies, the deepening of my mathematical accomplishments, the mastery of computer skills, and especially the growth of experience, I gradually realized the huge difference between scientific thinking and engineering thinking: the purpose of science is to explain nature and pursue the truth, while the purpose of engineering is to transform nature and promote development. The end of engineering is science, and science changes the world through engineering.
The end of engineering is science
At that time, Academician Sun Jiaguang of the Department of Computer Science was responsible for a major national science and technology project: the research and development of domestic CAD systems. Therefore, the department offered a large number of courses related to computer-aided design, teaching geometric modeling, spline representation, graphics rendering, computational geometry and other related theories and algorithms in detail. Although the course content is rich and in-depth, I found that the most fundamental issue was not covered. It was like the elephant in the room that everyone turned a blind eye to: First of all, why can't we naturally construct high-order smooth spline surfaces on high-genus surfaces? Secondly, tensor product spline relies on regular quadrilateral meshing of the surface. Why does regular mesh generation have to be done manually rather than automatically? And this is equivalent to another mathematical problem: regular meshing depends on the configuration of singular points, so is there a governing equation for the configuration of singular points of the grid? At that time, many students were working on CAD projects, discussing various underlying algorithms enthusiastically every day, and writing codes with passion, but they turned a blind eye to the theoretical flaws of the CAD system. At first, I thought that the questions I asked were too naive and superficial, so that no one would pay attention to them. Later, I discovered that these questions were too difficult and beyond the cognitive scope of engineering. The entire automobile, aviation, and shipbuilding industries are based on CAD systems. Countless people have devoted themselves to them and are busy dealing with various practical engineering problems that arise in practice. However, no one has considered the theoretical foundation of CAD in depth. My knowledge at the time was too limited to answer these fundamental questions, and I later realized that the answers to these questions required modern mathematics.
The real opportunity came from a visit by Mr. Chen Shengshen. Mr. Chen specially gave a report to the students in the theory class and mathematics department. At the beginning of the report, Mr. Chen criticized the huge Tsinghua University for having no one teach algebraic topology, and then explained the theory of representational classes of fiber bundles, from topological obstacle classes to Gauss-Bonnet theorem, to Chen Shengshen's representational classes. Only then did I know that there was a subject called "Algebraic Topology", but I searched the Tsinghua Library at that time and could not find any suitable teaching materials. Finally, I found a copy of Jiang Zehan's "Theory of Fixed Point Classes" in the garbage pile of the library. The title page was stamped with a huge seal of "Cancelled". It was through self-study of this book that I began to come into contact with the theory of algebraic topology, but I swallowed it all and my understanding was superficial. Later, I was fortunate enough to become a disciple of Mr. Qiu Chengtong, and received Mr. Qiu's personal teachings, and gradually understood it deeply. In particular, Mr. Qiu taught me Hodge theory in differential topology many times, which became the beginning of computational conformal geometry. To this day, in my daily scientific research and teaching, algebraic topology algorithm and Hodge decomposition algorithm are inseparable. Many years later, I realized that the answer to the above basic questions about CAD theory lies in the representative theory of fiber bundles. In CAD systems, the construction of traditional splines relies on affine geometric invariants (blossom theory). However, general closed surfaces do not have an overall affine structure due to topological obstacles, so traditional spline theory cannot be naturally extended to curved surfaces. The quadrilateral mesh of a curved surface is essentially an overall meromorphic section of a certain kind of holopure line bundle on the Riemann surface. The grid singular point configuration is exactly the representative class of the holopure line bundle and satisfies the classic Abel-Jacobi equation. In the past 30 years, these issues have not attracted enough attention in the CAD/CAE field. However, in recent times, industrial software has returned to the center of the technology stage. It is believed that these foundational theoretical breakthroughs can give birth to a new generation of CAD/CAE algorithms. This experience made me realize that the end of engineering is science.
Science changes the world through engineering
On the other hand, during my studies at Tsinghua University, I discovered that although modern mathematical theories are broad, profound and abstract, there is a huge gap between them and real applications. There are a large number of beautiful and grand theorems in topology and geometry, which have unparalleled significance in theory, but cannot be directly applied. Those who do not understand the theory think it is an exaggeration, while those who understand the theory deeply deplore it. It is of great significance to extend the abstract geometric theory from smooth manifolds to discrete structures and convert the theory into algorithms. I once experienced something personally, and I was deeply moved. Students who have studied modern geometry know that the core concept in geometry is the Riemannian metric. Most differential geometry problems (including low-dimensional topology, such as the Poincaré conjecture) ultimately boil down to finding some special Riemannian metric. Students who have studied engineering mechanics and applied mathematics also know that the current mainstream methods for solving numerical partial differential equations are finite elements, finite volumes, level set methods, etc. However, these methods cannot be used to effectively calculate the Riemannian metric, so a large number of theories of Riemannian geometry and differential geometry cannot be applied in practice. When I was studying at Tsinghua University, I was always puzzled about how to design an algorithm to obtain the Riemannian metric through Gauss curvature, but I had no clue. I don't know if there is a corresponding mathematical theory, and I can't imagine the practical scenario of this algorithm. I don't know who to ask for advice, and no one can discuss it.
Later, I was fortunate enough to become a disciple of Mr. Qiu Chengtong and began to come into contact with Ricci flow theory. Mr. Qiu founded the school of geometric analysis, which solved geometric problems by solving differential equations. Many years later, I finally realized that geometric analysis methods are more suitable for computer science than algebraic geometry methods. At that time, for several years, a Hamilton professor flew from New York to Boston almost every weekend to discuss with Mr. Qiu day and night in the mathematics department of Harvard University. This professor is bright-eyed, elegant and suave, which is very different from the traditional image of a mathematician. I was very curious at the time, and later I found out that he was the inventor of the Ricci flow. At that time, there were different schools trying to prove the Poincaré conjecture, such as the partial topology method of the Thurston school, the geometric analysis method insisted by Mr. Qiu and Hamilton, etc. Both Mr. Qiu and Professor Hamilton believe that Ricci flow is the ultimate tool to solve the Poincaré conjecture, and they are close to the final proof. The final obstacle is the so-called "cigar" type singularity, so they study day and night. Later, Perelman ruled out the existence of cigar-shaped singular points and gave an outline of the proof of Poincaré's conjecture. This is another great victory for the school of geometric analysis, thus once again proving Mr. Qiu's amazing mathematical insights. I learned about the great power of Ricci flow and firmly believed that it would be a key theoretical tool for calculating Riemannian metrics. The traditional Ricci flow theory is based on smooth manifolds, and all manifolds in the computer field are approximated by discrete combination structures. Therefore, extending Ricci flow theory to discrete manifolds has become the focus of my research.
Under the guidance of Mr. Qiu Chengtong, I worked in depth with many mathematicians and spent more than ten years establishing the theory of discrete surface Ricci flow and inventing practical algorithms. At that time, Academician Sun Jiaguang of Tsinghua University was explaining the principles of CAD to everyone. He mentioned that the most difficult program in the world was to calculate the intersection of two curves on a plane. At the time, this puzzled me. Academician Sun explained that ideal geometric algorithms assume that each calculation step has infinite accuracy, but in reality, the accuracy of each calculation step is limited, which brings substantial difficulties to geometric algorithms. Although we proved the existence, uniqueness and convergence of the discrete Ricci flow solution, the biggest difficulty we encountered in writing practical algorithms was the limited machine accuracy. In order to overcome this difficulty, the algorithm design is very circuitous and requires a lot of engineering skills. Currently, the discrete surface Ricci flow algorithm is the only algorithm that can calculate the Riemannian metric through curvature, thus substantially surpassing the limitations of traditional finite element algorithms. Surface Ricci flow has been widely used in medical imaging, computer graphics, CAD/CAE and other fields, such as virtual colonoscopy, surface parameterization, mesh generation, etc. The virtual colonoscopy patents of my collaborators and I were once licensed to Siemens. Through Siemens’ marketing network, our virtual colonoscopy algorithm has spread all over the world and has truly saved many people’s lives. At present, industrial software is on the rise again. My collaborators and I all realize that the solution to the underlying core problems of industrial software cannot be separated from the fiber cluster theory and Ricci flow theory. On the one hand, we expect that the discrete surface Ricci flow algorithm and the Abel-Jacobi algorithm can fundamentally promote the development of this field; on the other hand, we also find that in the field of industrial software, Some of the core problems require more powerful mathematical methods. If we use relatively superficial topological theory, the algorithm will have exponential complexity, making it unsolvable in reality. If we analyze the problem in detail, we will encounter a key group that is a non-Abelian group. The traditional fiber cluster theory cannot be directly applied, and we must invent new mathematical tools to completely solve it. Over the years, we have been tracing our roots and pursuing our goals diligently.
Expectations and suggestions for junior students
Over the years, I have met many Tsinghua classmates and alumni around the world. No matter where you are, Tsinghua people give others the impression that they have terrifying engineering capabilities and strong and tough psychological ability to withstand stress. I think this is related to Tsinghua’s special humanistic environment and ultra-high-intensity engineering training. When I first entered Tsinghua University, the classmates around me were top scorers, second picks, and top picks from various provinces and cities. They were all outstanding academic elites. Everyone is the best in the world and has never lost in life. Suddenly arriving at Tsinghua University, where there are many masters, various mental blows came one after another, and everyone was under tremendous psychological pressure. At that time, a local classmate in Beijing declared: "It was a mistake to bring so many outstanding people together." At the first freshman meeting in college, President Chang Xiaowen's enlightenment was enlightening. Principal Zhang used his own ups and downs in life as an example to advise everyone to "be rough with yourself." I think the education system at that time introduced relatively brutal competition in the college entrance examination, which strengthened students' sense of competition and weakened everyone's original intention to pursue knowledge. Our ultimate goal in entering university is to pursue knowledge and learn knowledge, not to compete with others for ranking. I hope that the junior students can effectively learn difficult modern theories in a relaxed and natural state, which requires breaking through their own vanity.
At the same time, many students chose majors not out of natural love, but because of the secular values at that time. The real purpose was the interests and reputation behind the major, so the academic level could not move forward too far. When I was studying, computer science was extremely popular and basic mathematics was not available. However, through a period of study, I realized that the structure of the mathematical world is exquisite and profound, magnificent and unified, and its depth exceeds ordinary engineering thinking. Therefore, I quickly determined that geometry was my “true love.” But it brought up a deeper anxiety: whether I was talented enough to really make a difference. In the field of basic science, those who can leave their names in history are geniuses. If a young man with mediocre qualifications devotes himself to it with only passion, will he accomplish nothing and have regrets for the rest of his life? At that time, I asked my teacher for advice. Teacher Chen Tianquan said that as long as a young man keeps working hard, he will always achieve something. After years of insights, I now feel that mankind will face new challenges in every era, and emerging disciplines will emerge. Everyone has their own unique talents. As long as they work hard to pursue them, they will eventually respond to the call of the times and make substantial contributions. The purpose of traditional mathematicians is to "see through natural structures that others cannot see through." However, the era I live in is the era of the rise of computers. The development of the times requires us to "calculate natural structures that others cannot calculate", and this has become the goal I pursue. Later, I had the honor to follow Mr. Yau Shing-tung. When Mr. Yau once talked about how to measure a scholar's contribution, he mentioned that "either he has proved a new mathematical theorem, or he has written a practical computer program that cannot be replaced." However, a large number of mediocre people are somewhere in between, unable to reach either end. This gave me great spiritual encouragement and helped me determine the direction of my efforts: applying modern mathematics, inventing new algorithms, and substantively advancing the frontiers of computer science! Therefore, I hope that young students will not be too anxious about their talents and the uncertainty of the future. They should try to broaden their horizons, learn modern scientific theories, cultivate engineering skills, proactively discover the direction they are truly passionate about, and look for emerging fields of their own.
I later met a large number of young people in academia who were very eager to publish articles but did not work hard to improve their academic cultivation. Especially in recent years, the rise of artificial intelligence in the field of computer science has made "eliminating the intervention of human thought" and allowing "machines to discover laws independently" a new value concept. This denies the minimum requirement for humans to strengthen their own academic cultivation from the basic level of methodology. In fact, this has caused huge ideological confusion in the academic world. "In the era of AI, do humans need Einstein anymore?" has become a mystery. Therefore, a large number of young people have devoted themselves to the infinite parameter adjustment movement with great enthusiasm, mass-producing papers in an industrial production line model. As for these papers, humans cannot understand their working mechanisms, and there is no way to judge their quality. It seems that academic paper publishing is confused with the commercial advertising model. Many traditional disciplines are facing the crisis of having no successors. I still uphold traditional academic values and firmly believe that in the foreseeable future, computers cannot replace humans in academic fields that require profound thinking and tremendous creativity. I hope that fellow students will not be superstitious and follow trends too much, but will think independently, have a long-term vision, and pursue profundity. In particular, they will learn difficult and cutting-edge science, seek truth and beauty, and develop core competencies that are difficult to replace.
thanks
Thanks to Mr. Qiu Chengtong for his teachings, and to many Tsinghua professors such as Huang Liansheng, Academician Sun Jiaguang, Chen Tianquan, and Lu Kaicheng for their painstaking cultivation! Thank you to Yue Mengkun from the Student Affairs Department of my alma mater for your letter! I wish more students will make major breakthroughs in different fields of science and technology, substantially promote the development of human civilization, and help China stand among the nations of the world!
Machine-translated from the Chinese original 回复《一封清华学子的来信》 (published 2024-03-28). WeChat original ↗