Gail van der, who foresaw a story that had not yet ended

“Mathematics is part of culture, just like music, poetry and philosophy.”

—Israel Gelfand

Written by Chen Guanrong (City University of Hong Kong)

Israel Moiseevich Gelfand (September 2, 1913 - October 5, 2009) was born into an ordinary Jewish family in the town of Okny (now called Krasni Okny) in southern Ukraine.

After graduating from junior high school, Gelfand entered a vocational and technical school to train as a chemical laboratory technician, but dropped out before graduation. In early 1930, at the age of 16, he came to Moscow from his hometown alone to seek a living with his parents. He worked as a temporary worker everywhere in Moscow, but was often unemployed. Later, he found a job as a lending librarian at the Lenin Library. There, he eagerly taught himself the knowledge he had not learned before. He got to know several college students and often arranged time to follow them to Moscow University to attend classes. His greatest gain was to audit the complex function discussion class organized by mathematician Mikhail A. Lavrentyev (1900-1980).

In 1932, Gelfand, who had no university degree, was admitted as a graduate student of Moscow State University due to his outstanding mathematical performance. He studied under the famous mathematician Andrey N. Kolmogorov (1903-1987). From then on, in Gelfand's own words, his academic career was "ordinary and regular, and entered the usual track of mathematicians."

Figure 1 Gelfand (1913-2009)

1

A boy who likes mathematics

Before introducing how Gelfand "entered the usual track of mathematicians" and grew up to become "one of the greatest mathematicians of the 20th century", let's review how he learned mathematics as a child.

There was only one middle school in Okny, Gelfand's hometown. When he was 12 years old and in junior high school, he figured out that some geometry problems could not be solved by algebra. Once, he calculated the length of a chord every 5 degrees for a sine curve and made a table. Later, of course, he realized that he had "created" a trigonometric function table. During that time, in the absence of a corresponding algebra textbook, he also completed a book of elementary algebra exercises. This small success made him understand and remember a truth, that is, by solving problems, you can enter a new field of mathematics.

At the age of 13, he developed a special interest in plane geometry. He noticed that some right triangles had sides of 3, 4, 5, or 5, 12, 13, and wanted to find all the right triangles with sides of integers. As a result, he "invented" the Pythagorean theorem.

At that time, his family was poor and his environment was bad, and he was often sick. But whenever he fell ill in bed, especially during school holidays, he would study mathematics on his own, and the results were very fruitful. Many years later, he often asked his son to stay at home for a few more days after he recovered from his illness, saying that capable students could do a lot of things when they were sick.

When Gelfand was a child, his family was extremely poor, and it was hard for others to imagine that his parents would buy him a workbook. Later, he finally got one, and he copied the statements and proofs of mathematical theorems on every page. He later recalled that this taught him how to write a mathematical book.

For Gelfand, who was very talented in mathematics, the lack of mathematics books was a serious obstacle to his further development. He often saw advertisements for advanced mathematics books and guessed that advanced mathematics would be very interesting. Unfortunately, his poor parents could not buy these books for him. When he was 15 years old, he had appendicitis and needed to go to the hospital in the big city of Odessa for resection. He took the opportunity to cheat his parents and said that if you don’t buy me advanced mathematics books, I will not go to the hospital. His parents had no choice but to buy him a Ukrainian "Advanced Mathematics Course". Poor parents only had enough money to buy the first volume, which included plane analytic geometry and elementary calculus.

Gelfand felt very lucky to have the opportunity to learn advanced mathematics from a formal university textbook. He couldn't wait to start reading this book on the third day after the operation, and also read a novel by French writer Emile Zola (1840-1902). In the nine days in the hospital, he finished the first volume of this advanced mathematics. During this period, he also independently derived the Euler-Maclaurin formula, the Bernoulli number recursion formula, and the formula for the sum of the first n natural numbers to the power of p. For him, the biggest gain was to exercise the ability to solve problems independently and to develop a good habit of continuing to explore further results after solving problems.

In 1930, the 16-year-old Gelfand said goodbye to his parents and went to Moscow to work and make a living.

2

Outstanding contributions of mathematicians

In 1932, Gelfand became a graduate student at Moscow State University, studying under the great mathematician Kolmogorov.

During his postgraduate study, under the guidance of his supervisor, Gelfand entered the then emerging field of functional analysis. In 1935, Gelfand received his Ph.D. with a thesis on abstract functions and linear operators. He proved many basic properties of functional analysis, especially complete normed spaces, and established some universal methods to transform many of these problems into classical analysis problems through continuous linear functionals.

In 1940, Gelfand received a doctorate in physics and mathematical sciences from the Soviet Union. In his dissertation, he established the theory of commutative normed rings, laying the foundation for Banach algebra. In this field, he later established a completely new representation theory and extended the spectral theory of linear operators in Hilbert space to normed algebra. An interesting example is that he applied the theory and techniques of normed rings and used only five lines to deduce a famous theorem proved by Norbert Wiener (1894-1964) in an earlier long article: If a function that does not take zero values ​​can be expanded into an absolutely convergent Fourier series, then its reciprocal can also be expanded into an absolutely convergent Fourier series. Next, Gelfand pioneered the study of C* algebra.

In 1943, Gelfand became a professor at Moscow State University and later worked at the Institute of Applied Mathematics of the USSR Academy of Sciences. In the 1950s, Gelfand carried out a lot of fruitful research in many branches of pure and applied mathematics, with fruitful results. His main contributions covered the fields of functional analysis, harmonic analysis, group representation theory, integral geometry, generalized functions, differential equations, mathematical physics, etc. In addition, after 1958, he also carried out in-depth research on biology and physiology, and established and directed an Institute of Biophysics at the USSR Academy of Sciences. In the field of biomedicine, he studied the mathematical problems of X-rays and CT scans, and improved the image transformation of Johann KA Radon (1887-1956), thus creating a new integral geometry.

Gelfand published nearly 500 papers, of which 33 were published in his own name, accounting for only 7% of the total, while there were 206 co-authors, including Chinese mathematician Xia Daoxing. He also published 18 textbooks and monographs. In the late 1980s, Springer published the three-volume "Selected Works of Gelfand", which included 167 papers selected by the author. Between 1958 and 1966, Gelfand took the lead in editing and publishing the six-volume masterpiece "Generalized Functions". The first volume discusses the definition, basic properties and Fourier transform of generalized functions and various special types of generalized functions; the second volume examines various types of basic function spaces and generalized functions on them and the corresponding Fourier transforms; the third volume applies generalized functions to study the existence, uniqueness and well-posedness of solutions to the Cauchy problem of partial differential equations and the expansion of self-adjoint differential operators according to characteristic functions; the fourth volume studies kernel space and its applications and introduces Hilbert space, as well as positive definite generalized functions, generalized random processes and measure theory on linear topological spaces; the fifth volume discusses harmonic analysis on Lorentz groups and related homogeneous spaces based on integral geometry; the sixth volume introduces representation theory and automorphic functions. This set of monographs enjoys a very high international reputation and has Chinese, English, French and German translations. It is a basic textbook and reference for analytical mathematicians.

Figure 2 Gelfand (Moscow State University)

Gelfand's mathematical research was closely linked to mathematical teaching. He often taught junior undergraduates at Moscow State University. In 1944, he started a functional analysis seminar for young teachers and graduate students, and later a theoretical physics seminar. The seminars organized and directed by Gelfand continued until his later years, becoming a major base for the development of functional analysis and training of new mathematical talents in the Soviet Union. Gelfand was always humorous and witty. He joked many times: "This seminar is for ordinary high school students, good undergraduates, excellent graduate students and outstanding professors." A group of young people who worked with him came from his seminars, many of whom later became famous mathematicians, including F. Berezin, J. Bernstein, E. Dynkin, A. Goncharov, D. Kazhdan, A. Kirillov, M. Kontsevich, A. Zelevinsky, etc., who are well-known in the mathematics community, especially Ilya Piatetski-Shapiro (1929-2009), winner of the 1990 Wolf Prize in Mathematics. Piatetski-Shapiro believes that there were three masters in the Soviet mathematics community at that time, namely Kolmogorov, Gelfand and Igor R. Shafarevich (1923-2017), three generations of master and apprentice. He said: "Gailvand is the most outstanding. He has the profound mathematical attainments of Shafarevich and the extensive knowledge of Kolmogorov. In addition, Gelfand has a special talent: he can engage in research in several basic fields at the same time without feeling overwhelmed... In this regard, Gelfand is unparalleled."

It is worth mentioning that, perhaps because of his own poverty and lack of education when he was young, Gelfand was particularly concerned about the mathematics education of middle school students. He was one of the initiators of the Moscow Mathematical Olympiad in the 1930s and also participated in the establishment of a distance learning mathematics correspondence school. He took the lead and worked with several mathematician friends to compile five basic mathematics books for middle school students: "Algebra", "Geometry", "Trigonometric Functions", "Functions and Graphs", and "Coordinate Methods". The English version was published by Springer in the 2000s, and the Chinese translation is included in the "Gelfand Middle School Mathematical Thinking Series".

When Gelfand was 90 years old, he recalled the past and was very grateful for the teachers at various stages of his life: "For me, the most important teacher was Schnirelman, a genius mathematician who died young. Then there were Kolmogorov, Lavrentyev, Plesner, Petrovsky, Pontriagin, Vinogradov, Lusternik, they were all different... but they were all great mathematicians. I am very grateful to all of them, and I learned a lot from them."

Figure 3 Some of Gelfand’s works

3

Awards and Honors

Gelfand has given three plenary lectures at the International Congress of Mathematicians (1954, 1962, 1970), which is enough to show his important position in the development of contemporary mathematics. In fact, to date, only Vito Volterra (1860-1940) has given four plenary lectures, and Élie Cartan (1869-1961), Lars Ahlfors (1907-1996) and André Weil (1906-1998) have given three plenary lectures.

Gelfand was elected a corresponding member of the USSR Academy of Sciences in 1953 and a member in 1984. He served as president of the Moscow Mathematical Society from 1968 to 1970. Later, he was awarded the Lenin Medal three times (the first in 1973), the Kyoto Prize (1989), known as the "Japanese Nobel Prize", the highest honor in the American cultural world, the MacArthur Fellow (1994), and the American Mathematical Society's Lifetime Achievement Award (LP Steele Prize in 2005).

Gelfand is also a member of the Royal Society of London, the National Academy of Sciences of the United States, the American Academy of Arts and Sciences, the Academy of Sciences of Paris, and the Royal Swedish Academy of Sciences. He has been awarded honorary doctorates by Oxford University, Harvard University, and the University of Paris.

In 1978, Gelfand and German mathematician Carl L. Siegel (1896-1981) shared the first Wolf Prize in Mathematics established by the Wolf Foundation in Israel. However, because he participated in an open letter signed by 99 Soviet mathematicians in 1968, demanding that the government release Alexander Esenin-Volpin (1924-2016), a mathematician who was imprisoned in a mental hospital for his dissident views, and because the Soviet Union and Israel had previously severed diplomatic relations, Gelfand was prohibited from attending the award ceremony that year, making the first Wolf Prize presentation embarrassing. It was not until 1988 that Gelfand was able to go to Israel to receive the medal he had won ten years earlier.

By the way, the only Chinese mathematicians who have won the Wolf Prize in Mathematics so far are Shiing-Shen Chern (1983) and Shing-Tung Yau (2010).

Figure 4 Gelfand giving a lecture at MIT

4

The story is not over yet

In 1989, at the age of 76, Gelfand moved to the United States. After visiting Harvard University and MIT for a period of time, he was hired as a lifetime distinguished visiting professor at Rutgers University.

Gelfand died on October 5, 2009 in New Brunswick, New Jersey at the age of 96.

Gelfand and his first wife Zorya Shapiro had three sons, Sergei and Vladimir, but the youngest son Aleksandr died of leukemia in 1958. The pain of that year became the main motivation for him to start studying biomedicine. He and his later wife Tatiana had a daughter, also named Tatiana.

Ukrainian mathematician Gelfand spent his extraordinary life in this way, leaving behind a large amount of valuable mathematical treasures for mankind.

On September 2, 2003, Gelfand made a short speech at a dinner celebrating his 90th birthday (see "Speech by the Ukrainian-born Great Mathematician Gelfand at His 90th Birthday Party"), the theme of which was his views on mathematics and why he was still doing mathematics at such an old age. At the end, he changed the subject and said something off topic:

"Finally, I want to give an example outside of mathematics. There is a short and pithy sentence that combines the characteristics of simplicity and precision that I mentioned earlier. It is a sentence by Isaac Bashevis Singer, a Nobel Prize winner in literature: 'As long as men use swords to destroy the weak, there will be no justice.'"

Today, he foresaw a story that has yet to end.

Figure 5 Gelfand giving a lecture at Rutgers

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