Brief life of a mathematician

On the occasion of the 113th birth anniversary of Srinivasa Ramanujan, KRISHNASWAMI ALLADI pays tribute by describing the life and contributions of Evariste Galois, the mathematical genius who founded Group Theory, and who, like Ramanujan, died very young.

G. H. HARDY of Cambridge University, Ramanujan's mentor, said that the real tragedy of Ramanujan's life was not his early death, but that in his formative years, the genius spent much time proving results which were rediscoveries of past work due to lack of proper guidance in India. Hardy argued that in mathematics especially, the most brilliant work is done when one is very young, so had Ramanujan lived longer, he would have proved more theorems, but might not have produced work of higher quality. Hardy cited as an example Evariste Galois, founder of Group Theory, who met an untimely end in a duel at the age of twenty. Hardy pointed out that Galois, like mathematicians Abel who died at 22, and Riemann at 40, had done his best work by then. Even the great Gauss, the prince of mathematicians, who lived a full life, made most of his discoveries in his teens, and spent the rest of his life polishing up his results for suitable presentation.

In this article I shall describe the all too brief life of Galois and the path breaking discoveries he made. I shall also make a comparison with the life and contributions of Ramanujan. Finally I shall describe the impact that Group Theory has made in various fields and the present state of its research. In doing so, I will talk briefly about the contributions of my distinguished colleague Professor John G. Thompson of the University of Florida, arguably the greatest group theorist since Galois.

For biographical details pertaining to the life of Galois, I have relied on MacTutor History and for the development of Group Theory, I have consulted an article by O'Connor and Robertson. With regard to the work of Ramanujan, I profited from the article entitled "Ramanujan's Association with Radicals in India" by Berndt, Chan, and Zhang, that appeared in 1997 in the American Mathematical Monthly.

Life of Galois

Evariste Galois was born on October 25, 1811 in Bourg La Reine near Paris, France. His parents were well educated but there is no indication of mathematical talent in his family. Galois's father was well known in the community and was elected mayor of his township in 1815. This period in France was quite tumultous and saw rapid changes in leadership. Napoleon who was at the height of his power in 1811 was thrown out in 1815 after the defeat at Waterloo. The political instability that followed was to have a devasting effect on the young Galois.

Galois's performance in elementary school at the Lycee in Louis- le-Grand was very good and he won several prizes. However, in 1826 he was asked to repeat a year because his work in rhetoric was not up to the standard. Age 16 was a turning point in his career because it was then that he took his first mathematics course under M. Vernier. Like Ramanujan, he became engrossed in mathematics to such an extent that the Director of Studies wrote, "It is the passion for mathematics that dominates him. I think it would be best if his parents would allow him to study nothing but this. He is wasting his time here and does nothing but torment his teachers and overwhelm himself with punishment."

The Ecole Polytechnique was the premier university in Paris, but Galois failed the entrance exam in 1828. So he was back at the Louis-le-Grand and enrolled in a mathematics course by Loius Richard. It was at this time that he read Legendre's classic treatise on Geometry. As Richard wrote "This student works only in the highest realms of mathematics." In April 1829 Galois published his first paper on continued fractions (a favourite topic of Ramanujan's) in the Annales de Mathematiques. Shortly thereafter he started submitting articles on algebraic solutions to equations, a topic for which he would soon contribute revolutionary and far reaching ideas. Unfortunately, as a result of a politically based conspiracy, Galois' father committed suicide on July 2, 1829. This tragedy that struck the family had a telling effect on the young Galois. For one thing, Galois failed the entrance exam to the Ecole Polytechnique the second time he took it shortly after his father's death. This forced him to enter the Ecole Normale which was an annex to the Louis-le- Grand. His total immersion in mathematics like Ramanujan's, cost him in his performance in other subjects. His literature examiner wrote "This is the only student who answered me poorly, he knows absolutely nothing. I was told that this student has an extraordinary capacity for mathematics. This astonishes me greatly, because after the examination, I believed him to have but little intelligence." Whatever be the opinion of the literature examiner, Galois at that time wrote a beautiful mathematical paper entitled "On the condition that an equation be soluble by radicals" that was being considered for the Grand Prize by the Academy. Unfortunately, the paper was in the possession of Fourier who died in April; the paper was subsequently never found and so was not considered for the prize after all.

Sophie Germain (known now for a major contribution in connection with Fermat's Last Theorem) wrote a letter to a mathematical friend describing Galois' situation: "... the death of M. Fourier has been too much for this student Galois .... He has been expelled from the Ecole Normale. He is without money... They say that he will go completely mad. I fear that this is true."

To make matters worse, in the midst of this mental depression following Fourier's death and that of his father, Galois got involved in political controversies that were raging in France. He was imprisoned for open demonstrations, and even attempted to commit suicide in prison. In March 1832, a cholera epidemic swept through Paris, and all prisoners including Galois were transferred to the Pension Sieur Faultrier. There he fell in love with Stephanie-Felice du Montel, daughter of the resident physician. Although he was released shortly thereafter, his freedom was short lived. Once again, for political reasons he was imprisoned, but this time he was to fight a duel to get out. Galois was aware that he was fighting a superior adversary, and that most likely he would be killed in the duel. So on the night before the duel, he wrote a letter to a friend outlining the wonderful new ideas he had in connection with the solvability of algebraic equations. Galois died in the duel on May 31, 1832, at the tender age of 20. The reasons for the duel are not entirely clear, but Stephanie's name appears as a marginal note in the manuscript that Galois wrote the night before he was killed. Fortunately, his letter was preserved. The revolutionary mathematical ideas in this letter led to the birth of Group Theory, a central branch of mathematics with important applications in several other fields as well.

The birth and growth of Group Theory

The subject of Group Theory deals with symmetries in general such as those that arise in geometry and in solutions to polynomial equations. Although certain key properties associated with groups can be traced to earlier mathematicians, it was only with the work of Galois that the concept of a group crystallised and concrete applications of the concept emerged. Three different streams that gave rise to group theory were

(i) geometry at the beginning of the 19th century, (ii) number theory at the end of the 18th century, and (iii) the theory of algebraic equations at the end of the 18th century leading to the study of permutations. Since the study of geometry goes back to antiquity, it is natural to ask what was the reason for the emergence of the group concept via geometry. During the 19th century, a mathematical revolution was taking place with the emergence of non-Euclidean geometry and synthetic geometry. Suddenly, instead of just angles and lengths dominating the discussions, invariances under transformations were the key to geometrical study, and indeed, this eventually led to the study of transformation groups. Euler, the most prolific mathematician in history, systematically studied remainder arithmetic in number theory during the mid 18th century.This was subsequently called modular arithmetic by Gauss who took it several steps further. In the work of Euler and Gauss in number theory, group theoretical properties (as we know today) associated with remainders were crucial, but neither of them formulated the group concept in generality.

We all learn the quadratic formula in school, namely the formula which gives the solutions to the general quadratic equation. We are told that there are similar but more complicated formulae for roots of the general cubic and quartic equations, but we are not given these formulae. The French mathematician Lagrange wanted to find out why the cubic and quartic equations could be solved algebraically. In this connection, permutations were first studied by Lagrange in a classic paper of 1770 on the theory of algebraic equations. Although the beginnings of permutation groups can be seen in this work, Lagrange does not discuss the general group concept at all. The first person to claim that there is no general quintic formula, namely, the non-existence of a formula to solve all quintics, was Ruffini in 1799. Ruffini's work on quintics was based on Lagrange's permutation approach, but had gaps in his reasoning. It was Abel in 1824 who gave the first complete proof of the insolvability of general quintics. It is here that Galois enters the picture.

Galois in 1831 was the first to really understand that the algebraic solvability of a polynomial equation was intimately related to the group structure of certain permutations associated with the equation. In his now famous letter of 1832 written on the eve of his death, Galois had demonstrated by the study of groups, that there is no general formula that will give the roots of all polynomials of degree n, when n is at least five. Galois' work was not known until Liouville published it posthumously in 1846. But even then, Liouville failed to grasp the group concept that was the key to Galois' work. The understanding of the general group concept, and the realisation that it was the basis of Galois' work, came only in the second half of the 19th century. Thus like Ramanujan, Galois was much ahead of his time, and a full grasp of his ideas came only decades later.

By 1872, Group Theory was becoming the centre stage of mathematics because Felix Klein of Gottingen in his famous Erlangen Programme called for the group theoretic classification of geometry. Group Theory really came of age with the publication of the book Theory of Groups of Finite Order by Burnside in 1897. Also the two volume book Lehrbuch der Algebra by Weber in 1895 and 1896 became a standard text. These books influenced the next generation of mathematicians to make group theory perhaps the most major single branch in 20th century mathematics.

With advanced and abstract mathematics playing a prominent role in the sciences during the 20th century, group theory became a crucial tool outside of mathematics as well - in quantum mechanics in physics, and crystal structure in chemistry, for instance.

Group Theory today

In the modern era, the most prominent figure in Group Theory is John Griggs Thompson. Born in Ottawa, Kansas, in 1932, Thompson entered Yale University in the early 1950s to earn his B.A. in Theology with the desire to become a Presbyterian minister. His interest in mathematics was sparked when his roommate drew his attention to George Gamow's book, One, Two, Three, Infinity. From then on, the rest is history. The call of mathematics was too strong to resist. Thompson changed his major and received a Bachelors in Mathematics at Yale in 1955 and moved to the University of Chicago for his Ph.D. His Ph.D. thesis of 1959 was a masterpiece. It was not just an extension of known techniques, but full of new and powerful ideas that soon led to major developments in group theory. The most sensational of these was the resolution of a long standing conjecture that all finite groups with an odd number of elements are solvable. Thompson proved this in collaboration with Walter Feit, and their 253 page proof in 1963 occupied one entire issue of the Pacific Journal of Mathematics!

For this magnum opus, Feit and Thompson received the Cole Prize of the American Mathematical Society in 1966. Thompson continued establishing further fundamental results, and in 1970 was awarded the Fields Medal, the highest prize in mathematics equivalent in prestige to the Nobel Prize, at the International Congress of Mathematicians in Nice. That year he was also appointed Rouse Ball Professor of Mathematics at Cambridge University, a position that he held until 1993, when he moved over to The University of Florida as Graduate Research Professor.

Thompson's name is also closely associated with one of the monumental achievements of the 20th century, namely, the classification of finite simple groups. In any field of study, one tries to understand complex objects in terms of those simpler in structure. For example, in number theory, we try to understand properties of integers by decomposition into prime factors. In group theory, finite simple groups are basic building blocks. Starting at the time of Thompson's thesis, group theory leapt into prominence as the mathematical topic undergoing the most rapid development.

The main reason for this was that it became clear that the classification of all finite simple groups was now realisable, and not just a dream. The classification was completed only in the early 1980s as a collective effort of many noted mathematicians, and Thompson's ideas were crucial in this effort.

In the past few years, Thompson has been working on, and made major contributions to, the famous Inverse Galois problem which has remained unsolved. Certain special types of groups that Galois investigated in connection with the algebraic solvability of polynomial equations are called Galois groups today. The Inverse Galois Problem states that given an arbitrary finite group, one can produce a setting in which the given group is the Galois group of a certain polynomial.

Since Thompson's productivity over the years at the highest level has remained unabated, he has received numerous awards and recognitions in a steady stream. He was elected to the U.S. National Academy of Sciences in 1971 and made Fellow of the Royal Society in 1979. He was awarded the Sylvester Medal of the Royal Society in 1987 and the Wolf Prize of Israel in 1992.

That year he received the Poincare golden medal by the Academie des Sciences, Paris. This medal is awarded only on exceptional occasions, the two previous recipients being Jaques Hadamard (1962) and Pierre Deligne (1974). And on December 1, 2000, Thompson was awarded the National Medal of Science by President Clinton for his lifelong contributions to mathematics. I had the honour of representing the University of Florida at the Medals Ceremony, and the pleasure of seeing Thompson receiving the medal. We at the University of Florida feel priviledged to have Thompson as a colleague and to know personally one of the greatest mathematicians of the 20th century.

Ramanujan and radicals

Ramanujan's interest in algebraic solutions to polynomial equations can be seen by his work on radicals. A radical is an expression involving combinations of various n-th roots of integers. When one solves a polynomial equation algebraically, such as with the quadratic formula, one expresses the solution in terms of radicals. Out of the 58 problems that Ramanujan submitted to the Journal of the Indian Mathematical Society, ten of them involve equalities between radicals.

During Ramanujan's time, especially among British mathematicians, establishing identities involving radicals was quite common. Ramanujan investigated radicals in connection with the study of class invariants. German mathematician Weber had studied class invariants extensively, but Ramanujan found an astonishing number of new ones. It was only after his arrival in Cambridge that Ramanujan knew of Weber's work. Ramanujan used class invariants to find excellent approximations to pi, as well as determine explicitly values of theta functions at certain points. It is still a puzzle as to what methods Ramanujan used to compute these class invariants. He has left no clues in his notebooks. Since Weber's methods were highly algebraic, it is unlikely that Ramanujan pursued such techniques. Thus, as Berndt, Chan, and Zhang say in their paper on Ramanujan and radicals, "Ramanujan's ideas still remain hidden behind an opaque curtain."

In summary there are many similarities in the life stories of Galois and Ramanujan. Both faced numerous obstacles. Undaunted, both continued to pursue mathematics with a passion and made outstanding discoveries marked with supreme originality. The tragedy is that both died very young, and we can only contemplate what more they might have accomplished had they lived longer. Finally, what a remarkable coincidence, that both communicated their most important findings in letters just before their death. Galois's letter gave birth to Group Theory, and Ramanujan's last letter to Hardy created the subject of mock theta functions. We should be thankful that these genuises have left behind ideas for succeeding generations to ponder on and develop and that their legacy remains strong even in this new millenium.

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