Duan Xuefu,1965438+was born on July 29th, 2004 in hua county, Shaanxi Province. His father Duan Da was really a scholar in Shen Jia during Guangxu 10 (1884) in Qing Dynasty, and his mother Lei Yongni was also a learned man. /kloc-before the age of 0/0, Duan Xuefu had been taught Chinese by his father at home, recognized Chinese characters and read books on classics and history. The educational thought of "being educated by talented people in the world and enjoying it" has a great influence on Duan Xuefu. At the same time, I completed elementary arithmetic with my cousin who was studying medicine in Beijing at that time.
At that time, the education quality of the affiliated middle school was very high, the teaching materials were advanced, the requirements were strict, and elective courses were also offered. The mathematics teaching represented by Mr. Fu made Duan Xuefu have a strong interest in mathematics. Because of his love for mathematics, Duan Xuefu graduated from high school 1932 and was admitted to the Department of Mathematics of Tsinghua University (then called the Department of Mathematics).
During his four years in Tsinghua University, Duan Xuefu studied under Xiong Qinglai, Zheng, Zhao Fangxiong and Zeng successively. These teachers have their own characteristics, which makes Duan Xuefu lay a solid foundation in analysis, algebra and geometry. During this period, Duan Xuefu also took the courses of Fourier series and Fourier integral taught by Professor N. Wiener of Massachusetts Institute of Technology, and participated in the course of partial differential equations taught by Academician J. Hadamard of French College. These have broadened Duan Xuefu's horizons.
Physical education class teacher Ma Yuehan is another teacher Duan Xuefu will never forget. With the enthusiastic encouragement and scientific training of Professor Ma, Duan Xuefu, who was originally very thin, has been able to run a mile long-distance race successfully after a semester of training. It was because of his great improvement in physical condition that he endured thousands of miles of long-distance bumps during his stay in War of Resistance against Japanese Aggression.
Duan Xuefu knew Hua as soon as he entered school, and they became familiar with him from the second year. They often take a walk on campus with Wang Jing, a countryman of the Chinese Department of China, after dinner, talking about mathematics and the current situation. Hua's method and practice of learning mathematics and the extracurricular mathematics books recommended by Duan Xuefu have great influence and help on Duan Xuefu.
At that time, the Japanese invaders were expanding their war of aggression against China. At the critical juncture of the country's life and death, Duan Xuefu was also baptized by patriotism. He took part in two demonstrations on February 9 1935, 16 and 1936, and held a collective sit-in with lights off in the new gymnasium in Tsinghua University on February 29 1936 to protest that a large number of military and police officers broke into the campus and arrested students.
1936 In the summer, Duan Xuefu obtained a bachelor's degree in science and stayed as a teaching assistant after graduation. 1On July 7, 937, the Japanese aggressor troops brazenly invaded Beiping under the pretext of the so-called Lugouqiao Incident, and provoked a full-scale war of aggression against China. On the evening of July 29, Duan Xuefu left Beiping with his mother and other three people, and it was bumpy all the way. In June of that year, he came to Changsha Temporary University, which was jointly composed of Peking University, Tsinghua University and Nankai University. On April 6th of the following year, Duan Xuefu married Ms. Lei Binru, a middle school Chinese teacher, in Xi 'an. Shortly thereafter, Duan Xuefu went to Kunming alone and taught in The National SouthWest Associated University-Tsinghua University.
In the autumn of the same year, Hua returned from a visit to Cambridge University in England and became a professor in Tsinghua University, The National SouthWest Associated University. The course Modern Algebra he taught was based on the first volume of Modern Algebra by B.L. Vander Waals, which was just published at that time, but it was revised a lot. Duan Xuefu undertook the task of writing lecture notes and correcting students' homework. Hua also made a series of reports on "Eight Lectures in the Field" among teachers. All these make Duan Xuefu's algebra skills to a new level. In addition, Hua also hosted a limited group discussion class with Duan Xuefu, Sun and Fan? And Xu Xianxiu. Everyone takes turns to report, and the materials are the important paper "Contribution to P- Group Theory" just published by P. Hall and "Group Theory Course I" by H. Zassen-haus. From then on, Hua and Duan Xuefu began to cooperate to study the counting theorem of P groups. This is also the beginning of Duan Xuefu's theoretical research and talent training in algebra, especially in finite groups.
1939 In the first half of the year, Duan Xuefu was admitted as a public school student studying in the UK. Due to the outbreak of World War II and the expansion of Japan's war of aggression against China, he arrived in Canada in September of the following year and entered the University of Toronto. Meanwhile, Guo Yonghuai, Qian Weichang, Lin Jiaqiao, etc. The Department of Mathematics at the University of Toronto was the largest in Canada at that time, and Duan Xuefu's tutor, R. Brauer, was establishing the modular representation theory of finite groups. There are also famous professors in algebra, such as G. Deb Robinson, H. S.M. Coxeter. Duan Xuefu took four courses in Toronto, including brower and Robinson's Group Theory. In addition, Duan Xuefu mainly conducted research under the guidance of brower, and soon made some achievements on the P group, and obtained a master's degree at 194 1. After that, he entered the Mathematics Department of Princeton University in August, 194 1 studying for a doctorate.
At that time, Princeton was known as the world mathematics center, and Wedderburn, a famous algebra scientist, taught there. C chevali, a young assistant professor in his thirties, is very active academically. Princeton Institute for Advanced Studies has gathered a group of world-renowned scientists such as A Einstein, h·Weyl and J von Neumann. Here, Duan Xuefu attended many courses and seminars, including Algebraic Geometry Basis and Integral Equation by Chevalier, Algebraic Number Theory and Quadratic Arithmetic Theory by Will, and Analytic Number Theory and Transcendental Number Theory by C.L. Siegel He also listened to Lefschetz's topology class and Church's logic class. In scientific research, Duan Xuefu attended lectures and seminars under the guidance of brower and Chevalier, especially by studying their published papers and unpublished manuscripts and manuscripts, and finally cooperated with them to complete the important work of modular representation theory of finite groups, lie groups and algebraic groups. 1943 Duan Xuefu received a doctorate in philosophy from Princeton University. After that, I stayed in school for two years as a postdoctoral fellow, and visited and studied in E. Artin for four months. During this period, he worked as a research assistant in the Department of Mathematics. From September of 1945, Duan Xuefu worked as a teaching assistant at the Princeton Institute for Advanced Studies, helping him to set up a distinctive group theory course and help to revise his classic masterpiece "Typical Group" (1939), which Wail influenced by him until 1946 returned to China.
These six years abroad are a very important period in Duan Xuefu's mathematics career. In the past six years, his courses have covered almost all major fields of basic mathematics. It goes without saying that brower and Chevalier were lucky enough to learn from these two masters and cooperate with them, and their influence on Duan Xuefu was even more obvious.
After the victory of the Anti-Japanese War, Duan Xuefu politely declined to stay and decided to return to China. He believes: after all, the motherland will always go back; Anyway, my career can only be in China! Duan Xuefu returned to Shanghai in July 1946. In Shanghai, he met Hua, whose family is going to the United States, and attended the memorial service of two martyrs, Wen Yiduo. At the same time, Duan Xuefu also met with Chen Shengshen, who is preparing for the establishment of the Institute of Mathematics of Academia Sinica. Chen Shengshen hired Duan Xuefu as a part-time researcher at the Institute of Mathematics, responsible for guiding new graduates from Zhejiang University to work in Cao Xihua.
1946 10 Duan Xuefu returned to Tsinghua campus after 9 years' absence and became a professor in the Department of Mathematics of Tsinghua University. Since the second year, he has served as the acting director of the department. During this period, courses such as advanced algebra, advanced calculus, modern algebra and point set topology were offered. 1946- 1947 school year, guiding the graduation thesis of recent graduate Wan Zhexian. 1947 instructed Cao Xihua, who had transferred to Tsinghua at that time, to study abstract algebra and modular representation theory in the first half of the year. 1948 recommended him to go to the United States to study as a doctoral student in brower, who was teaching at the University of Michigan at that time. Now Cao Xihua has established an active algebra group research group in East China Normal University. During his tenure as Dean, Duan Xuefu hired Xu, Shen Youcheng, Zhuang Qitai and other Peking University professors to work part-time in Tsinghua, and also hired Min Sihe, who came back from England, to teach in Tsinghua.
1948 12 13 Tsinghua campus was first liberated from Beiping. Duan Xuefu was appointed as the head of the Department of Mathematics. Encouraged by the new atmosphere of the Chinese people, he devoted himself to heavy teaching, scientific research and administrative leadership with great enthusiasm, regardless of his recovery from a serious illness. 1In the spring of 950, Hua returned to Tsinghua University from the United States, and Cheng Minde, who returned home at the same time, was invited to teach in Tsinghua University. With the joint efforts of all the teachers, from 1949 to 1952, the Department of Mathematics of Tsinghua University has trained a number of outstanding talents for the people of China, including Wan Zhexian, Ding, Zeng Kencheng, Qiu Guangming and Wang Erfang, who are engaged in algebra and other similar fields.
Duan Xuefu served as the executive director of the Chinese Mathematical Society from 1950 to 1987, participated in the preparatory work of the Institute of Mathematics of China Academy of Sciences from 1950- 1952, served as the head of the department of mathematical mechanics in Peking University from 1952, and was elected from 1955. Participate in the formulation and terminology review of national scientific planning and mathematics discipline planning such as the National "Twelve-year Scientific Vision Plan" from 65438 to 0956, and participate in the formulation of scientific research planning and teaching plan and textbook editing of the Ministry of Education and the Ministry of Higher Education.
198 1 In the first half of the year, Duan Xuefu voluntarily resigned as the head of the Department of Mathematics of Peking University. But his workload has not decreased much. 1981-1984 He was a member and one of the conveners of the first mathematics evaluation group of the State Council Academic Degrees Committee, and also served as the director of the academic committees of the Peking University Mathematics Department and Institute. He used to be the editorial board member of China Science, Science Bulletin, Mathematics Magazine, Mathematics Bulletin and Mathematics Yearbook, and the editor-in-chief and honorary editor-in-chief of advances in mathematics (1980- 1987). He is also the editorial board member of Encyclopedia of China, the executive deputy editor of Mathematical Volume, and the editor-in-chief of Number Theory and Algebra.
Founder of China's group representation theory
For decades, Duan Xuefu has published about 30 academic papers and some other works. As a mathematician, Duan Xuefu's research field is mainly algebra. His earliest and most important achievement is the modular representation theory of finite groups, especially the exponential block and its application in the construction of finite simple groups and finite linear groups.
The modular representation theory of finite groups studies the representation of finite groups on a field characterized by prime number P. When P can bisect the order of a group, its representation is very different from the representation of finite groups on a field characterized by 0, and the theory is more complicated and profound. This theory was founded by brower in 1935, and had begun to take shape in the 1940s. At this time, Duan Xuefu began his research work in this field. Under the guidance of brower's important paper "Finite Groups with Order Exactly Containing a Power of Prime Number" published in 1942, in his doctoral thesis on the same topic (Princeton University, 1943), he achieved some important achievements in two papers completed in cooperation with brower, and continued Brinell's work. They are mainly:
(1) obtained some structures of simple groups of order pqbm, where p and q are different prime numbers, b and m are positive integers, and m ≤ P- 1.
(2) It is proved that the list of simple groups listed by L.E. Dickson in the book Linear Groups is complete in order.
(3) For a linear group of order pg', where p is a prime number and (p, g')= 1, its structure is determined when its dimension is ≤(2p 1 1)/3.
In order to get these results, Duan Xuefu proved some basic facts of modular representation theory, for example, he determined the important properties of the brower tree of P block of pg' ordered group. The three lemmas he proved are called (brower-Duan-) Stanton principle, brower-Duan exponential block separation principle and brower-Duan theorem respectively.
Decades have passed, but these achievements have not lost their luster. On the one hand, the problems they involve have always been the mainstream of finite group theory research, and these works are the starting point of later development; On the other hand, the existing new achievements still cannot bypass or replace the above results obtained by Duan Xuefu himself and his cooperation with brower. Because of this, these results were written in detail in W. Feit's masterpiece "Representation of Finite Groups" and widely quoted by group theorists. According to incomplete statistics, since 1945, Duan Xuefu's papers have been cited by more than 30 mathematical works.
In 1950s and 1960s, Duan Xuefu continued to make efforts along this research direction. During this period, he organized two seminars on finite group module representation theory in Peking University to guide young teachers and graduate students. In particular, Hong Jiawei and Li Huiling, graduate students trained by 1964- 1966 seminar, identified some special types of monomorphism. Just when they were likely to make a breakthrough, the "Cultural Revolution" began and China's research in this direction was interrupted. It was also at this time that the classification of finite simple groups began to flourish around the world.
After the Cultural Revolution, Duan Xuefu instructed students to continue their research work in this field, among which Zhang Jiping, a doctoral student, was the best. He thoroughly solved the structure problem of complex linear groups with dimension less than p by using representation theory and simple group classification theorem.
Duan Xuefu also has excellent work in algebraic Lie groups. Algebraic Lie groups over complex fields are complex matrix groups, in which the matrix is determined by algebraic equations satisfied by its coefficients. The germination of this concept appeared as early as the end of last century and was forgotten for 50 years. However, during this period, E. Cartan and Weil made a thorough study of Lie algebras of Lie groups. In 1943, Chevalier first introduced the definition of matrix complex obtained by using tensor invariants of matrices in his paper entitled "A new relationship between matrices", and then further gave the definition that the subalgebra of an N-dimensional matrix Lie algebra over a field with characteristic 0 is an algebraic Lie algebra by using matrix complex. At this time, Duan Xuefu followed him to study Lie Groups and Lie Algebras, and co-published a paper outlining the proof clues of the cooperation between Knight and Duan Xuefu. However, due to the contact between the two authors, the full text was interrupted for a time and was not published until six years later. In this paper, they proved the following basic theorem of algebraic lie groups: "The lie algebra of every algebraic lie group is an algebraic lie algebra, and the algebraic lie algebra of every complex field must be a lie algebra of an algebraic lie group". With this theorem, we can use the method of Lie algebra to extend algebraic Lie groups to any field with characteristic 0. The famous mathematician A Borel once pointed out that in the mid-1940s, Chevalier and Duan Xuefu extended algebraic Lie groups to any field with zero characteristics by means of Lie algebras, which was the prelude to the birth of the general theory of 1955 linear algebraic groups. In the mid-1980s, Lan did some research related to algebra Lie algebra.
In fact, one of Duan Xuefu's earliest papers in this field is a note on the complex form of nilpotent matrix. In this paper, Theorem 6 in Chevalier's first article is calculated by using matrix Jordan canonical form, and a direct and much simpler proof is given, which is strengthened and extended to the field of characteristic p≠0.
Due to the establishment of P.L.M. Sylow theorem, the study of P groups has special significance in finite group theory. As early as the end of 1930s, shortly after the publication of Hall's important paper on P- groups, Duan Xuefu began to study this field. Content index of cooperative research with China p2(p & gt；; 2) P- groups of cyclic subgroups of, give the relevant counting theorems, and classify this group completely, and the results are published in English. On this basis, Hua extended it, introduced the concepts of rank of P group (that is, A in the index pn-A of the largest cyclic subgroup contained in a group of order PN) and pseudo-base, proved that any odd number of P groups must have pseudo-base, and proved the generalized Miller theorem of the number of cyclic subgroups (see "Some Counting Theorems of P Groups"). Duan Xuefu made use of the above results of Hua, and through careful analysis and calculation, generalized Kurakov theorem on the number of subgroups in odd P groups, and proved that the number of subgroups of order pm N(m)modp3 in odd P groups with rank A must be 65,438+when+1≤ m ≤ n. Later, many mathematicians abroad, especially the Soviet Union, made research in this field, which still attracted researchers' attention. Duan Xuefu also gave a more concise proof of Hua's pseudo-base theorem.
Mainly from the early 1980s, Duan Xuefu, Wang Efang's students Xu, Tang Shouwen and others, under the guidance and development of the above papers, have made research achievements in the relationship between the power structure and the arrangement structure of P groups. In 1939, Tang Shouwen continued Duan Xuefu's work on finite P groups with cyclic Frattini subgroups, and finally gave a complete classification of such P groups.
In a theorem about P group, Duan Xuefu proved that if P group G contains a maximal commutative normal subgroup A and G/A is a cyclic group, then A/Z≌K, where Z is the center of G and K is its commutative subgroup. For the corresponding subgroups in the sequence of upper and lower central groups of G, he also proved the existence of corresponding isomorphism. This book is quoted by some Chinese and foreign scholars. Brower and Duan Xuefu also have some unpublished works about P groups, and the manuscripts have been preserved to this day.
Combinatorial mathematics and discrete mathematics have flourished with the appearance of electronic computers, and finite group theory is closely related to combinatorial mathematics (including block design, finite geometry, graph theory, etc.). ), coding theory and cryptography. At the same time, the computer method, algorithm complexity and practical software development of finite groups have developed rapidly since 1960s because of its theoretical significance and practical value. In the early 1970s, Duan Xuefu studied several application problems for a scientific research department, and the given method increased the calculation time several times in practical work. He also held workshops with other comrades, and trained a group of professionals for the practice department, which was commended by the relevant departments. After the "Cultural Revolution", he further carried out research work in computational group theory and combinatorial mathematics, and cooperated with Wang Efang to train five doctoral and master students in this field, including Wang Jie.
From 65438 to 0985, the scientific research project "Finite Groups and Their Representation Theory and Combinatorial Mathematics" led by Duan Xuefu was rated as excellent scientific and technological achievements by the State Education Commission, and its "Application of Finite Groups in a Class of Combinatorial Problems" won the scientific and technological achievement award of a scientific research department. 198519 October, Duan Xuefu was awarded the honorary certificate of 50 years' scientific work by China Academy of Sciences, 198919 October, and165438 was awarded the honorary seal by China Academy of Sciences. 1990 12 was awarded the honorary certificate of "40 years of scientific and technological work" by the State Science and Technology Commission and the State Education Commission.
Struggle for building a department and cultivating talents all his life.
From 65438 to 0952, in order to better adapt to the rapid development of various undertakings brought about by the national liberation, the Ministry of Education made reasonable adjustments to the regional distribution, specialty setting and the allocation of teaching and scientific research forces of its colleges and universities on a large scale. The new Department of Mathematical Mechanics in Peking University is composed of the former Mathematics Departments of Peking University, Tsinghua University and yenching university, with Duan Xuefu as the head of the department.
New China needs a lot of talents. Peking University has a long way to go. Only the Department of Mathematical Mechanics will recruit nearly 200 college students every year and train about 10 graduate students. However, after the adjustment, there are only about 30 teachers in the Department of Mathematical Mechanics of Peking University, and there are only two teaching and research departments, namely Analysis and Advanced Mathematics. Not only the teaching and scientific research strength of important disciplines such as differential equations, probability statistics and computational mathematics is insufficient, but also the teaching materials used in classroom teaching are very scarce due to the total blockade of China by the west.
Facing this arduous task, Duan Xuefu, who is only 38 years old, unites all the teachers, especially after 1955. With the assistance of Professor Cheng Minde, deputy director of the Department, he supported his weak body and devoted a lot of efforts to the establishment of the new Department of Mathematical Mechanics. First of all, I tried to learn from the Soviet Union. First, I sent four teachers to study in the Soviet Union. Second, please come in. Belova (белва), an adviser to the head of the department and a Soviet mechanic, came to the department to personally guide five mechanics graduate students. Soon after, with the support and help of Professor Zhou Peiyuan, the head of the department, the first mechanics major in China was established. As the head of the department, Duan Xuefu attaches great importance to the role of experts. No matter how busy he is at work, he will arrange talks with Belova, even once a week in the 1953- 1954 school year. It is worth mentioning that it is at this time that the exercise class, an important teaching link in China, began to spread. At that time, e.б Duncan (дынкин), an expert in algebra and probability theory in the Soviet Union, and Fitz, a Polish mathematical statistician, gave lectures in this department, which contributed to the establishment of the teaching and research section of probability theory in China. Secondly, create conditions to give full play to the strength of domestic experts. For example, in cooperation with the Ministry of Higher Education, a seminar on ordinary differential equations and partial differential equations was held in the Department of Mathematical Mechanics of Peking University on 1954, which was attended by more than 100 teachers from all over the country, which brought a good start for expanding the professional team and establishing the teaching and research department of differential equations. 1955 the department of mathematical mechanics in Peking University has established the teaching and research section of computational mathematics, which has laid a foundation for developing this direction in China. Since then, the Department of Mathematical Mechanics of Peking University has also prepared a scientific research cooperation plan with Moscow University.
As for the construction of teaching materials, since 1950s, with the personal participation of Duan Xuefu and the efforts of all teachers, Advanced Algebra has been written by A.г.. Qin Xin of Kypoш and хиниич has been translated one after another.
In this way, with the joint efforts of all cadres and workers, from 1952 to 1966, the Department of Mathematical Mechanics of Peking University has trained about 2,000 undergraduate graduates and dozens of graduate students for the country, and has also achieved good scientific research results.
Dean's work is very busy. Duan Xuefu is in poor health and suffers from gastrointestinal ulcer for a long time. But in this case, his teaching and scientific research work never stopped. He has taught advanced algebra, modern algebra, Lie algebra and other courses for many times, with graduate students and guidance in writing papers. 1952- 1966 Duan Xuefu, with the help of other comrades, trained undergraduates majoring in algebra such as Xu Yichao, Shen Guangyu, Lan, Xu and Lu, and postgraduates such as Hong Jiawei. In particular, Duan Xuefu held two seminars on the representation theory of finite group modules during 1952- 1966. The first time was in 1954- 1955. He cooperated with Nie and Wan Zhexian to write a lecture, put forward his own important research topic without reservation, and guided Wang Erfang to study finite simple groups with order ≤27000, and achieved results. The second time was held in 1964- 1966, which was entrusted by the Ministry of Education. Duan Xuefu cooperated with Wang Efang in compiling the lecture notes, and teachers from other units such as Chen Zhongmu and graduate students from our department participated in it, conducting special research and writing papers. Now Chen Zhongmu has established an active limited scientific research group in Southwest Normal University.
1960 ——1966 Duan Xuefu also served as the head of the Department of Mathematics of Beijing TV University. He participated in the mathematics competition for middle school students in Beijing twice, wrote articles and made reports, and wrote the book Symmetry, which made great efforts for general education and adult education.
Since 1978, with the establishment of China's academic degree system, Duan Xuefu, with the assistance of other comrades, concentrated his efforts on training five doctors and 14 master students in the fields of finite groups and their representation theory, computational group theory and combinatorial mathematics, and also instructed a postdoctoral fellow. At the same time, it also trained young and middle-aged teachers such as Qiu and advanced teachers from other schools. 65438-0988, Advanced Algebra, which he participated in, won the "National Excellent Textbook Award for Colleges and Universities" by the State Education Commission.
Duan Xuefu participated in many academic conferences at home and abroad and made academic reports at the conference. 1982 presided over the first national algebra academic exchange meeting in chinese mathematical society, 1984 presided over the Beijing international group theory seminar and edited the conference proceedings. Until 1988 retired, Duan Xuefu was still working hard for China's mathematics career. In recent years, he has undertaken scientific research projects funded by the National Natural Science Foundation and the Doctoral Program Fund of the State Education Commission to train doctoral students and guide postdoctoral students. At the same time, he still cares about and helps the growth of young teachers with considerable energy.
Duan Xuefu's motto is "seeking truth from facts, being serious and rigorous". Years of academic experience made him deeply realize that science is honest learning, and any naughty is not enough. You must study hard, practice more, lay a good foundation, learn deeply, learn thoroughly, use it flexibly, and at least have one pair of hands. After grasping the problem, we should broaden our thinking, explore in many aspects and persevere on the basis of mastering the main work of our predecessors, so as to get through at once and get results. He takes pleasure in discovering and cultivating talents, and thinks that a teacher is not necessarily superior to his disciples, but a teacher can cultivate students who are better than teachers is the greatest happiness.
Duan Xuefu fell ill many times, especially in the summer of 1959 when he had rectal cancer resection, which brought him permanent inconvenience and difficulty. However, he always maintained an optimistic attitude, fought tenaciously against the disease and persisted in his work.