Mathematics For All

“What one fool can do, another can.”
— Calculus Made Easy (1910), Silvanus P. Thompson 📘

Calculus Made Easy: being a Very Simplest Introduction to those Beautiful Methods of Reckoning which are Generally Called by the Terrifying Names of the Differential Calculus and the Integral Calculus by Silvanus Phillips Thompson was first published in 1910 and remains one of the most classic, elegant, and beginner-friendly introductions to calculus.
It simplifies the ideas of Differential Calculus and Integral Calculus in a remarkably clear and engaging way, proving that calculus is not as frightening as it sounds.

✨ A timeless book for students, teachers, and anyone who wants to understand calculus with clarity and confidence.

Mathematics is not only formulas — it is form.

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📸 Image by Yung Cheng Lin

Daniel Quillen (June 22, 1940 – April 30, 2011) - The Fields Medalist of 1978

Daniel Quillen earned his B.A. degree in 1961 and completed his Master’s degree the following year. He subsequently commenced doctoral research at Harvard University under the supervision of Raoul Bott. Quillen consistently acknowledged Bott’s profound influence on his mathematical formation and intellectual maturation. As Graeme Segal remarks (see [E. Friedlander and D. Grayson (eds.), Daniel Quillen, Notices Amer. Math. Soc. 59 (10) (2012), 1392-1406.]):-

"He said that Bott - a large, outgoing man universally beloved for his warmth and personal magnetism, outwardly quite the opposite of his shy and reticent student - was a crucial model for him, showing him that one did not have to be quick to be an outstanding mathematician. Unlike Bott, who made a performance of having everything explained to him many times over, Quillen did not seem at all slow to others, yet he saw himself as someone who had to think things out very slowly and carefully from first principles and had to work hard for every scrap of progress he made. He was truly modest about his abilities - very charmingly so - though at the same time ambitious and driven."

In 1964, Daniel Quillen was awarded his Ph.D. for his thesis on partial differential equations, Formal Properties of Over-Determined Systems of Linear Partial Differential Equations. Following the completion of his doctorate, he joined the faculty at Massachusetts Institute of Technology. Over the next several years, he conducted research visits at various universities, experiences that proved decisive in shaping the future direction of his work. During the academic year 1968–69, he held a Sloan Fellowship at the Institut des Hautes Études Scientifiques near Paris, where he was deeply influenced by Alexander Grothendieck. As Hyman Bass writes [E. Friedlander and D. Grayson (eds.), Daniel Quillen, Notices Amer. Math. Soc. 59 (10) (2012), 1392-1406.]:-

"During the year in Paris, Quillen presented his typical personal characteristics: a gentle good nature, modesty, a casual and boyish appearance unaltered by his prematurely graying hair, and his already ample family life. In that brilliant, and often flamboyant, mathematical milieu, Quillen seemed to listen more than he spoke, and he spoke only when he had something substantial to say. His later work showed him to be a deep listener."

During the 1960s, Daniel Quillen developed a general framework for defining the homology of simplicial objects across a wide range of categories, including sets, algebras over a ring, and unstable algebras over the Steenrod algebra. He also worked on a conjecture in homotopy theory proposed by Frank Adams. Quillen attacked the Adams conjecture by two markedly different methods: one drawing on ideas from algebraic geometry and the other using techniques from the modular representation theory of finite groups. Both strategies proved effective, the algebraic–geometric approach was ultimately completed by one of his students, while Quillen himself established the result via modular representation theory.The latter methods were later applied with great success to the cohomology of groups and to algebraic K-theory. In particular, his work on group cohomology led to a structure theorem describing mod-p cohomology rings of finite groups, resolving several open problems in the field.

Quillen was awarded a Fields Medal at the International Congress of Mathematicians 1978, held in Helsinki, in recognition of his role as the principal architect of higher algebraic K-theory (introduced in 1972). This theory provided powerful new tools that combined geometric and topological ideas to address deep problems in algebra, particularly in ring and module theory. Algebraic K-theory extended concepts introduced by Alexander Grothendieck for commutative rings, which had earlier inspired Michael Atiyah and Friedrich Hirzebruch in their development of topological K-theory. Quillen’s formative periods in Paris under Grothendieck’s influence and later at Princeton University working with Atiyah were therefore crucial in shaping his creation of higher algebraic K-theory.

Among the many distinctions awarded to Daniel Quillen, in addition to the Fields Medal, was his invitation to deliver a plenary lecture at the International Congress of Mathematicians 1974, held in Vancouver, in August 1974, where he presented his address Higher Algebraic K-Theory. He was also awarded the American Mathematical Society’s Cole Prize in Algebra in 1975 for his paper "Higher algebraic K-theories".

Let us conclude this biographical sketch by quoting the tribute to Daniel Quillen written by the editors of the Journal of K-Theory [Editors, Hommage to Daniel Gray Quillen, J. K-Theory 8 (2011), 1.]:-

"More than anyone else, he was responsible for creating the subject of algebraic K-theory as it is pursued today, and for demonstrating its power and elegance. He also made fundamental contributions to many other aspects of mathematics: rational homotopy, model categories, formal groups, and cyclic homology, to mention a few. All of the ideas he has developed will survive him and give him the stature of a great mathematician of the 20th century. Many mathematicians including all of the members of our Board were greatly inspired and influenced by his vision, his teaching and his writing. As editors devoted to the subject that Quillen largely created, we are highly appreciative of his crucial support for the journal "K-Theory" and its successor the "Journal of K-Theory", and of all that he has done for our area of mathematics. He will be greatly missed and fondly remembered."

(Source: MacTutor)
(Photo courtesy of Cypora Cohen)

Grigory Margulis (born February 24, 1946) - The Fields Medalist of 1978

Gregori Margulis began his undergraduate studies at Moscow University in 1962, earning his first degree in 1967 and continuing for postgraduate studies. He demonstrated significant mathematical potential, winning the young mathematicians prize from the Moscow Mathematical Society in 1968 during his postgraduate tenure. Margulis completed his graduate studies in 1970, receiving the degree of Candidate of Science for his thesis "On some problems in the theory of U-systems."

Margulis, after earning the Candidate of Science degree, worked at the Institute for Problems in Information Transmission as a Junior scientific worker from 1970 to 1974, then became a Senior scientific worker. He was awarded a Fields Medal at the International Congress in Helsinki but could not attend due to restrictions from Soviet authorities. Jacques T**s expressed sadness over Margulis's absence during his address. He addresses Margulis' contributions to combinatorics, differential geometry, ergodic theory, dynamical systems, and discrete subgroups of Lie groups, emphasizing that the Fields Medal was mostly granted for his work in the latter area. He mentioned in [J. T**s, The work of Gregori Aleksandrovitch Margulis, Proceedings of the International Congress of Mathematicians, Helsinki, 1978 (Helsinki, 1980), 57-63.] that:-

"Already Poincaré wondered about the possibility of describing all discrete subgroups of finite covolume in a Lie group G. The profusion of such subgroups in
G=PSL_2(R) makes one at first doubt of any such possibility. However, PSL_2(R) was for a long time the only simple Lie group which was known to contain non-arithmetic discrete subgroups of finite covolume, and further examples discovered in 1965 by Makarov and Vinberg involved only few other Lie groups, thus adding credit to conjectures of Selberg and Pyatetski-Shapiro to the effect that "for most semisimple Lie groups" discrete subgroups of finite covolume are necessarily arithmetic. Margulis's most spectacular achievement has been the complete solution of that problem and, in particular, the proof of the conjecture in question."

Margulis was soon able to leave the Soviet bloc and, in 1979, he was able to spend three months at the University of Bonn. Between 1988 and 1991 Margulis made a number of visits to the Max Planck Institute in Bonn, to the Institut des Hautes Études and to the Collège de France, to Harvard and to the Institute for Advanced study in Princeton. From 1991 he has held a chair at Yale University.

The Oppenheim conjecture was made in 1929 and concerns values of indefinite irrational quadratic forms at integer points. Early work was based on results of Jarnik and Walfisz. In the 1940s Davenport and Heilbronn contributed by proving special cases and in 1946 Watson extended their results showing the conjecture to be true for further special cases. Margulis proved the full conjecture in 1986 and gives a beautiful survey of the work leading to this solution in [G .A. Margulis, Oppenheim conjecture, in M Atiyah and D Iagolnitzer (eds.), Fields Medallists Lectures (Singapore, 1997), 272-327.].

Margulis has garnered numerous accolades for his contributions, including the Fields Medal, the Medal of the Collège de France (1991), and honorary membership in the American Academy of Arts and Science in the same year. He received the Humboldt Prize in 1995 and became a member of the Tata Institute of Fundamental Research in 1996. Additionally, he was awarded the Lobachevsky International Prize by the Russian Academy of Sciences and has been elected to the United States National Academy of Sciences. In 2005 he was awarded the Wolf Prize for Mathematics:-

"... for his monumental contributions to algebra, in particular to the theory of lattices in semi-simple Lie groups, and striking applications of this to ergodic theory, representation theory, number theory, combinatorics and measure theory."

(Source: MacTutor)

Charles Fefferman (born April 18, 1949) - The Fields Medalist of 1978

Charles Fefferman mastered calculus before the age of twelve. He explained how he got to that level so quickly:-

"I read the 4th grade Mathematics textbook in a day or two. My father didn't believe me, so he asked me a few questions and realised that I understood. Then he bought me a 5th grade textbook, and I read it in a day or two, and so on until I studied Calculus and that took longer than a day or two. But I was just a little boy studying Calculus so it was obvious that I had talent."

Fefferman's parents, realizing that their twelve-year-old son had a remarkable mathematical talent, took him to the University of Maryland, which was close to their home, for tutoring. The story is described in [C. Bode, One of the Great Mathematicians of the 20th Century, The Washington Post (20 May, 1979).]:-

"Jim Hummel, an authority on complex variables, was sitting in his tiny office one morning in July 1961. He got a phone call from the Feffermans. They had this avid 6th grader who was consuming mathematics textbooks like candy. Would Professor Hummel talk with him? Professor Hummel would. So Mrs Fefferman drove Charlie from Silver Spring to College Park in their trusty Dodge. "I wasn't scared," Charlie recalls, "but I was really excited."
Once a week for the rest of the summer, the two met for mathematical talk. Charlie was bright, indeed, so Hummel invited him back for the next summer. At a certain point during that second summer, Hummel realised that Charlie was a genuine mathematician, that he was already "thinking like a mathematician." The point came when Charlie asked a searching question about the Peano axioms for integers."

Charles Fefferman showed exceptional talent from a young age, entering Eastern Junior High in 1962 and then the University of Maryland at just fourteen in 1963, despite university rules. His professors advocated for his admission, recognizing his abilities. That same year, his father changed jobs to become director of economic analysis at the American Council of Life Insurance. Charles quickly excelled at university, moving from a basic calculus class to an honors course within two weeks. By age 15, during his second year, he had already written his first mathematics paper.
At 17, he graduated with a B.S. with the highest distinction in Mathematics and Physics and received a 3-year National Science Foundation Fellowship for research. Although he skipped high school, Montgomery Blair High School awarded him an honorary diploma. He then pursued postgraduate studies at Princeton University under Elias Stein.

Fefferman was full of praise for his thesis advisor [A Timón, Interview: Charles Fefferman, Fields medalist in 1978 and director of a ICMAT-Laboratory, ICMAT newsletter (Third quarter, 2013), 6-8.]:-

"My thesis advisor, Elias Stein, has had a tremendous influence on me; I think he was probably the best teacher of advanced mathematics in the world. He influenced me about what I learnt and formed my taste, and he taught me his spirit of optimism when facing hard problems."

Fefferman earned his PhD in 1969 with a thesis entitled Inequalities for Strongly Regular Convolution Operators. He published two papers based on the material of his thesis, On some singular convolution operators (1969) and Inequalities for strongly singular convolution operators (1970).

Fefferman lectured at Princeton from 1969 to 1970 before joining the University of Chicago as an assistant professor in 1970. He became the youngest full professor in the U.S. in 1971. He received the Alfred P. Sloan Foundation Fellowship in 1970 and a NATO Postdoctoral Fellowship in 1971.

In 1976, Fefferman became the first recipient of the Alan T. Waterman Award, the United States' highest honor for scientists under 35, which included a medal and a $50,000 annual grant for three years. This allowed him to focus on research at Princeton without teaching duties. He made significant contributions to multidimensional complex analysis and partial differential equations. His work on partial differential equations, Fourier analysis, in particular convergence, multipliers, divergence, singular integrals and Hardy spaces earned him a Fields Medal at the International Congress of Mathematicians at Helsinki in 1978.

Charles Fefferman has received several prestigious awards throughout his career, including the Bergman Prize (1992), the Bôcher Memorial Prize (2008), honorary membership in the London Mathematical Society (2009), the Wolf Prize (2017), and the BBVA Frontiers of Knowledge award (2022).

(Source: MacTutor)

Pierre Deligne (born 3 October 1944) - The Fields Medalist of 1978

Deligne obtained his Licence en mathématiques in November 1966, equivalent to a bachelor’s degree. He then pursued doctoral studies at the Free University of Brussels. In September 1967, he became a junior researcher at the Fonds National de la Recherche Scientifique in Brussels, while simultaneously serving as a guest at the Institut des Hautes Études Scientifiques (IHES) in Bures-sur-Yvette, France, where he worked with Alexandre Grothendieck. He was awarded the Doctorat en mathématiques by the Free University of Brussels in November 1968.

Following the completion of his doctorate, Deligne joined the IHES as a visiting member, a position he held until February 1970, after which he became a permanent member of the Institute. During his early years at the IHES, he worked with Grothendieck on the generalization of Zariski’s main theorem. He also collaborated closely with Jean-Pierre Serre, resulting in significant advances in the study of ℓ-adic representations associated with modular forms and in the conjectural functional equations of L-functions.

Deligne’s interactions with both Grothendieck and Serre proved especially influential, as their mathematical philosophies differed markedly. Grothendieck sought the utmost generality and preferred to develop results independently of existing literature, whereas Serre possessed a deep command of the literature and emphasized elegant and well-chosen special cases. Deligne believed that their contrasting approaches complemented one another and that their collaboration was mutually beneficial. For himself, he found that navigating between these perspectives was an invaluable learning experience. He later remarked that while Grothendieck’s lectures were inspiring, attending Serre’s lectures helped him remain grounded. During this period at the IHES, Deligne also collaborated with David Mumford on a new description of moduli spaces of curves, work that has since played an important role in developments related to string theory.

Deligne’s exceptional contributions were rapidly recognized through several prestigious awards. In 1974, he received both the François Deruyts Prize from the Royal Academy of Sciences of Belgium and the Henri Poincaré Medal from the French Academy of Sciences. In 1975, he was further honored with the A. De Leeuw–Damry–Bourlart Prize from the Belgian National Science Foundation.

Deligne resolved the three Weil conjectures in 1974, producing a landmark achievement that unified ideas from algebraic geometry and algebraic number theory. This breakthrough led to his being awarded the Fields Medal at the International Congress of Mathematicians in Helsinki in 1978. Addressing these conjectures required the creation of a new form of algebraic topology. As Jacques T**s remarked [Pierre Deligne, in Chris R Somerville and Elliot M Meyerowitz, Premi Balzan (Fondazione internazionale Balzan, 2004).]:–

"These conjectures were both exceptionally hard to settle (the best specialists, including A Grothendieck, had worked on them) and most interesting in view of the far-reaching consequences of their solution."

Deligne has made significant contributions to a wide range of major mathematical problems. Beyond algebraic geometry, his work spans Hilbert’s 21st problem, Hodge theory, moduli theory, modular forms, Galois representations, L-series and the Langlands conjectures, as well as the representation theory of algebraic groups.

In addition to the Fields Medal, Deligne was awarded the Crafoord Prize by the Royal Swedish Academy of Sciences in 1988, in recognition of his fundamental contributions to algebraic geometry.

In 2004, Deligne was elected an honorary member of the London Mathematical Society. In the same year, he was awarded the Balzan Prize in Mathematics by the International Balzan Foundation. The prize carried an award of one million Swiss francs (approximately US$800,000), half of which was designated to support research projects involving young scholars in his field. The award ceremony was held on 18 November 2004 at the Accademia dei Lincei in Rome.

In February 2008, Deligne received the Wolf Prize, which on that occasion was shared with Phillip Griffiths and David Mumford. Later that year, he became Emeritus at the Institute for Advanced Study in Princeton. His distinguished career continued to be recognized with further honours: he was elected a Foreign Member of the Royal Swedish Academy of Sciences in February 2009 and a member of the American Philosophical Society in April 2009. Among the many distinctions he received, perhaps the most significant was the Abel Prize, awarded to him in May 2013.

(Source: MacTutor)

David Mumford (born 11 June 1937) - The Fields Medalist of 1974

After attending Exeter School, Mumford enrolled at Harvard University, where he first developed an interest in algebraic varieties. He relates in [D. Mumford, Autobiography of David Mumford, in M Atiyah and D Iagolnitzer (eds.), Fields Medallists Lectures (Singapore, 1997), 225.] that

"... a classmate said "Come with me to hear Professor Zariski's first lecture, even though we won't understand a word" and Oscar Zariski bewitched me. When he spoke the words "algebraic variety", there was a certain resonance in his voice that said distinctly that he was looking into a secret garden. I immediately wanted to be able to do this too. It led me to 25 years of struggling to make this world tangible and visible."

After graduating from Harvard, Mumford joined the faculty there. He was appointed Professor of Mathematics in 1967 and, ten years later, became Higgins Professor. He served as Chair of the Mathematics Department at Harvard from 1981 to 1984 and was a MacArthur Fellow from 1987 to 1992.

Mumford’s greatest honour was the award of the Fields Medal at the International Congress of Mathematicians in Vancouver in 1974. In [J. Tate, The work of David Mumford, Proceedings of the International Congress of Mathematicians, Vancouver, 1974 1 (Montreal, Que., 1975), 11-15.], Tate describes the work for which Mumford received the Fields Medal, writing that

"Mumford's major work has been a tremendously successful multi-pronged attack on problems of the existence and structure of varieties of moduli, that is, varieties whose points parameterise isomorphism classes of some type of geometric object. Besides this he has made several important contributions to the theory of algebraic surfaces. ... Mumford has carried forward, after Zariski, the project of making algebraic and rigorous the work of the Italian school on algebraic surfaces. He has done much to extend Enriques' theory of classification to characteristic p > 0, where many new difficulties appear."

Mumford has received numerous honours in addition to the Fields Medal. He was awarded honorary D.Sc. degrees by the University of Warwick (1983), the Norwegian University of Science and Technology (2000), and Rockefeller University (2001). He was elected to the National Academy of Sciences in 1975 and became an Honorary Fellow of the Tata Institute of Fundamental Research in 1978. Further distinctions include election as a Foreign Member of the Accademia Nazionale dei Lincei, Rome (1991), Honorary Member of the London Mathematical Society (1995), and membership in the American Philosophical Society (1997), as well as election as a Foreign Member of the Royal Society in 2008.

In recognition of his contributions, Mumford received the Shaw Prize in 2006, the Steele Prize for Mathematical Exposition from the American Mathematical Society in 2007, and the Wolf Prize in 2008. He also served as President of the International Mathematical Union from 1995 to 1999.

Finally, we mention a lecture delivered by Mumford on 11 February 2008, which reflects his interests at that stage of his career. The lecture, entitled What’s an Infinite Dimensional Manifold and How Can It Be Useful in Hospitals?, was accompanied by the following abstract:

"Morphing faces has become a popular game but what is the math behind it? One way to view it is as the construction of geodesics on an infinite dimensional manifold of shapes. I will try to explain what this means, using simple examples and then go on show why it is proposed as a new tool in the diagnosis of medical conditions."

(Source: MacTutor)

Enrico Bombieri (born 26 November 1940) - The Fields Medalist of 1974

Enrico Bombieri was awarded the Fields Medal at the International Congress of Mathematicians in Vancouver in 1974 in recognition of his outstanding contributions. The award honored his major work on prime number theory, univalent functions and the local Bieberbach conjecture, functions of several complex variables, as well as partial differential equations and minimal surfaces, particularly his work on Sergei Bernstein’s problem in higher dimensions.

K. Chandrasekharan highlights Bombieri’s significant contributions to the theory of prime distribution, univalent functions and the local Bieberbach conjecture, and functions of several complex variables. He writes

"First among Bombieri's achievements is his remarkable theorem on the distribution of primes in arithmetical progressions, which is obtained by an application of the methods of the large sieve."

The large sieve method was first introduced by Linnik in 1941 while addressing problems posed by Vinogradov. Given an arithmetic progression, the large sieve gives information about the distribution of an arbitrary finite set of integers. Rényi further developed Linnik’s ideas in 1950, and in 1965, Klaus Roth and Enrico Bombieri independently refined Rényi’s results. Bombieri later used his improved large sieve technique to establish what is now known as Bombieri’s mean value theorem, which deals with the distribution of prime numbers in arithmetic progressions.

In 1966, Bombieri was appointed to a professorship at the University of Pisa. During this period, he became interested in the work of De Giorgi and his school of geometric measure theory at the Scuola Normale Superiore in Pisa. Their research focused on Plateau-type problems in spaces of dimension higher than three.

Bombieri was awarded the Balzan International Prize in 1980 and was elected a foreign member of the French Academy of Sciences in 1984. He now works in the United States. In 1996, he was elected to the National Academy of Sciences, and the citation for his election stated:

"Bombieri is one of the world's most versatile and distinguished mathematicians. He has significantly influenced number theory, algebraic geometry, partial differential equations, several complex variables, and the theory of finite groups. His remarkable technical strength is complemented by an unerring instinct for the crucial problems in key areas of mathematics."

In addition to the honors mentioned earlier, Bombieri received the Feltrinelli Prize in 1976, the Cavaliere di Gran Croce al Merito della Repubblica Italiana in 2002, and the Premio Internazionale Pitagora from the City of Crotone in 2006. In January 2008, he was jointly awarded the Doob Prize with Walter Gubler at the 114th Annual Meeting of the American Mathematical Society in San Diego, in recognition of their book Heights in Diophantine Geometry, which they co-authored. The citation for the prize reads as follows:

"The book is a research monograph on all aspects of Diophantine geometry, both from the perspective of arithmetic geometry and of transcendental number theory. ... One gets the sense that every lemma, every theorem, every remark has been carefully considered, and every proof has been thought through in every detail. There are well-chosen illuminating examples throughout every chapter. The book is a masterpiece in terms of its original approach, its unrivalled comprehensiveness, and the sheer elegance of the exposition. There can be no doubt that this book will become the basis for the future development of this central subject of modern mathematics."

(Source: MacTutor)
(PC: https://shorturl.at/JJCqf)

"Great spirits have always encountered violent opposition from mediocre minds..."

~Albert Einstein

(Pic. Credit: https://shorturl.at/42hWK)

In 1986, math teachers protested against the use of calculators in classrooms. They feared students would stop thinking and learning basic concepts.

Today, history seems to be repeating itself.
Now, it’s AI that worry educators. Essays can be written in seconds, homework can be generated instantly, and schools are still figuring out how to respond.

Back then, calculators didn’t destroy mathematics. Instead, they changed how we learned it.

Will AI do the same?

Should we expect new protests from teachers, or will we learn how to use this tool wisely?

What do you think — threat or opportunity?

John G. Thompson (born October 13, 1932) - The Fields Medalist of 1970

John Thompson went to the University of Chicago to pursue research and completed his doctorate in 1959. His doctoral thesis, entitled “A Proof that a Finite Group with a Fixed-Point-Free Automorphism of Prime Order is Nilpotent,” was supervised by Saunders Mac Lane. In this work, Thompson solved a conjecture of Frobenius that had remained open for nearly 60 years. As reflected in its title, his thesis proved that a finite group admitting a fixed-point-free automorphism of prime order must necessarily be nilpotent.

The resolution of Frobenius’s conjecture was not achieved by merely extending existing methods. Instead, Thompson introduced several highly original ideas that led to significant developments in group theory and profoundly influenced the field.

It is no coincidence that, beginning with Thompson’s thesis, group theory rose to prominence as one of the most active and rapidly developing areas of mathematics. During this period, major advances were made on one of the central problems of finite group theory: the classification of finite simple groups.

Every finite group can be regarded as being constructed from a finite collection of finite simple groups. These simple groups serve as the fundamental building blocks of all finite groups. Consequently, the classification of finite groups reduces to two main tasks: first, the classification of finite simple groups themselves, and second, the resolution of the extension problem, which concerns how these building blocks can be combined to form more complex groups.

In 1963, Thompson, in collaboration with Walter Feit, proved that every nonabelian finite simple group has even order. Their result was published in the monumental paper “Solvability of Groups of Odd Order,” a 250-page work that appeared in Pacific Journal of Mathematics, Volume 13 (1963), pages 775–1029. Owing to its extraordinary length, several journals initially declined to publish the paper. Ultimately, it occupied an entire part of Volume 13 of the journal.

This remarkable achievement astonished the mathematical community and strengthened the belief that a complete classification of finite simple groups might be attainable. In recognition of this groundbreaking work, Thompson and Feit were awarded the Frank Nelson Cole Prize in 1965, when the thirteenth award was conferred upon them for their joint paper.

Another important early contribution by Thompson to the classification of finite simple groups was his classification of those finite simple groups in which every soluble subgroup has a soluble normalizer.

In recognition of his outstanding work, Thompson was awarded the Fields Medal at the International Congress of Mathematicians in Nice in 1970. Speaking at the Congress, Brauer referred first to Thompson’s celebrated “odd order paper,” highlighting its profound impact on the development of group theory. He mentioned that:-

"The first paper I have to mention is a joint paper by Walter Feit and John Thompson and, of course, Feit's part in it should not be overlooked. Here, the authors proved a famous conjecture, to the effect that all non-cyclic finite simple groups have even order. I am not sure who was the first to observe this. Fifty years ago [1920] this was already referred to as a very old conjecture. While it was usually mentioned in courses on algebra, it is only fair to say that nobody ever did anything about it, simply because nobody had any idea how to get started. It was not even clear that the whole problem made sense. Was the role of the prime 2 simply a little accident; did 2 play an entirely exceptional role, or were there properties of other prime divisors of the group order which bore at least some resemblance to those of 2? It was only after the Feit-Thompson paper that one could be sure that the whole question was a reasonable one."

Thompson received numerous honors in recognition of his exceptional contributions to mathematics. In addition to the Cole Prize from the American Mathematical Society and the Fields Medal in 1970, he was awarded the Senior Berwick Prize by the London Mathematical Society in 1982 and the Sylvester Medal from the Royal Society in 1985. In 1992, he received both the Wolf Prize and the Poincaré Prize.
He was elected to the National Academy of Sciences of the United States in 1971 and to the Royal Society of London in 1979. In 2000, he was honored with the National Medal of Science.

In 2008 the Norwegian Academy of Science and Letters awarded the Abel Prize to John Griggs Thompson and Jacques T**s:-

"... for their profound achievements in algebra and in particular for shaping modern group theory."

(Source: MacTutor)