I drafted a book for my supervisor, but unexpectedly got a doctoral dissertation...

Some graduate students are not good at finding research topics on their own, but rather "wait for the rice to cook." Ding Jiu, a professor in the Department of Mathematics at the University of Southern Mississippi, recalled his special experience of studying and selecting and writing thesis topics.

Written by Ding Jiu (Professor of Mathematics at the University of Southern Mississippi)

Before the outbreak of the COVID-19 pandemic, when I returned to China for academic visits and communicated with professors, they mentioned that some graduate students would not find topics for their own research, but would wait for their supervisors to assign them a research topic, just like handing out test papers in class. "Food given out of pity" is never as comfortable as a buffet. We all know that free-range chickens raised by farmers are delicious and juicy, far from being comparable to collectively raised broiler chickens. This is because broiler chickens eat when they open their mouths, and their diet is monotonous; while free-range chickens forage everywhere and are rich in nutrition. The market price of free-range chickens is therefore much more expensive than broiler chickens. When I was a child, I lived in the staff dormitory of my parents' school. Before it rained, I loved to use a big broom to catch short-flying dragonflies on the playground to feed the chickens, because they laid eggs for us to eat. Just like when you are studying, your vision should not be limited to the small pages of textbooks. When choosing a research topic for a degree thesis, it is also best to pay attention to everything and take the initiative. In my case, the topic of my doctoral dissertation was chosen by chance, and even my supervisor, Professor Li Tianyan (1945-2020), might not have thought of it beforehand. It was the result of "learning everywhere you look". In fact, the first research article I wrote after going to the United States benefited from the pioneering work of my old classmate Dr. Wei Musheng from Nanjing University, and had nothing to do with the several fields that my supervisor was involved in. Although it was not included in my doctoral dissertation, its birth opportunity was similar to that of the later degree thesis.

For the sake of accuracy, this article will introduce some mathematical concepts. I will use elementary or geometric language, as well as metaphors and analogies, to describe the concepts, even if the reader does not fully understand the mathematical connotations. Impatient readers should not be discouraged and reduce their enthusiasm to continue reading. I hope that the drama and inspiration of the story will ignite their greater desire to read.

“Finding rice to cook”

The first research I did during my doctoral studies was on the perturbation theory of least square solutions for rank-deficient matrices. Carl Friedrich Gauss (1777-1855), one of the founders of the least squares method, had come up with the idea of ​​the method when he was 18 years old. As the director of the Observatory of the University of Göttingen, he invented the least squares method while studying astronomical observation data. This is geometrically related to the curve fitting of given experimental data points on a plane. For example, imagine that there are ten points on a rectangular coordinate plane, which can be regarded as ten sets of data for a certain experimental result. They generally do not happen to be arranged along a straight line. However, can we draw a straight line so that the sum of the squares of its "perpendicular distance" from these points is the smallest? This is a simple example of the "least squares problem". Its answer is yes, and its solution is the "least squares method". The least squares problem is determined by a matrix, which is a set of numbers arranged in several rows and columns. For "full rank" matrices (that is, the "rank" of the matrix is ​​equal to the smaller value of the number of rows and columns), the theory and algorithms of least squares are already very mature and constitute part of numerical linear algebra, a sub-discipline of computational mathematics.

My college classmate Wei Musheng ranked first in the province in mathematics in the 1977 Jiangsu College Entrance Examination - he scored full marks in both the main questions and the supplementary questions. After graduating from undergraduate studies, he went to Brown University in the United States on public funds and received his doctorate in 1986. His doctoral dissertation was about scattered wave calculations, which required the consideration of least squares problems. But at this time, the matrix was no longer full rank, but "deficient rank", that is, the rank of the matrix was less than the number of rows and columns of the matrix. He could not find a ready-made perturbation theory for deficient rank problems in the literature for reference. Once at an academic conference, Wei Musheng met Professor Gene Howard Golub (1932-2007), a great figure in numerical algebra, a dual academician of the U.S. Academy of Sciences and the Academy of Engineering, and a professor in the Department of Computer Science at Stanford University, and asked him for advice. The other party's answer surprised him: no one had seriously studied such problems. So Wei Musheng decided to lay the first stake in this new field himself. In 1989, his first paper on the least squares perturbation theory of deficient rank matrices was published in the journal Linear Algebra and Its Applications.

In the 1986-87 school year, Wei Musheng worked as a postdoctoral fellow at the Institute of Mathematics and Its Applications at the University of Minnesota. In the fall of 1987, Wei Musheng came to the Department of Mathematics at Michigan State University, where I was doing my doctoral studies, to continue his postdoctoral research. Professor Li Tianyan selected him from six or seven applicants because Professor Peter Lax (1926-), a great mathematician at the Courant Institute of Mathematical Sciences, wrote a strong letter of recommendation. Of course, someone who can make Lax write a letter is not an ordinary person. Indeed, the reason why Wei Musheng received this honor is that in his doctoral thesis, he overturned a point of view in Lax's monograph on scattered wave theory. For a whole school year, the two of us old classmates often drove to go shopping with our families and spent many happy times together. In that autumn school season, I read several articles written by Wei Musheng and found them very interesting.

Dr. Wei Musheng's pioneering work is essentially to prove the "upper semi-continuity" of the general least squares solution by estimating the upper bound of the error of the perturbation solution of the rank-deficient least squares problem. Many phenomena in nature are continuous, such as water flowing continuously. "The continuity of the solution" probably means that when the data of the problem being solved changes slightly, the solution does not change much. In order to be able to apply the famous "singular value decomposition theorem" in matrix theory, he had to use a matrix norm named after the German mathematician Ferdinand Georg Frobenius (1849-1917) a hundred years ago. This norm is to arrange all the elements of the matrix of m rows and n columns into a vector with mn components, and then calculate the Euclidean norm of the vector (that is, sum the squares of all components and then take the square root). After reading the full text, a strong feeling quickly came to my mind: the conclusion of the article is naturally beautiful, and the mathematical analysis is also very insightful, but using this norm is not as natural as the vector Euclidean norm used in the definition of the least squares problem itself. So I took the initiative, concentrated my energy and thought hard, and soon I got a clue. Using only the Euclidean norm, I obtained a relatively simple perturbation bound.

Because this is the first article I wrote after coming to the United States, I was quite excited when I finished it. I never submitted my master's thesis for publication. On the one hand, my wife was pregnant at the time, so I had to do my part. In addition, I was busy studying abroad and had no time to organize it. On the other hand, I no longer valued that work. After coming to the United States, I found that the international research on simple fixed point algorithms based on triangulation has become quiet, not as popular as it was in the 1970s and early 1980s. However, the modern homotopy algorithm based on differential topology has always been vigorous. The doctoral dissertations of several of my brothers are closely related to this method. Later, I also used the idea of ​​homotopy continuation method in the research of optimization.

In my excitement, I sent the first draft of the article to my advisor, Li Tianyan, who was a lecturer at the Institute of Mathematical Analysis at Kyoto University in Japan at the time, and asked him for advice. Professor Li Tianyan quickly replied, and in a three-page letter, he expressed specific opinions on the main theorems of my article and gave enlightening comments on the methodology of reading and research. He usually doesn't praise students in person, but this time he encouraged me enthusiastically because in the matter of doing academic research, I did not "wait for the rice to cook", but "looked for the rice to cook". According to his point of view, this is a graduate student's "obligation".

Teachers preach

Professor Li Tianyan's own study experience is the best example of how doctoral students should conduct research. His three major academic contributions before the age of 30 were: the eight-page short article "Period Three Means Chaos" gave the first precise definition of the concept of "chaos" in mathematics; he was the first to construct Brouwer fixed points computationally, which was the pioneering work of modern homotopy continuation method; and he proved the Ulam conjecture in computational ergodic theory for the first time in history. The most famous of these works is the result of his collaborative research with his doctoral thesis supervisor, James Yorke (1941-), which has been cited more than 5,900 times to date. This is among the best in the group of mathematical papers whose citations are generally much lower than those in the fields of experimental science and engineering. In addition to Yorke, Bruce Kellogg (1930-2012) was also an author of the second paper. The third achievement he completed alone provided the source of inspiration for my doctoral thesis.

Let's first review how he was "lucky" enough to write the world's first article that used the modern homotopy continuation method to calculate Brouwer's fixed points. This great theorem in topology, named after a Dutch mathematician, is the intermediate value theorem in elementary calculus in the simplest one-dimensional case. Its geometric properties are well known to everyone: any continuous curve connecting points on both sides of a straight line must intersect the straight line. In the two-dimensional case, Brouwer's fixed point theorem is: any continuous self-mapping (i.e., the value range is included in the definition domain) on a closed disk must have a fixed point, that is, the point is mapped to itself. Li Tianyan graduated from National Tsing Hua University in Hsinchu, Taiwan in 1968. After serving in the military for a year, he went to the Department of Mathematics at the University of Maryland in the United States to study for a doctorate, under the tutelage of Professor York. In 1973, a year before graduation, he took Professor Kellogg's course "Numerical Solutions of Nonlinear Equations". In the course, the professor talked about the new proof of Brouwer's fixed point theorem published ten years ago by Professor Morris Hirsch (1933-) of the Department of Mathematics at the University of California, Berkeley.

The idea behind this concise proof by contradiction is: assuming that fixed points do not exist, it will lead to a contradiction with a certain theorem in topology. This latter theorem says that there is no smooth mapping that maps a closed disk to its circumference so that all points on the circumference remain fixed. These interesting and profound theorems in topology can explain why there is a vortex on the top of a person's head where hair does not grow. When Li Tianyan heard such a novel proof, he, who likes to think, suddenly came up with an idea: he could use this idea to calculate the fixed points guaranteed by the theorem. Because the closed disk is a two-dimensional area, and the circumference is only a one-dimensional curve, for the mapping that Hirsch considered to map the disk to the circumference, the domain has one more dimension than the range, so there is an "inverse image" curve that starts at a point on the circumference and ends at the fixed point set of the original mapping. As long as this "homotopy curve" can be followed numerically, the fixed points can be calculated. Active and independent thinking leads to such a wonderful new algorithm! Creative thinking is a miracle that those readers who rely on rote memorization of definitions, theorems, and proofs can hardly imagine. But for explorers who like to trace the roots and find original ideas, this is the most natural outcome.

When Li Tianyan told York about his idea, the latter fully supported him to continue, even though he had other research projects in hand. The far-sighted mentor knew the value of this topic. After two months of programming and calculation, Li Tianyan's algorithm idea was finally realized - a thin page of printed paper recorded the numerical results of the first modern homotopy algorithm in history. When the organizing committee of the fixed point algorithm and its application conference to be held at Clemson University heard that they used differential topology ideas to construct a new homotopy fixed point algorithm, instead of following the route of the simple fixed point algorithm based on simple decomposition and combination techniques pioneered by Yale University economics professor Herbert Scarf (1930-2015) in 1967, they immediately provided two tickets and invited them to attend the conference to read the paper. Later, in the preface to the conference proceedings, Scarf praised the new ideas of the Kellogg-Lee-York article. From then on, the modern homotopy continuation method entered the big stage of computational mathematics.

"With a strong will"

As mentioned above, the three most famous and outstanding works in Professor Li Tianyan's academic career were all completed during his doctoral student period. The third paper on the "Ulam conjecture" was completed by him independently and published in the American Journal of Approximation Theory in 1976. How did this article come about? In 1973, York and his collaborator, Andrzej Lasota (1932-2006), a member of the Polish Academy of Sciences, published a paper in the journal Transactions of the American Mathematical Society that has now become a classic document of ergodic theory, in which a theorem on the existence of absolutely continuous invariant measures was proved. It asserts that a class of piecewise elongated self-mappings defined on an interval has an "invariant density function". The density function is a mathematical object that often appears in probability theory. It is a function with non-negative values ​​and an overall integral of 1. That is, the area of ​​the "curved rectangle" below its image and above the interval is equal to 1. After the existence of the invariant density function was guaranteed, Li Tianyan began to consider how to calculate it. In other words, how to effectively approximate it numerically. He proposed an approximation method using piecewise constant functions, and proved the convergence of the algorithm for the type of interval mapping considered by Losuda and York. As the name suggests, the piecewise constant function takes constant values ​​on the subintervals that divide the domain interval.

However, Li Tianyan was completely unaware that the father of the US hydrogen bomb, the Polish-born outstanding mathematician Stanislaw Ulam (1909-1984), had already proposed this method to calculate the invariant density function in his 1960 book A Collection of Mathematical Problems, which was only 150 pages long. After writing the article, Li Tianyan heard that this was the Ulam method that had existed for more than a decade. And Ulam speculated in the book that as long as the invariant density function exists, the algorithm will converge. The "Ulam conjecture" gave birth to the discipline of "computational ergodic theory" with important application value in physics and engineering. The "historical coincidence" between Li Tianyan's article and Ulam's method also led to the change of the title of the article, adding "a solution to Ulam's conjecture". This milestone work in the field of computational ergodic theory was ultimately "Finite approximation by Frobenius-Perron operators - a solution to Ulam's conjecture".

Many years later, Professor Li Tianyan recalled to me how he came up with this masterpiece, and said with great emotion: "If I had known in advance that even Ulam, a mathematician of the same level as von Neumann, had not given a proof of the convergence of this algorithm, I might not have dared to take on this task." However, when he was young, Li Tianyan was a brave warrior who was "not afraid of tigers." According to himself, he "relying on a strong will, he always persists to the end and never gives up easily." He believes that problems that big people cannot solve do not mean that small people cannot solve them either, and the way big people think about problems is not necessarily the only way to solve problems. On the road of learning, as long as you have an independent spirit and free thoughts, as long as you spend one more minute thinking than others, you can get to the bottom of seemingly difficult problems.

Before the early summer of 1987, after passing the exams in two foreign languages ​​(neither English nor Chinese counts as foreign languages), I continued to take classes while actively keeping up with a brand new field - interior point algorithms for linear programming. It was related to the optimization direction of my master's degree at Nanjing University, and it started with a groundbreaking paper published by Indian Narendra Karmarkar (1956-) in 1984. This field had already begun to gain popularity in the international optimization community at that time, and researchers followed up in droves. Many people even predicted that Karmarkar would win the Nobel Prize in Economics before the end of the last century, just like Soviet mathematician Leonid Kantorovich (1912-1986), who was the first to propose an effective computational method for linear programming. However, this prediction did not come true. Considering that my main field was mathematical programming, Professor Li suggested that I keep up with the rapid development of interior point algorithms. Some of his friends who had academic exchanges with him, such as Professor Masakazo Kojima (1944-), a famous Japanese optimization theory scholar, often sent preprints of articles on this subject. Several Chinese scholars, such as Ye Yinyu, a doctoral student in operations research at Stanford University, also began to emerge. I tried to learn more about these latest research results, gradually approached the academic frontier, and completed several articles on interior point algorithms for linear complementarity problems. The first one I collaborated with my supervisor was fortunately published in the inaugural issue of the SIAM Journal on Optimization, which was newly established by the American Society for Industrial and Applied Mathematics in 1991. I had planned to organize these contents into my doctoral thesis, but the outcome was unexpected.

West Coast Tour

In March 1989, my three-year-old daughter came to the United States with her grandmother. This was the first time we met, although when she saw me at the Detroit airport, she said to me in pure Yangzhou dialect: "I've seen Dad in photos." The spring semester had just ended in early June of that year, and the academic year came to an end. Professor Li's three-quarter course "Ergodic Theory on [0, 1]" also came to a successful conclusion. Although his main interest at that time was no longer chaos and ergodic theory, but in matrix eigenvalues ​​and homotopy solutions of multivariate polynomial equations, we disciples gained knowledge and broadened our horizons, and had a clearer understanding of his research results until the mid-1980s. Indeed, if you know nothing about your mentor's past work, what kind of student are you? To be a good student, you must not only understand your mentor's current work, but also know what your mentor has done in the past, otherwise you can be called a lame disciple. This is the same as how to treat the history of science. The great all-round mathematician Henri Poincaré (1854-1912) warned us: "If we want to foresee the future of mathematics, the appropriate way is to study the history and current situation of this science." This sentence was placed at the beginning of the preface by Morris Kline (1908-1992), the author of the famous mathematical history book "Mathematical Thought from Ancient to Modern Times". It is equally important to understand history and to understand the current situation, because history is a mirror. Hermann Weyl (1885-1955), the most outstanding disciple of German mathematician David Hilbert (1962-1943), once mentioned that he "likes to teach the history of mathematics", which makes a lot of sense.

While I was in Northern California for a meeting, my family and I planned to tour the west coast of the American continent in June. This would be my first long-distance trip since coming to the United States.

During the month from Michigan to San Francisco, our family left footprints in many places along the way. I reunited with many old classmates and old acquaintances from Nanjing University. At the first stop of the journey, the University of Illinois at Urbana-Champaign, I unexpectedly met my college classmate Hu Zhuoxin and chatted all night; I also reunited with Li Qiaoying, a student of the Department of Chemistry of Nanjing University in the seventh and eighth grades, who went to the United States to study for a doctorate with me on the same plane. Then my family went to Kansas City and was warmly welcomed by the family of Korean senior brother Li Hongjiu, who teaches at the University of Missouri-Kansas City. After arriving in Salt Lake City, the Mormon stronghold, Yin Guangyan, a doctoral student at the University of Utah, drove us to see the scenery of the Salt Lake. Now he and two other "Jiangsu College Entrance Examination Champions" among my former classmates have begun to enjoy the "beautiful sunset" retirement scenery.

When I arrived in the San Francisco Bay Area, I met Zhang Yanning, who had obtained a doctorate degree in statistics from Stanford University. Born in Beijing, he had excellent academic performance and loved long-distance running. He was admitted to the Chinese Academy of Sciences Center for Computing and studied abroad. The first letter I received from my old American classmate after I came to the United States to study was his enthusiastic "welcome postcard". In the second letter, he also praised me for naming my newborn daughter "Yizhi": "You are worthy of your literary skills." Later, Zhang Yanning, who had a successful career, did not lose his passion for long-distance running. He participated in several marathons, including the famous Boston International Marathon.

I also met again with Dai Jiangang, a mathematical genius from my class of 1978 at Nanjing University. He is writing his doctoral dissertation at the Department of Mathematics at Stanford University (he is currently a professor at the School of Operations Research and Information Engineering at Cornell University and the dean of the School of Data Science at the Chinese University of Hong Kong (Shenzhen). In early March, he went to San Francisco International Airport to pick me up and sent my mother and daughter to the boarding gate for Detroit. We strolled around the beautiful Stanford campus and toured this world-renowned university. My mother took a photo in front of the famous cathedral in the center of the campus. This was built by Mrs. Stanford in memory of her husband, the railroad tycoon who co-founded the school with her in 1885. A quarter of a century later, before Thanksgiving in 2013, I went to visit my daughter who was already working there. Under the cloudless blue sky, the two of us took a photo in front of this beautiful church, allowing my 85-year-old mother to see the magnificent Stanford architecture again.

Flashes in the Manuscript

Before leaving for the trip, Professor Li Tianyan asked me if I was interested in helping him complete the first draft of a Chinese book based on the ready-made lecture notes of the course he had just finished. A certain academic foundation in Taiwan wanted him to publish this book. Last year, he met Professor Mo Zongjian (1940-) of the Department of Mathematics at Purdue University in Japan, and the two discussed this matter, which was one of the original intentions of his opening this course. Professor Li promised to allocate a portion of the summer research grant awarded to him by the National Science Foundation to me, so that I can concentrate on writing the book without being distracted by teaching. Of course I am willing to do so. This is not only an excellent opportunity to consolidate the knowledge I have learned, but also provides me with a training ground for future academic writing.

After returning to Michigan, I quickly got into the groove and started drafting the book assigned by my supervisor. The basic framework of the book was already in place, and I only needed to add the basic parts as preliminary knowledge and unify the written expressions and language symbols. I worked non-stop for two months. This was also a process for me to reorganize my knowledge and practice academic writing, which provided me with an excellent opportunity to practice writing my own books in the future. More importantly, at a certain moment when I was writing the chapter on the Ulam method for calculating absolutely continuous invariant measures, I accidentally had a flash of inspiration, which created an opportunity for a new research.

The Chinese book that Professor Li plans to publish mainly talks about a type of positive operator widely used in ergodic theory - the Frobenius-Perron operator, which is a linear operator that maps non-negative functions to non-negative functions and keeps the integral unchanged. The former property is the definition of a "positive operator". The name of the operator is borrowed from the names of two German mathematicians, but in fact they have nothing to do with them. It is only because this infinite-dimensional operator inherits several good properties of non-negative matrices, and because Perron (Oskar Perron, 1880-1975) in 1907 and Frobenius in 1912 established the general theory of non-negative matrices, that Ulam borrowed their names in "Mathematical Problems" to name the operator. This phenomenon of "putting the wrong name on someone else's head" is not uncommon in the history of mathematics. For example, the Newton method for solving nonlinear equations was not formally proposed by Newton. He just used it to approximate the roots of a polynomial equation. The systematic study of the convergence theory of Newton's method is attributed to the 20th century Russian mathematician Kantonovich. L'Hôpital's rule for finding indeterminate limits in calculus is even more of a "fraudulent" result. This rule, which was placed in L'Hôpital's (Guillaume de l'Hôpital, 1661-1704) book in 1696, was actually discovered by Swiss mathematician Johann Bernoulli (1667-1748). Invariant density functions are fixed points of Frobenius-Perron operators. The first few chapters of Professor Li's manuscript are all about the existence theorems and properties of operator fixed points, and the last chapter involves their calculation. The title is "Finite Approximation of Frobenius-Perron Operators", which contains Ulam's method and Li Tianyan's beautiful proof of Ulam's conjecture and the convergence of the L'Hôpital-York interval mapping family.

When I was about to finish writing this chapter and the whole manuscript was about to be completed, a question suddenly popped up in my mind. From the perspective of computational mathematics, approximating general functions with piecewise constant functions is the simplest and crudest approach. Why can't we use piecewise linear or even piecewise quadratic functions to approximate? Common sense tells us that if we use a horizontal line to approximate a catenary, the accuracy is much lower than using a line segment connecting two points on the curve to approximate it. So I was very curious and immediately picked up a pen and paper to draw pictures and calculate. For more than 30 years, I have been committed to the research of computational ergodic theory, and this journey of thousands of miles started from here. It has nothing to do with my previous research on interior point algorithms, and they belong to two worlds that are 10,800 miles apart. Since I have mastered the basic knowledge of ergodic theory, a branch of pure mathematics, in the United States, and have a good foundation in computational mathematics, my old profession of Nanjing University, my ideas are relatively clear and my progress is quite smooth. I noticed that the Ulam method is a structure-preserving algorithm that preserves positivity and integrals, and it also belongs to the category of traditional projection algorithms. So I extended it along these two directions. Soon, I constructed two new algorithms based on piecewise linear or piecewise quadratic polynomials. The first type relies on the Galerkin projection principle, and the other type preserves structure by using finite-dimensional Markov operators, which I named Markov finite approximation methods. The Markov operator, named after the Russian mathematician, has a wider range than the Frobenius-Perron operator and is defined as a positive operator that preserves the integral of non-negative functions. For the Losuda-York type interval mappings that Ulam's method converges to, I proved the convergence of the new algorithm. To prove that the higher-order numerical method converges faster, I used the Sun-workstation computer in the department (referring to the workstation launched by the computer company Sun Microsystems - Editor's Note) and input my own Fortran program to perform calculations and comparisons. The results of numerical experiments showed that they converged much faster than Ulam's method, and the difference was as big as Lu Bu's Red Hare racing with Lu Bu himself. Later, two Chinese professors theoretically studied the convergence rate of the piece-by-piece linear Markov finite approximation method. Together with the subsequent articles published by Professor Li and I in 1998, we finally established the convergence rate theory and error estimation of this type of algorithm.

At the end of August 1989, I finished the first draft of Professor Li Tianyan's Chinese monograph. As a byproduct, I also produced two research articles. What surprised me was that I didn't even write a draft for this book. Each chapter and each section was basically based on the outline of Professor Li's course lecture notes. I first thought about it in my mind and then wrote it down in one go, which greatly saved writing time. In two months, I not only drafted a Chinese book, but also did some meaningful research. When I handed the manuscript to my supervisor and presented the first draft of the paper, he was surprised at first, but was satisfied after reading it. From then on, I never asked about the fate of the manuscript. Unfortunately, Professor Li Tianyan has been busy with his grand research plan on numerical solutions of polynomial equations. He has basically left the field of chaotic dynamical systems and ergodic theory, which made him famous in his early years. Therefore, he never had time to revise and complete this book, which is a pity. On the contrary, the interest I developed from learning ergodic theory in my tutor's course, coupled with this unique experience of writing and research, made me leave the interior point algorithm and devote myself to computational ergodic theory. I have been doing this for many years and have collaborated with Dr. Zhou Aihui from the Institute of Computational Mathematics and Scientific Engineering Computing of the Chinese Academy of Sciences to publish a Chinese graduate textbook "Statistical Properties of Deterministic Systems" through Tsinghua University Press in 2006. Its corresponding English version was jointly published by the press and Springer in Germany at the end of 2008.

Graduation and work

One morning in October 1989, Professor Li Tianyan came to my teaching assistant's office and said to me kindly: "You can consider graduating next year and compile these two latest articles into your doctoral dissertation." I was grateful and agreed to his arrangement. Among our batch of doctoral students recruited directly from the mainland, except for me who came to the United States in January 1986, the rest of us were from famous domestic universities in the 1977 class ("1977" refers to the first batch of college students after the resumption of the college entrance examination - Editor's note), graduated from Jilin University, Wuhan University, Xiamen University, etc., plus a doctoral student who came to him from Northwestern University, a famous private university in the United States, and transformed from a visiting scholar to a doctoral student. All of them entered the school in August 1986. Naturally, I could graduate earlier than them. The previous year, my senior sister from the 1977 class of Beijing Normal University had already graduated with a doctorate and found a university teaching position. She was an assistant professor in the Department of Mathematics at Clemson University, where Professor Li's Brouwer fixed point calculation paper began to become famous. At that time, the US economy was still relatively strong, and there were many new university teaching positions.

The difficulty of finding a job is linearly related to the economic situation and even the social environment. In 1957, the Soviet Union launched a satellite into space, which shocked the United States, thinking that its technological level had fallen behind that of the Soviet Union. At the command of the top leaders, American universities began to expand immediately, resulting in the popularity of new PhDs in the 1960s, and everyone could find a good job teaching at a university. As a result, some of them, due to congenital deficiencies or acquired laziness, failed in the fiercely competitive academic environment, and their treatment deteriorated, especially in research universities. There are also such people among the professors of Michigan State University, and the proportion among the old professors is not too low. I remember one time, the tutor pointed to the office of a professor with a low academic status from a distance and joked to his students: When I was still in high school, he was a professor, but his current salary is almost half of mine. When Professor Li received his doctorate in 1974, the good days of American PhDs were over, and many people could not find a job. He was lucky to find a university teaching position, while many Taiwanese PhDs like him had to go home. However, many people later made a lot of money due to Taiwan's economic boom. He also told us that when his first doctoral student, Zhu Tianzhao from Taiwan, received his degree in 1982, the situation was reversed again. He was overwhelmed by the number of campus interview opportunities. In the end, Dr. Zhu chose North Carolina State University. His research performance was very outstanding, and he was promoted to full professor six years later.

However, I didn’t expect that I would graduate in 1990, just in time for the most severe round of the US university teaching job market! Fortunately, I found a formal assistant professor position, but that’s another story.

Written on Sunday, March 24, 2024

Hattiesburg Summer House

Note: This article is modified based on Chapter 6 "Doctoral Dissertation" of "Personal Experience of American Education: Thirty Years of Experience and Reflection", published by the Commercial Press in 2016.

Acknowledgements: We would like to thank Dr. Huijian Zhu for providing two revision suggestions that improved the accuracy of the narrative.

This article is supported by the Science Popularization China Starry Sky Project

Produced by: China Association for Science and Technology Department of Science Popularization

Producer: China Science and Technology Press Co., Ltd., Beijing Zhongke Xinghe Culture Media Co., Ltd.

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