D'Alembert was both a brilliant mathematician and a famous writer and thinker of the 18th century, known as the ''century of light''. The great writer Voltaire wrote:

What I love most about D'Alembert is the clarity of his writing and speaking. He can be considered the leading writer of the century

Saint-Beuve called him ''one of the great men of the 18th century''. The poet Chateaubriant wrote:

Diderot and D'Alembert are the most outstanding geniuses that France has produced

D'Alembert was born on November 18, 1717 in Paris, the capital of France. He was the illegitimate son of Mrs. Jancin and an artillery officer, Mr. Destouches. When he was born, he was abandoned on the steps of the Saint Jean la Rond church, so he was later named Jean le Rond. Madame Rousseau, the wife of a poor glazier, took him in and raised him. Throughout his life, D’Alembert regarded his foster mother as his own. Thanks to his biological father’s constant financial support, he was able to study well at the famous Mazarin High School in Paris. He was an excellent student, and although he studied for a philosophy degree, he was fascinated by geometry and mathematics. This growing interest led to his first work, A Treatise on Integrals, published at the age of 22 (in 1739), which made him famous. Two years later (1741), his work On the Refraction of Solids not only consolidated his reputation in the scientific community but also opened the doors of the Academy of Sciences for him, when he was just 24 years old. However, it was two years later, in 1743, that his main work appeared, entitled A Treatise on Dynamics (published in 1743, when the author was 26). In this treatise, he first proposed the basic principles of classical mechanics, including a famous principle commonly known as d'Alembert's Principle. He also proposed the idea that statics is a special case of dynamics. D'Alembert's treatise truly opened a revolution in the study of the laws of nature.

In 1746, his work: Monograph on the General Origin of the Wind was awarded the Prize of the Berlin Academy of Sciences and D'Alembert was invited to join this Academy. His next works were: Investigations on Precessions (1749); Essay on the Resistance of Fluids (1752).

D'Alembert's reputation would probably have been limited to the scientific community had he not participated in the life and struggles of his century. A friend of Voltaire and Diderot, he was drawn into the compilation and publication of the world-famous Encyclopedia. In addition to compiling many articles on science and philosophy, D’Alembert was responsible for revising and editing the entire mathematical section. D’Alembert’s important contribution was to compose the “Preface” at the beginning of Volume I of the Encyclopedia, in which, with a strong and clear writing style and a wonderful synthesis, he drew a picture of all human knowledge, proving his encyclopedic erudition.

With an independent and straightforward character, glory did not make him stray from a simple life. He refused the invitation of King Frédéric le Grand to be President of the Berlin Academy of Sciences, as well as the invitation of Empress Catherine II to go to the Russian court to teach the Prince. Although he was received with great honor in the castles, he did not leave the poor house where he had lived with his foster mother since childhood. Loyal to his family and friends, he was also loyal for twenty years to his lover, Miss De Lespinasse, a woman of frivolous and capricious temperament, who brought him no small amount of suffering.

An outstanding mathematician and encyclopedist, he was also a talented writer. In 1754, he was elected to the French Academy of Literature and in 1772 was appointed its permanent Secretary. He wrote many literary and philosophical works such as Monographs on Philosophy, History and Literature (1753-1783); Principles of Human Knowledge (1759); Essays on Society – Writers and Great Men (1753), in which he fiercely attacked writers who hid in the shadow of the powerful.

Karl Theodor Wilhelm Weierstrass (Weierstraß) (31 October 1815 – 19 February 1897) was a German mathematician who is considered the "father of mathematical analysis".

Weierstrass was born in Ostenfelde, in the Ennigerloh district of the state of North Rhine-Westphalia.

Weierstrass was the son of Wilhelm Weierstrass, a government employee, and Theodora Vonderforst. He became interested in mathematics as a student at the Gymnasium. He went on to study at the University of Bonn in preparation for a government position. Since he also studied other subjects such as law, economics, and finance, he struggled with whether to choose mathematics over those subjects. He eventually decided to pay some attention to those subjects and taught himself mathematics at the same time. As a result, he did not receive a university degree. He then continued his studies at the then prestigious University of Münster, where his father had found him a teaching position. During his studies, Weierstrass attended the lectures of Christoph Gudermann and became interested in elliptic functions.

From 1850 Weierstrass suffered from frequent illnesses, but he still wrote papers that made him famous and prominent. He also held the chair of the Technical University of Berlin (Gewerbeinstitut). He was bedridden for the last three years of his life and died in Berlin of pneumonia.

Mathematical works

Weierstrass was very interested in the logic of analysis. At this time, there were many unclear definitions of the foundations of analysis, and some important theorems could not be rigorously proven. While Bernard Bolzano had given a rigorous definition of a limit as early as 1817 (or earlier), it remained unnoticed by the mathematical community for many years, and there have been many ambiguous definitions of limits and continuity of functions.

Cauchy had given a formal definition of a limit , while giving a formal definition of the derivative in the 1820s, but did not properly distinguish between continuity at a point and uniform continuity on an interval, due to lack of rigor. In particular, in Cours d'analyse, (1821) Cauchy gave an incorrect proof that the pointwise limit of pointwise continuous functions is pointwise continuous. The correct statement is that the uniform limit of uniformly continuous functions is uniformly continuous.

This requires the notion of uniform convergence, first noted by Weierstrass's teacher Christoph Gudermann, in a paper (1838). Gudermann noted this but did not define or elaborate on it. Weierstrass saw its importance and formalized it and applied it extensively to the foundations of analysis.

Weierstrass's definition of a limit in terms of is as follows:
is continuous at if for every such that
Using this definition and the concept of uniform convergence, Weierstrass proved a number of previously unproven theorems such as the 'Mean Value Theorem', the 'Bolzano-Weierstrass Theorem', and the 'Heine-Borel Theorem'.

Calculus of Variations

Weierstrass also made an important contribution to the development of the calculus of variations. Using the tools of analysis he had developed, he refined the formalism of the theory for the present-day study of the calculus of variations.

Kunihiko's father, Gon-ichi, studied agriculture and politics at Tokyo Imperial University, and while his son was born, he worked at the Ministry of Agriculture. He retired from the Ministry of Agriculture in 1939 and was elected to the Japanese Diet, where he served throughout World War II. After Japan was defeated, the Allies dismissed him. In addition to these activities, he wrote about 40 professional books and 350 professional articles. Kunihiko's mother, Ichi, was the daughter of principal Kyuji Kanai. Kunihiko was the eldest child in the family, with a younger brother named Nobuhiko (born in 1919).

Kunihiko entered junior high school in 1921, but those years were not easy for him. He was quite shy and often stuttered, especially under pressure. He was not the athletic type, so he hated physical education class. In his autobiography, he said that he was a poor student in elementary school, although he was generally modest, perhaps he really did not shine during this period. However, he had a passion for numbers from a very young age, liked to count beans, at the age of ten he tried to test whether dogs could count. When the dog gave birth to puppies, he hid them and waited for the mother dog to get upset and search for them until he returned them. However, when he hid a few puppies, the mother dog seemed happy with the puppies, so at the age of ten Kunihiko came to the conclusion: dogs cannot count. Gon-ichi, Kunihiko's father, was in Germany in 1921-22, when inflation was out of control, he realized that the Japanese Yen could buy a large number of goods very cheaply. He brought back many gifts to Japan for his children and the young Kunihiko was very fond of the German construction tool set his father gave him. The toy made him decide that he wanted to become an engineer.

Kodaira finished elementary school in 1927 and entered junior high school. He said in his autobiography that he was a poor student in junior high school but this was not true. He did very well in English and Mathematics, soon surpassing his peers. By the time he was halfway through the three-year course, he had learned the entire textbook on number theory, algebra, two-dimensional and three-dimensional geometry and solved all the problems in the workbooks. He bought M Fujiwara's Algebra, a textbook for university students, and began to learn about matrices, determinants, continuous functions, quadratic reciprocity and many other things. We note that Fujiwara Matsusaburo (1881-1946) was a learned and famous author who published his two-volume treatise Daisugaku (Algebra) in 1928-29. Some have placed the quality of this book on a par with the classics of Joseph Serret and Heinrich Weber, and it is worth noting that he had studied in Paris, Göttingen, and Berlin.

One of the many things Kodaira's father brought back from his trip to Germany in 1921-22 was a piano. At the age of fifteen, Kodaira began to practice the piano and was taught by Nakajima, a student at Tokyo University. When Nakajima graduated from university and moved away, his sister Tazuku Nakajima became Kodaira's piano teacher, although she was a violinist rather than a pianist. After junior high school, Kodaira went to high school, where he was taught by Hideo Aramata (190-1947), a brilliant mathematician who wrote books on matrices and determinants as well as interesting papers on the zeta function. Kodaira sensed Aramata's enthusiasm for mathematics and realized that it was the subject for him. He decided at that moment that he wanted to become a mathematics teacher.

In 1935, Kodaira entered university at Tokyo University. During his first year, he took the course 'Introduction to Analysis' taught by Teiji Takagi. It was Takagi's last year of teaching before he retired. Shokichi Iyanaga, who censored the assignments for this course, wrote:

I have given an example problem to prove that e is not the square root of a rational number that is not a perfect square (after proving that e is an irrational number). Kodaira stepped up to the podium and wrote his proof in a few lines without saying anything. As we read these lines with the other students, we praised his perfect proof, every word written was on point!Zyoiti Suetuna was appointed chair of the algebra-number theory department at Tokyo University in 1936, when Takagi retired. In 1936–37 Kodaira attended Iyanaga's course on modern analysis, which was based on von Neumann's ideas. He also attended Suetuna's lectures, and towards the end of 1937 he asked Suetuna if he could attend his seminar the following year. He was accepted, but Suetuna then suggested that he would be more suitable for studying geometry in Iyanaga's seminar. He attended Iyanaga's seminars and, in 1937–38, he often visited Iyanaga's house. There he played the piano and showed his talent as a pianist. Iyanaga's sister Seiko was also a musician, and was a violin student with Tazuku Nakajima. Kodaira graduated from Tokyo University in March 1938 with a Bachelor of Science degree in mathematics. Not only that, he also graduated from the physics department of Tokyo University in March 1941 with a Bachelor of Science degree in physics. It should be noted that by 1941 he had ten published papers. During his years studying physics, he became closer to the Iyanaga family. In addition to Seiko, Shokichi Iyanaga's daughter, there were two other sons in the family. All of them had their own achievements in life: Kyoji Iyanaga became president of Nikon Optics and Teizo Iyanaga became professor of Japanese history at Tokyo University. Tazuku Nakajima organized concerts and Kodaira played the piano with other violinists. He accompanied Seiko, who played in these concerts, and the two became closer. They married in 1943 and spent their honeymoon in Gora. Gora is a hot spring resort near Hakone, in south-central Honshu, located on the southern side of Lake Ashino, in the extinct volcano of Mount Hakone. The honeymoon was not as peaceful as its name suggests, as wartime Japan had little food and the couple had to bring their own rice to the hotel they were staying at as the kitchen was empty. In March 1944 their first child, a boy named Kazuhiko, was born, but conditions in Tokyo became more difficult as Japan was attacked. Sadly Kazuhiko suffered from kidney problems and died in 1946. The Kodairas also had two daughters, Yasuko and Mariko.

Kodaira was appointed as a lecturer in the Physics Department of Tokyo Imperial University in April 1941 and then as an associate professor in the Mathematics Department of Tokyo Bunri University in April 1942. He was then promoted to an associate professor in the Physics Department of Tokyo Imperial University in April 1944. In the fall of 1944, the situation in Tokyo became dangerous, and women and children were moved to the safety of Karuizawa Town, in the mountains north of Tokyo. After Kodaira finished teaching in Tokyo in the fall semester, he reunited with his family in Karuizawa. Because Tokyo was heavily attacked by nearly 1,000 American aircraft bombing the city in January 1945, the Institute of Physics and Mathematics was evacuated. On April 13, an air raid destroyed their home in Tokyo, and Kodaira and his family moved to Yonezawa, where his father had a house. On August 6, 1945, an atomic bomb was dropped on Hiroshima, and on August 9, another atomic bomb was dropped on Nagasaki. Japan surrendered to the Allies on August 14, and the Institute of Physics and Mathematics was reopened in Tokyo. Kodaira returned to the Institute two weeks later, leaving his family in Yonezawa. Incredibly, Kodaira was able to quickly restart his seminar and began producing remarkable results. However, he wrote in his autobiography:

I had thought of living in Japan forever, enjoying mathematics and music. This idea was completely destroyed by the war.

During this time, Kodaira became interested in topology, Hilbert spaces, Haar measures, Lie groups, and periodic functions. World War II had a serious impact on Japan, especially isolating Japanese scientists from contact with foreign colleagues around the world. Despite this, Kodaira continued to receive papers on new mathematical developments, and he was greatly influenced by the work of Weyl, Stone, von Neumann, Hodge, Weil, and Zariski. Kodaira received his doctorate from the University of Tokyo in April 1949 for his thesis Harmonic Fields on Riemannian Manifolds and published it in an 80-page paper in the Annals of Mathematics in 1949. This paper brought him international fame, and in particular, he received an invitation from Weyl to Princeton. Donald Spencer writes:

The paper made a strong impression on many people, including me, and I invited Kodaira to lecture on his paper at Princeton University during the 1949–59 academic year. This was the beginning of a collaboration that resulted in twelve papers and a close friendship that lasted until his death.

Kodaira accepted Weyl's invitation and from September 1949 he worked as an academician at the Academy.advanced research at Princeton. He then became a visiting professor at Johns Hopkins University from September 1950 to June 1951, when he returned to the Institute for Advanced Study at Princeton. During this time, his wife Seiko and their two daughters, Yasuko and Mariko, who had remained in Japan when he left, joined him at Princeton. He was appointed associate professor at Princeton University in September 1952 and promoted to full professor in September 1955. He had retained his position in Tokyo until this time, but after being appointed full professor at Princeton, he resigned from the Tokyo position. Michael Atiyah writes of Kodaira's notable work during this period:

During his time at Princeton, Kodaira continued to be interested in harmonic forms, especially their applications to algebraic geometry, the area that provided much of the impetus for Hodge's work. The 1950s saw a great success in complex algebraic geometry, where sheaf theory, with its French origins in Leray, Cartan, and Serre, opened up a whole new set of tools for dealing with global problems. Sheaf theory naturally fitted into Hodge theory, so it was natural for Kodaira to exploit these lines of research. And Kodaira collaborated with Spencer to produce a series of papers. These papers changed the face of algebraic geometry, including the foundations on which Hirzebruch and others of the younger generation could make great advances. A large number of problems that had not been solved or were not completed by the Italian algebraic geometers were now reasonably solved. This work led to Kodaira being nominated for the Fields Medal in 1954. He sailed from New York in August to the International Congress of Mathematicians held in Amsterdam in September 1954. He was awarded the Fields Medal by Hermann Weyl at the opening ceremony on 2 September, as was Jean-Pierre Serre. Kodaira delivered his lecture ‘Some Results in the Transcendental Theory of Algebraic Varieties’ to the Congress on 3 September. However, upon his return to the United States, he did not feel well liked at Princeton. He wrote:

Since Lefschetz retired, I have gradually come to realize that the older professors at Princeton dislike me.

After a year as a visiting professor at Harvard from September 1961 at the invitation of Oscar Zariski, in September 1962 he was appointed chairman of the mathematics department at Johns Hopkins University. In 1965, Kodaira left Johns Hopkins to become chair of the mathematics department at Stanford University. Donald Spencer was so angry that Princeton did not make any effort to retain Kodaira on the faculty that he resigned from Princeton and moved to Stanford with Kodaira. While at Stanford, Kodaira gave an introduction to abstract complex analytic manifolds, and his course was written up as Complex manifolds (1971). After two years at Stanford, he returned to Japan and served as chair of the mathematics department at the University of Tokyo from 1967:

After Kodaira returned to Japan, he taught and gave seminars that attracted many talented students. Kodaira's influence was so pronounced that it is said that he founded a new school of Japanese algebraic geometers.

At the University of Tokyo he served as dean of the science departments of the university before retiring in March 1975. Note that he was a reluctant dean, as he had been assured by the mathematics department that he would not be given any administrative duties when he returned to the University of Tokyo in 1967. The faculties did not support and vote for Kodaira as much as he would have liked. He was an excellent dean but hated the role. As a result of his deanship, he stopped doing research. He did not return to research until he resigned as dean after two years. This explains the title of his autobiography Notes of an Idle Mathematician.

Kodaira's work covered a wide range of topics. Among them were applications of Hilbert space methods to differential equations, an important topic and one largely influenced by Weyl. During this time, under the influence of Hodge, he studied harmonic integrals and applied them to problems in algebraic geometry. An important part of Kodaira's work was the application of bundles to algebraic geometry. During the 1960s, he devoted himself to the classification of compact complex analytic spaces, complex analytic spaces. One of the themes that ran through his work was the Riemann-Roch theorem and it played an important role in his research.

Kodaira received many awards for his outstanding research. But perhaps the most valuable award was the Fields Medal in 1954 that we mentioned above, he also received the Japan Academy Prize from the Japanese Academy in 1957 and the Order of Cultural Merit from the Japanese government in the same year. He received the prestigious Fujiwara Prize in 1974 and the Wolf in Mathematics in 1984. Citation for the Wolf Prize in Mathematics awarded to Kunihiko Kodaira:

…for his outstanding contributions to complex and algebraic manifolds…Professor Kunihiko Kodaira has produced a beautiful study of harmonic integrals with incisive and sensitive applications to complex and algebraic geometry. These include the projective embedding theorem, deformations of complex structures (with D C Spencer), and the classification of complex analytic surfaces. His achievements have greatly influenced and inspired researchers in these fields throughout the world.

He is an honorary member of many major academies and academic societies worldwide, including the Göttingen Academy of Sciences (1974), the National Academy of Sciences (1975), the American Academy of Arts and Sciences (1978), and the London Mathematical Society (1979).

After retiring from the University of Tokyo in 1975, he was appointed professor at the Faculty of Science of Gakushuin University, a highly regarded university. This led him to write to the Ministry of Education:

… Accusing the Ministry of Education of crushing individualism, eliminating creativity and initiative in children and university students….

and writing books for students and universities in an attempt to improve the standard of mathematics teaching. For example, in 1977 he wrote Complex analysis (Japanese), which was translated into English and published in 2007. The publisher writes:

Written by a master, this book will be appreciated by students and professionals alike. The author develops the classical theory of complex functions in a clear and easy-to-understand style. Overall, the approach in the book emphasizes the geometric aspects of the theory, avoiding some of the topological aspects associated with the subject. Thus, the Cauchy integral formula in the simple topological case, from which the author deduces the basic properties of holomorphic functions. From these foundations, students are led to learn conformal mappings, the Riemann mapping theorem, analytic functions on Riemann surfaces, and finally the Riemann-Roch theorem and Abel's theorem. Well illustrated with many examples and exercises (including solutions to many of the exercises), this book is recommended for advanced courses in complex analysis.

In 1979, he published a five-volume Introduction to analysis in Japanese on real numbers, functions, differentiation, integration, infinite series, multivariable function theory, curves and surfaces, Fourier series, Fourier transforms, ordinary differential equations and distributions. In 1968, he published the monograph Complex manifolds and deformation of complex structures. Andrew Sommese reviews the book in detail:

In mathematics and science, it is common to encounter phenomena such as a system of equations that depend on a family of parameters. There are many names for this investigation, such as branching or unfolding or area-dependent deformations. Historically and conceptually, the theory of local deformations of compact complex manifolds has played a key role in the study of these phenomena. ‘Complex Manifolds and Deformations of Complex Structures’ is a careful exposition of the theory of deformations of compact analytic manifolds in local terms, written by one of the founders of this field.

James Carlson also reviews the book:

The author, who, together with Spencer, created the theory of deformations of complex manifolds, has written a book that will be useful to all interested in this vast field.

During the last ten years of his life, he struggled with health problems. He suffered from respiratory problems and became deaf, which made him sad because he could no longer enjoy music, which had been of great value to him throughout his life. He was too ill in 1990 to attend the International Congress of Mathematicians in Kyoto. Friedrich Hirzebruch recalled the last time he saw Kodaira:

Kunihiko Kodaira was both my friend and my teacher. My wife and I remember the last time we visited Kodaira's house in Tokyo. He was working at the kitchen table with books for middle school students. Seiko Kodaira had to put the newspapers aside to prepare the meal. In 1995, I wished him a happy 80th birthday. He responded in a charming way. But when we went to Kyoto in 1996, he was in the hospital. We could not speak to him again.

Kodaira's wife, Seiko, died in January 2000, two and a half years after her husband's death.

Jean-Pierre Serre was born in 1926 in France. He studied mathematics at the Ecole Normale Superieure. In 1954, at the age of 28, he was awarded a Fields Medal by the International Mathematical Union, the highest recognition for achievement in mathematics. Two years later he was appointed Professor of Algebra and Geometry at the College de France, where for about 15 years he was the youngest professor. He visited the Department of Mathematics, National University of Singapore, from 2 to 15 February 1985. His visit was sponsored by the French-Singapore Academic Exchange Programme. While in Singapore, Professor Serre delivered two lectures on algebraic curves over finite fields and one lecture on the Ramanujan function. He also gave a two-hour seminar talk on Faltings' proof of the Mordell conjecture, and a Colloquium lecture entitled "Discriminant = b2 - 4ac" on class numbers of imaginary quadratic fields. On February 14, 1985, he gave an interview in which he discussed various aspects of his mathematical career and his views of mathematics. What follows is a transcript of this interview, edited by C.T. Chong and Y.K. Leong, and revised by J.-P. Serre.

Q: What made you take up mathematics as your career?

A: I remember that I began to like mathematics when I was perhaps 7 or 8. In high school I used to do problems for more advanced classes. I was then in a boarding house in Nimes, staying with children older than I was, and they used to bully me. So to pacify them, I used to do their mathematics homework. It was as good a training as any.
My mother was a pharmacist (as was my father), and she liked mathematics. When she was a pharmacy student, at the University of Montpellier, she had taken a first-year course in calculus, just for fun, and passed the exam. And she had carefully kept her calculus books (by Fabry and Vogt, if I remember correctly). When I was 14 or 15, I used to look at these books, and study them. This is how I learned about derivatives, integrals, series and such (I did that in a purely formal manner -- Euler's style so to speak: I did not like, and did not understand, epsilons and deltas.) At that time, I had no idea one could make a living by being a mathematician. It was only later I discovered one could get paid for doing mathematics! What I thought at first was that I would become a high school teacher: this looked natural to me. Then, when I was 19, I took the competition to enter the Ecole Normale Superieure, and I succeeded. Once I was at "I'Ecole", it became clear that it was not a high school teacher I wanted to be, but a research mathematician.


Q: Did other subjects ever interest you, subjects like physics and chemistry?

A: Physics not much, but chemistry yes. As I said, my parents were pharmacists, so they had plenty of chemical products and test-tubes. I played with them a lot when I was about 15 or 16 besides doing mathematics. And I read my father's chemistry books (I still have one of them, a fascinating one, "Les Colloides" by Jacques Duclaux). However, when I learned more chemistry, I got disappointed by its almost mathematical aspect: there are long series of organic compounds like CH4, C2H6, etc, all looking more or less the same. I thought, if you have to have series, you might as well do mathematics! So, I quit chemistry -- but not entirely: I ended up marrying a chemist.

Q: Were you influenced by any school teacher in doing mathematics?

A: I had only one very good teacher. This was in my first year in high school (1943--1944), in Nimes. He was nicknamed "Le Barbu": beards were rare at the time. He was very clear, and strict; he demanded that every formula and proof be written neatly. And he gave me a thorough training for the mathematics national competition called "Concours General", where I eventually got first prize.
Speaking of Concours General, I also tried my hand at the one in physics, the same year (1944). The problem we were asked to solve was based entirely on some physical law I was supposed to know, but did not. Fortunately, only one formula seemed to me possible for that law. I assumed it was correct, and managed to do the whole 6-hour problem on that basis. I even thought I would get a prize. Unfortunately, my formula was wrong, and I got nothing -- as I deserved!


Q: How important is inspiration in the discovery of theorems?

A: I don't know what "inspiration" really means. Theorems, and theories, come up in funny ways. Sometimes, you are just not satisfied with existing proofs, and you look for better ones, which can be applied in different situations. A typical example for me was when I worked on the Riemann-Roch theorem (circa 1953), which I viewed as an "Euler-Poincare" formula (I did not know then that Kodaira-Spencer had had the same idea.) My first objective was to prove it for algebraic curves -- a case which was known for about a century! But I wanted a proof in a special style; and when I managed to find it, I remember it did not take me more than a minute or two to go from there to the 2-dimensional case (which had just been done by Kodaira). Six months later, the full result was established by Hirzebruch, and published in his well-known Habilitationsschrift.
Quite often, you don't really try to solve a specific question by a head-on attack. Rather you have some ideas in mind, which you feel should be useful, but you don't know exactly for what they are useful. So, you look around, and try to apply them. It's like having a bunch of keys, and trying them on several doors.


Q: Have you ever had the experience where you found a problem to be impossible to solve, and then after putting it aside for some time, an idea suddenly occured leading to the solution?

A: Yes, of course this happens quite often. For instance, when I was working on homotopy groups (~1950), I convinced myself that, for a given space X, there should exist a fibre space E, with base X, which is contractible; such a space would indeed allow me (using Leray's methods) to do lots of computations on homotopy groups and Eilenberg-MacLane cohomology. But how to find it? It took me several weeks (a very long time, at the age I was then...) to realize that the space of "paths" on X had all the necessary properties -- if only I dared call it a "fibre space", which I did. This was the starting point of the loop-space method in algebraic topology, many results followed quickly.

Q: Do you usually work on only one problem at a time or several problems at the same time?

A: Mostly one problem at a time, but not always. And I work often at night (in half-sleep), where the fact that you don't have to write anything down gives to the mind a much greater concentration, and makes changing topics easier.

Q: In physics, there are a lot of discoveries which were made by accident, like X-rays, cosmic background radiation and so on. Did that happen to you in mathematics?

A: A genuine accident is rare. But sometimes you get a surprise because some argument you made for one purpose happens to solve a question in a different direction; however, one can hardly call this an "accident".

Q: What are the central problems in algebraic geometry or number theory?

A: I can't answer that. You see, some mathematicians have clear and far-ranging "programs". For instance, Grothendieck had such a program for algebraic geometry; now Langlands has one for representation theory, in relation to modular forms and arithmetic. I never had such a program, not even a small size one. I just work on things which happen to interest me at the moment. (Presently, the topic which amuses me most is counting points on algebraic curves over finite fields. It is a kind of applied mathematics: you try to use any tool in algebraic geometry and number theory that you know of ... and you don't quite succeed!)

Q: What would you consider to be the greatest developments in algebraic geometry or number theory within the past five years?

A: This is easier to answer. Faltings' proof of the Mordell conjecture, and of the Tate conjecture, is the first thing which comes to mind. I would also mention Gross-Zagier's work on the class number problem for quadratic fields (based on a previous theorem of Goldfeld), and Mazur-Wiles theorem on Iwasawa's theory, using modular curves. (The application of modular curves and modular functions to number theory are especially exciting: you use GL2 to study GL1, so to speak! There is clearly a lot more to come from that direction ... may be even a proof of the Riemann Hypothesis some day!)

Q: Some scientists have done fundamental work in one field and then quickly moved on to another field. You worked for three years in topology, then took up something else. How did this happen?

A: It was a continuous path, not a discrete change. In 1952, after my thesis on homotopy groups, I went to Princeton, where I lectured on it (and on its continuation: "C-theory"), and attended the celebrated Artin-Tate seminar on class field theory.
Then, I returned to Paris, where the Cartan seminar was discussing functions of several complex variables, and Stein manifolds. It turned out that the recent results of Cartan-Oka could be expressed much more efficiently (and proved in a simpler way) using cohomology and sheaves. This was quite exciting, and I worked for a short while on that topic, making applications of Cartan theory to Stein manifolds. However, a very interesting part of several complex variables is the study of projective varieties (as opposed to affine ones -- which are somewhat pathological for a geometer); so, I began working on these complex projective varieties, using sheaves: that's how I came to the circle of ideas around Riemann-Roch, in 1953. But projective varieties are algebraic (Chow's theorem), and it is a bit unnatural to study these algebraic objects using analytic functions, which may well have lots of essential singularities. Clearly, rational functions should be enough -- and indeed they are. This made me go (around 1954) into "abstract" algebraic geometry, over any algebraically closed field. But why assume the field is algebraically closed? Finite fields are more exciting, with Weil conjectures and such. And from there to number fields it is a natural enough transition ... This is more or less the path I followed.

Another direction of work came from my collaboration (and friendship) with Armand Borel. He told me about Lie groups, which he knows like nobody else. The connections of these groups with topology, algebraic geometry, number theory, ... are fascinating. Let me give you just one such example (of which I became aware about 1968):

Consider the most obvious discrete subgroup of SL2(R), namely G = SL2(Z). One can compute its "Euler-Poincare characteristic" X(G) which turns out to be -1/12 (it is not an integer: this is because G has torsion). Now -1/12 happens to be the value zeta(-1) of the Riemann's zeta-function at the point s = -1 (a result known already to Euler). And this is not a coincidence! It extends to any totally real number field K, and can be used to study the denominator of zetak(-1). (Better results can be obtained by using modular forms, as was found later.) Such questions are not group theory, nor topology, nor number theory: they are just mathematics.


Q: What are the prospects of achieving some unification of the diverse fields of mathematics?

A: I would say that this has been achieved already. I have given above a typical example where Lie groups, number theory, etc, come together, and cannot be separated from each other. Let me give you another such example (it would be easy to add many more):
There is a beautiful theorem proved recently by S. Donaldson on four-dimensional compact differentiable manifolds. It states that the quadratic form (on H2) of such a manifold is severely restricted; if it is positive definite, it is a sum of squares. And the crux of the proof is to construct some auxiliary manifold (a "cobordism") as the set of solutions of some partial differential equation (non linear, of course)! This is a completely new application of analysis to differential topology. And what makes it even more remarkable is that, if the differentiability assumption is dropped, the situation becomes quite different: by a theorem of M. Freedman, the H2-quadratic form can then be almost anything.


Q: How does one keep up with the explosion in mathematical knowledge?

A: You don't really have to keep up. When you are interested in a specific question, you find that very little of what is being done has any relevance to you; and if something does have relevance, then you learn it much faster, since you have an application in mind. It is also a good habit to look regularly at Math. Reviews (especially the collected volumes on number theory, group theory, etc). And you learn a lot from your friends, too: it is easier to have a proof explained to you at the blackboard, than to read it.
A more serious problem is the one on the "big theorems" which are both very useful and too long to check (unless you spend on them a sizable part of your lifetime...). A typical example is the Feit-Thompson Theorem: groups of odd order are solvable. (Chevalley once tried to take this as the topic of a seminar, with the idea of giving a complete account of the proof. After two years, he had to give up.) What should one do with such theorems, if one has to use them? Accept them on faith? Probably. But it is not a very comfortable situation.

I am also uneasy with some topics, mainly in differential topology, where the author draws a complicated picture (in 2 dimensions), and asks you to accept it as a proof of something taking place in 5 dimensions or more. Only the experts can "see" whether such a proof is correct or not -- if you can call this a proof.


Q: What do you think will be the impact of computers on the development of mathematics?

A: Computers have already done a lot of good in some parts of mathematics. In number theory, for instance, they are used in a variety of ways. First, of course, to suggest conjectures, or questions. But also to check general theorems on numerical examples -- which helps a lot with finding possible mistakes.
They are also very useful when there is a large search to be made (for instance, if you have to check 106 or 107 cases). A notorious example is the proof of the Four-Colour Theorem. There is however a problem there, somewhat similar to the one with Feit-Thompson: such a proof cannot be checked by hand; you need a computer (and a very subtle program). This is not very comfortable either.


Q: How could we encourage young people to take up mathematics, especially in the schools?

A: I have a theory on this, which is that one should first discourage people from doing mathematics; there is no need for too many mathematicians. But, if after that, they still insist on doing mathematics, then one should indeed encourage them, and help them.
As for high school students, the main point is to make them understand that mathematics exists, that it is not dead (they have a tendency to believe that only physics, or biology, has open questions). The defect in the traditional way of teaching mathematics is that the teacher never mentions these questions. It is a pity. There are many such, for instance in number theory, that teenagers could very well understand: Fermat of course, but also Goldbach, and the existence of infinitely many primes of the form n2 + 1. And one should also feel free to state theorems without proving them (for instance Dirichlet's theorem on primes in arithmetic progression).


Q: Would you say that the development of mathematics in the past thirty years was faster than that in the previous thirty years?

A: I am not quite sure this is true. The style is different. In the 50's and 60's, the emphasis was quite often on general methods: distributions, cohomology and the like. These methods were very successful, but nowadays people work on more specific questions (often, some quite old ones: for instance the classification of algebraic curves in 3-dimensional projective spaces!). They apply the tools which were made before; this is quite nice. (And they also make new tools: microlocal analysis, supervarieties, intersection cohomology...).

Q: In view of this explosion of mathematics, do you think that a beginning graduate student could absorb this large amount of mathematics in four, five or six years and begin original work immediately after that?

A: Why not? For a given problem you don't need to know that much, usually -- and, besides, very simple ideas will often work.
Some theories get simplified. Some just drop out of sight. For instance, in 1949, I remember I was depressed because every issue of Annals of Mathematics would contain another paper on topology which was more difficult to understand than the previous ones. But nobody looks at these papers any more; they are forgotten (and deservedly so: I don't think they contained anything deep ...). Forgetting is a very healthy activity.

Still, it is true that some topics need much more training than some others, because of the heavy technique which is used. Algebraic geometry is such a case; and also representation theory.

Anyway, it is not obvious that one should say "I am going to work in algebraic geometry", or anything like that. For most people, it is better to just follow seminars, read things, and ask questions to oneself; and then learn the amount of theory which is needed for these questions.


Q: In other words, one should aim at a problem first and then learn whatever tools that are necessary for the problem?

A: Something like that. But since I know I cannot give good advice to myself, I should not give advice to others. I don't have a ready-made technique for working.

Q: You mentioned papers which have been forgotten. What percentage of the papers published do you think will survive?

A: A non-zero percentage, I believe. After all, we still read with pleasure papers by Hurwitz, or Eisenstein, or even Gauss.

Q: Do you think that you will ever be interested in the history of mathematics?

A: I am already interested. But it is not easy; I do not have the linguistic ability in Latin or Greek, for instance. And I can see that it takes more time to write a paper on the history of mathematics than in mathematics itself. Still, history is very interesting; it puts things in the proper perspective.

Q: Do you believe in the classification of finite simple groups?

A: More or less -- and rather more than less. I would be amused if a new sporadic group were discovered, but I am afraid this will not happen.

More seriously, this classification theorem is a splendid thing. One may now check many properties by just going through the list of all groups (typical example: the classification of n-transitive groups, for n at least 4).

Q: What do you think of life after the classification of finite simple groups?

A: You are alluding to the fact that some finite group theorists were demoralized by the classification; they said (or so I was told) "there will be nothing more to do after that". I find this ridiculous. Of course there would be plenty to do! First of course, simplifying the proof (that's what Gorenstein calls "revisionism"). But also finding applications to other parts of mathematics; for instance, there have been very curious discoveries relating the Griess-Fischer monster group to modular forms (the so-called "Moonshine").
It is just like asking whether Faltings' proof of the Mordell conjecture killed the theory of rational points on curves. No! It is merely a starting point. Many questions remain open.

(Still, it is true that sometimes a theory can be killed. A well-known example is Hilbert's fifth problem: to prove that every locally euclidean topological group is a Lie group. When I was a young topologist, that was the problem I really wanted to solve -- but I could get nowhere. It was Gleason, and Montgomery-Zippin, who solved it, and their solution all but killed the problem. What else is there to find in this direction? I can think of one question: can the group of p-adic integers act effectively on a manifold? This seems quite hard -- but a solution would have no application whatsoever, as far as I can see.)


Q: But one would assume that most problems in mathematics are like these, namely that the problems themselves may be difficult and challenging, but after their solution they become useless. In fact there are very few problems like the Riemann Hypothesis where even before its solution, people already know many of its consequences.

A: Yes, the Riemann Hypothesis is a very nice case: it implies lots of things (including purely numerical inequalities, for instance on discriminants of number fields). But there are other such examples: Hironaka's desingularization theorem is one; and of course also the classification of finite simple groups we discussed before.
Sometimes it is the method used in the proof which has lots of applications: I am confident this will happen with Faltings. And sometimes, it is true, the problems are not meant to have applications; they are a kind of test on the existing theories; they force us to look further.


Q: Do you still go back to problems in topology?

A: No. I have not kept track of the recent techniques, and I don't know the latest computations of the homotopy groups of spheres pin + k(Sn) (I guess people have already reached up to k = 40 or 50. I used to know them up to k = 10 or so.)
But I still use ideas from topology in a broad sense, such as cohomology, obstructions, Stiefel-Whitney classes, etc.


Q: What has been the influence of Bourbaki on mathematics?

A: A very good one. I know it is fashionable to blame Bourbaki for everything ("New Math" for instance), but this is unfair. Bourbaki is not responsible. People just misused his books; they were never meant for university teaching, even less high school teaching.

Q: Maybe a warning sign should have been given?

A: Such a sign was indeed given by Bourbaki: it is the Seminaire Bourbaki. The seminaire is not at all formal like the books; it includes all sorts of mathematics, and even some physics. If you combine the seminaire and the books, you get a much more balanced view.

Q: Do you see a decreasing influence of Bourbaki on mathematics?

A: The influence is different from what it was. Forty years ago, Bourbaki had a point to make; he had to prove that an organized and systematic account of mathematics was possible. Now the point is made and Bourbaki has won. As a consequence, his books now have only technical interest; the question is just whether they give a good exposition of the topic they are on. Sometimes they do (the one on "root systems" has become standard reference in the field); sometimes they don't (I won't give an example: it is too much a matter of taste).

Q: Speaking of taste, can you say what kind of style (for books, or papers), you like most?

A: Precision combined with informality! That is the ideal, just as it is for lectures. You find this happy blend in authors like Atiyah or Milnor, and a few others. But it is hard to achieve. For instance, I find many of the French (myself included) a bit too formal, and some of the Russians a bit too imprecise...
A further point I want to make is that papers should include more side remarks, open questions, and such. Very often, these are more interesting than the theorems actually proved. Alas, most people are afraid to admit that they don't know the answer to some question, and as a consequence they refrain from mentioning the question, even if it is a very natural one. What a pity! As for myself, I enjoy saying "I do not know".

Joseph Ayoub, Professor of Mathematics at the University of Zurich, is the first holder of the “Alexzandria Figueroa and Robert Penner” Chair. He is interested in the cohomology of algebraic varieties and the theory of motives.

How did your interest in mathematics start?

I’ve always been very interested in maths. In my early teens, I had good grades in all subjects but maths was always a special interest of mine: in my spare time, I enjoyed solving maths problems. When I ran out of them, I made new ones up. I was particularly keen on plane geometry but I also liked calculating things and solving equations. During breaks, I often disappeared into the library to look through the Encyclopaedia Universalis in search of maths articles. This is how I became familiar with a number of modern concepts such as the classification of finite simple groups.
I was able to access bits of “advanced mathematics” at a very young age, when I found some papers in the storage room of our small apartment in Beyrouth. They were notes of the lectures on general topology which my father – a maths professor – had followed at the university. My mother, who was a librarian at the science faculty, knew someone who helped me lay my hands on a copy of Differential Geometry and Symmetric Spaces by Helgason. I remember having spent most of the summer holidays compulsively going through that book. I ended up reading it from start to finish and feeling I had understood everything!
In 1998, straight after my baccalaureate, I was lucky enough to be admitted to Lycée Louis‑le‑Grand in Paris. That’s when I understood that you could earn a living from mathematical research, which was a real revelation for me. It was my maths teacher, Hervé Gianella, who made me realise this and who encouraged me to take the École normale supérieure entrance examination. I had previously seen myself as becoming an engineer with a “proper” job and an “eccentric” hobby: reading maths books.

What is your connection with IHES?

The first time I heard about IHES was in connection with Alexandre Grothendieck. His name is inextricably linked with that of IHES. In a way, I first discovered IHES with the élément de Géométrie Algébrique and the “Séminaire de Géométrie Algébrique”, which were largely prepared and drafted at IHES. It was much later that I came to IHES, and that was for a conference in honour of Luc Illusie.
I am very grateful to the scientific council for having chosen me as the first holder of Alexzandria Figueroa and Robert Penner Chair. It is a great honour of course and I am already looking forward to the time I will be spending at IHES. I don’t yet know what impact my visits will have on my work but I will try to extract the maximum benefit from them.

How would you summarise your main contributions?

For a long time, I worked on a particular and crucial conjecture in motive theory called the “conservativity conjecture.” The conjecture is very easy to state and offers a bridge, or rather a return path to two different kinds of objects. One is a motive, which is a very rich algebraic geometric object, the other is its realisation which is a topological object with no additional structure.
The conservativity structure turned out to be very difficult. Nonetheless, I devised a strategy to demonstrate it. Even if I haven’t managed to make it work yet, I consider this unfinished business to be my most important contribution.

What inspired you so much to pursue your research and what do you find most exciting in what you do?

What I love most in mathematics is the coherence that emanates from a well‑constructed theory. Once the right point of view has been identified, the right definition, the right context, what follows is more or less inevitable and the result is very coherent. I think I really value that coherence. Luckily, there is no shortage of well-constructed theories in algebraic geometry, which is probably one of Grothendieck’s legacies.
I also like the writing stage. In fact, I think doing and writing maths are activities which cannot be separated. It’s only when I write an article that I really understand the demonstration of a problem and the cogs and wheels in a theory. Unfortunately, the big questions I’ve addressed so far have turned out to be very tough. This is naturally the source of some disappointment but I am an optimist. What inspires me to carry on is definitely the hope of seeing the solution to these great questions one day. Another source of hope and inspiration is to have been witness to spectacular progress on other topics and in other mathematics fields.

The 2018 Fields Medals have been awarded to four researchers in number theory, geometry and network analysis.

The 2018 Fields Medalists : Caucher Birkar, Alessio Figalli, Akshay Venkatesh, Peter Scholze. Photo: Handout/TheGuardian

Number theorist Peter Scholze, who was promoted to professor at the age of 24 and is now Germany's youngest professor; geometer Caucher Birkar - a Kurdish refugee; network analysis researcher Alessio Figalli and number theorist Akshay Venkatesh - have been awarded the 2018 Fields Medals at the International Mathematical Union congress in Rio de Janeiro, Brazil.

The Fields Medals - the most prestigious award in mathematics - are awarded every four years to mathematicians under the age of 40. For the first time in the 82-year history of the Fields Medal, none of the winners are from the United States or France, two countries that have traditionally won the prize, accounting for nearly half of all Fields Medals awarded. Maryam Mirzakhani, who won in 2014, remains the only woman to have ever received the award (she died of cancer in 2017).

Scholze was almost a foregone conclusion in the mathematics community that he would win this year’s Fields Medal, to the point where people often ask, “Who else could win the prize, other than Scholze?” He had already made a name for himself at the age of 22 for finding a way to dramatically shorten a proof in algebraic geometry.

Much of Scholze’s research has involved “p-adic fields,” exotic extensions of the normal number system that are useful tools for studying prime numbers. On p-adics, he has built Perfectoid spaces—fratal-like structures that have helped him and other researchers solve problems across a wide range of mathematical fields, including geometry and topology.

In a 2016 profile, a colleague described Scholze’s ability to focus on the essence of a problem as “a mixture of fear and excitement.” In recent months, Scholze has been working on a major proof of the abc conjecture, one of the most important problems in number theory. In 2012, Shinichi Mochizuki (a mysterious Japanese mathematician) posted a proof online, but no one could say for sure whether it was correct. Now Scholze and a colleague are said to have found a significant flaw in the proof. Scholze is a professor at the University of Bonn and also at the Max-Planck Institute for Mathematics.

Caucher Birkar, 40, has made breakthroughs in the classification of algebraic varieties — geometric objects that arise from polynomial equations, such as y = x2. He was born in 1978 in an ethnic Kurdish region of southern Iran. In a series of videos for the Simons Foundation, which funds mathematics and basic science research, Birkar talks about his childhood: “My parents were farmers, so I spent a lot of time farming. It was not the ideal place for a kid to be interested in something like mathematics.”

After studying at the University of Tehran, Birkar moved to the UK in 2000 as a refugee and was later granted British citizenship. In the video, Birkar says he hopes the Fields Medal will put “a smile on the faces” of the world’s 40 million Kurds.

Akshay Venkatesh, 36, works on classical problems in number theory, including rational number systems of integers and roots like √2. He is one of the few mathematicians to have made progress on a question posed by Carl Friedrich Gauss in the 1800s. Venkatesh was born in New Delhi, grew up in Australia, and is now at the Institute for Advanced Study in Princeton, New Jersey.

Alessio Figalli, 34, studies optimal transport, which seeks to find the most efficient way to distribute elements in networks, a field closer to the real world. Figalli applies it to partial differential equations—equations with several variables that often appear in physics. Figalli is Italian and works at the Swiss Institute of Technology (ETH) in Zurich.

A Rutgers University-New Brunswick professor who has devoted his career to resolving the mysteries of higher mathematics has solved two separate, fundamental problems that have perplexed mathematicians for decades.

The solutions to these long-standing problems could further enhance our understanding of symmetries of structures and objects in nature and science, and of long-term behavior of various random processes arising in fields ranging from chemistry and physics to engineering, computer science and economics.

Pham Tiep, the Joshua Barlaz Distinguished Professor of Mathematics in the Rutgers School of Arts and Science's Department of Mathematics, has completed a proof of the 1955 Height Zero Conjecture posed by Richard Brauer, a leading German-American mathematician who died in 1977.

Proof of the conjecture—commonly viewed as one of the most outstanding challenges in a field of math known as the representation theory of finite groups—is published in the Annals of Mathematics.

"A conjecture is an idea that you believe has some validity," said Tiep, who has thought about the Brauer problem for most of his career and worked on it intensively for the past 10 years. "But conjectures have to be proven. I was hoping to advance the field. I never expected to be able to solve this one."

In a sense, Tiep and his colleagues have been following a blueprint of challenges Brauer laid out for them in a series of mathematical conjectures posed and published in the 1950-60s.

"Some mathematicians have this rare intellect," Tiep said of Brauer. "It's as though they came from another planet or from another world. They are capable of seeing hidden phenomena that others can't."

In the second advance, Tiep solved a difficult problem in what is known as the Deligne-Lusztig theory, part of the foundational machinery of representation theory. The breakthrough touches on traces, an important feature of a rectangular array known as a matrix. The trace of a matrix is the sum of its diagonal elements. The work is detailed in two papers. One was published in Inventiones mathematicae, the second in Annals of Mathematics.

"Tiep's high-quality work and expertise on finite groups has allowed Rutgers to maintain its status as a top world-wide center in the subject," said Stephen Miller, a Distinguished Professor and Chair of the Department of Mathematics.

"One of the great accomplishments in 20th century mathematics was the classification of the so-called but perhaps misleadingly named 'simple' finite groups, and it is synonymous with Rutgers—it was led from here and many of the most interesting examples were discovered here. Through his amazing stretch of strong work, Tiep brings international visibility to our department."

Insights from the solution are likely to greatly enhance mathematicians' understanding of traces, Tiep said. The solution also provides insights that could lead to breakthroughs in other important problems in mathematics, including conjectures posed by the University of Florida mathematician John Thompson and the Israeli mathematician Alexander Lubotzky, he added.

Both breakthroughs are advances in the field of representation theory of finite groups, a subset of algebra. Representation theory is an important tool in many areas of math, including number theory and algebraic geometry, as well as in the physical sciences, including particle physics. Through mathematical objects known as groups, representation theory has also been used to study symmetry in molecules, encrypt messages and produce error-correcting codes.

Following the principles of representation theory, mathematicians take abstract shapes that exist in Euclidean geometry—some of them extremely complex—and transform them into arrays of numbers. This can be achieved by identifying certain points that exist in each three- or higher-dimensional shape and converting them to numbers placed in rows and columns.

The reverse operation must work, too, Tiep said. One needs to be able to reconstitute the shape from the sequence of numbers.

Unlike many of his colleagues in the physical sciences who often employ complex devices to advance their work, Tiep said he uses only a pen and paper to conduct his research, which so far has resulted in five books and more than 200 papers in leading mathematical journals.

He jots down math formulas or sentences indicating chains of logic. He also engages in continual conversations—in person or on Zoom—with colleagues as they proceed step by step through a proof.

But progress can come from internal reflection, Tiep said, and ideas burst forth when he is least expecting it.

"Maybe I'm walking with our children or doing some gardening with my wife or just doing something in the kitchen," he said. "My wife says she always knows when I'm thinking about math."

On the first proof, Tiep collaborated with Gunter Malle of Technische Universität Kaiserslautern in Germany, Gabriel Navarro of Universitat de València in Spain and Amanda Schaeffer Fry, a former graduate student of Tiep who is now at the University of Denver.

For the second breakthrough, Tiep worked with Robert Guralnick of the University of Southern California and Michael Larsen of Indiana University. On the first of two papers that tackle the mathematical problems on traces and solve them, Tiep worked with Guralnick and Larsen. Tiep and Larsen are co-authors of the second paper.

"Tiep and co-authors have obtained bounds on traces that are about as good as we could ever expect to obtain," Miller said. "It's a mature subject which is important from many angles, so progress is hard—and applications are many."
Source: phys.org