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.

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)

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)

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)