George Polya was born in Budapest on December 13, 1887. His Jewish father Jacob worked as a lawyer and also had a position at university. Polya studied philology at the beginning of two years in university, and then tried other subjects but in the end he compromised to study mathematics. At that time there was the leading mathematician, Lipot Fejer as a full professor at the age of thirty-one at the university. Polya described him as a role model and praised his great bohemian characteristics.
In 1911 Polya spent a year at the University of Vienna where the geometer Wilhelm Wirtinger worked, and then he returned to Budapest and completed his Ph.D. Polya arranged to spend two years at the Georgia Augusta and there he was able to listen to Hilbert’s lectures on the theory of partial differential equations, on the mathematical foundations of physics and on the foundation of mathematics. He also heard Toeplitz lecture on invariant theory, Cartheodory on calculus of variations, Hecke on algebraic number fields and Weyl on integral equations and their applications in mathematical physics. He listened to Landau lecture on infinite series, especially Fourier series. Polya got his special study field during listening to Landau lecture on mathematical problems. In 1914 Polya moved to Paris for a short period. During that time he was able to attend Hadamard’s seminar. When Polya received an offer of the position of privatdozent at the Eidgenossische Technische Hochschule in Zurich he accepted it. Polya could get to know Adolf Hurwitz, the mathematician whom Hilbert regarded as a true teacher and mentor. Polya had kept touch with him and played a large role in editing Hurwitz works after his death. During the time in Zurich, Polya had met his compatriot Gabor Szego while on a visit to Budapest. Polya and Szeco shared interests in posing and solving problems, which lead to publish their classic two-volume books ‘Problems and Theorems in Analysis’ in 1925. In 1924, Hardy had invited Polya to spend a year in Oxford and Cambridge. Hardy and Polya wrote several joint papers but most notably the book ‘Inequalities’ collaborated with Littlewood, which appeared in 1934. Polya got many invitations to do lecture in different countries and the most significant of these invitations was from the United States, three months at Princeton followed by three at Stanford. Also he accepted offer of two-year appointment at Brown, later Stanford offered him an associate professorship. He became an American citizen in 1947. Polya had always been interested in the techniques of problem solving, especially in heuristical methods, he wrote a book ‘How to solve it’, which was a huge success when it was published in 1945. After retirement from Stanford in 1953 he wrote two more books of general interest: Mathematics and Plausible Reasoning(1954) and the two-volume Mathematical Discovery(1962, 1964). He was elected a corresponding member of Paris Academy in 1947, the National Academy of Sciences of Washington in 1976. He died on September 7, 1985 aged ninety-seven.
0 Comments
Amalie Emmy Noether was born on March, 1882 in Erlangen, Germany. Her father was professor of mathematics at the University and her mother, Ida Amalia, came from a wealthy Jewish family of Cologne. Noether attended the Municipal School for the Higher Education of Daughters until she was eighteen. Women had been allowed to enroll at universities in France since 1861, England since 1885 but Germany as late as 1900.
Emmy Noether was determined to study and succeeded in attending courses in mathematics and other science subjects at Erlangen and Gottingen. In 1908 she completed her doctoral dissertation ‘On complete systems of invariants for ternary biquadratic forms’. For the next seven years she mainly did her research. At that time the new relativity theory caused sensational excitement and Emmy was one of the first to understand the implication. Despite her gender she became a privatdozent at the age of thirty-seven and three years later she received the honorary title of unofficial associate professor. Many outstanding mathematicians often make great contributions in their early age but Emmy Noether began to produce her powerful work in her forties. She developed a very abstract and generalised approach to the axiomatic development of algebra. Her revolutionary paper in 1921 on ideal theory where the concept of Noetherian ring originated is believed to be her finest work. Although Dedekind may have the basic ideas it was Noether who flourished the concept into full richness of the theory. By this time Noether lectured at the international Congress in Bologna in 1928, gave course at the University of Moscow and the communist Academy. At last she had been appointed associate professor at Georgia Augusta. A year later the Nazis seized power and one of their first acts was deprive Jewish officials including university teachers of their positions. Most members of the mathematical faculty at the Georgia Augusta were Jewish so they were forbidden to teach. For a time Emmy Noether continued to meet privately with students and colleagues. Before the end of 1933 she arrived in America with a temporary position at Bryn Mawr College. She stated with four students under her wing and taught them with mixture of German and English. She gave a course of weekly lectures on Algebra at Princeton. She was appreciated at Bryn Mawr and Privceton as she had never been appreciated in her own country. She was remembered as the first woman mathematician and died on April 14, 1935. The village of Breselenz lies near the city of Dannenberg, George Friedrich Bernhard Riemann was born on September 17, 1826. At the age of 5 He was strongly interested in history and at fourteen he entered the senior class of the gymnasium at Hanover, but two years later he transferred to the one at Luneburg close to home where he studied until he was nineteen. The director of the school allowed him to use his personal library where Legendre’s treaties on the theory were included. The young Riemann devoured this knowledge and in later years its influence was great when he contributed to number theory.
In 1846, Riemann matriculated at the Georgia Augusta in the faculty of theology but he changed to the faculty of philosophy. After one year at the Georgia Augusta, Riemann moved to Berlin where Dirichlet, Eisenstein, Jacobi and Steiner were professors. Among them it was Dirichlet whose lectures influenced him greatly. Two years later he returned to Gottingen where he attended the course of electrodynamics given by Weber, and he worked in Weber’s laboratory as an assistant for over an year and half. At the end of November 1851, Riemann presented his doctoral thesis ‘Functions for a general theory of functions of one complex variable’. The results in his thesis arise general picrures. As trigonometric functions may be thought of as a real valued functions defined on a circle, so called circular functions, so elliptical functions as complex-valued functions defined on a torus. After two years of intense work in various fields of mathematics Riemann presented his Habilitationsschrift in December 1853, its title as ‘On the representability of a function by a trigonometric series’, but only published after his death. It is here that he introduced the type of integral we know as the Riemann integral. Riemann finally obtained the right to teach publicly. In his first course on partial differential equations only 8 students including Dedekind attended. When Gauss died in 1855, his successor Dirichlet helped Riemann by finding him a minor position in the university and a small rent-free apartment in the observatory. In the same year, his father died and soon after his sister Clara. There were further misfortunes on the way. His brother and one of his three sisters died. The death of Dirichlet eighteen month later was a further blow. In 1859 Riemann was appointed to succeed Dirichlet as full professor at the Georgia Augusta. He was elected a corresponding member of the Berilin Academy and further academic honours were conferred on him by the Paris Academy and the Royal Society of London. As a full professor Riemann got married in 1862, and it did brighten his remaining years of his life. Shortly later he developed pleurisy and tuberculosis had taken hold. The final years of his life were spent in Italy for health reason. He died on July 20, 1866 less than forty years old. In just fifteen years of activity he made an enormous contribution to almost all areas of mathematics. He studied on the theory of integral, complex variable function, geometry, the theory of electricity and much else. His exceptional fruitful ideas stimulated further development in mathematics as well as in mechanics, physics and the natural sciences. Arthur Cayley was born on August 16, 1821 in England. His father Henry was one of the merchants working in St Petersburg and his mother, Marie Antonia was believed to have Russian ancestors. After his father’s retirement, the family settled in Blackheath, England. Arthur entered the senior department of King’s college at the age of 14. Later Cayley went up to Trinity College, Cambridge in October 1838, and won a college scholarship. In the highly competitive examination Cayley came the first in his year and became Senior Wrangler of 1842. Not only mathematics but Cayley also was interested in reading novels, and he appreciated architecture and painting. He also loved travelling and enjoyed scrambling in the Alps.
Cayley’s academic successes led to his election to a Trinity minor fellowship which lasted 7 years. During this period, Cayley published wide range of mathematical subjects. After his fellowship he was admitted to Lincoln’s Inn in April 1846 and trained as a barrister. He was called to the bar in 1849 and had a peaceful office in Lincoln’s Inn where he pursued his mathematical work when not engaged in conveyancing. Cayley devoted in invariant theory studying the effect of linear transformation on algebraic expressions such as binary forms. For a number of years Cayley continued his research with his job. In 1863, he was elected to a new professorship at Cambridge. In September 1863 Cayley married Susan of Greenwich, who gave birth two children, Mary and Henry. Although Cayley researched many works and the list of his publications extends to over 800 items, he wrote only one book himself, A Treatise on Elliptic Functions, published in 1876. In his works there are many brilliant ideas which we know with exciting possibilities. He introduced the notion of abstract group as opposed to permutation group, also he wrote about determinants and matrices which went to Frobenius to deal with them. Cayley took a particular interest in the education of women. He influenced the council of the institution(Later Newnham College) as a chairmen to allow women to become a member of Cambridge University. When Charlotte Angas Scott entered Girton in 1876, the college had been open for 7 years. Her success was a break-through for the higher education of women in England. She sttended Cayley’s lectures and wrote her thesis under his supervision. Because of the rule at Cambridge, she achieved doctorate at the University of London in 1885. Cayley was elected an honorary fellow of Trinity in 1872 and served on the Council of Senate. He presided over the London Mathematical Society once and the British Association for the Advancement of Science at another. Honorary degrees arrived from universities and academic societies, notably first a Royal medal and then the Copley medal of the Royal Society. He died on January 26, 1895 at the age of 73. The greatest French mathematician, Augustin-Louise Cauchy was born in Paris on August 21, 1789. His father did legal work for the Paris police and his mother came from a well-to-do Partisian family. In 1794 the family fled to their country house at Arcueil to escape the terror of the French Revolution. Arcueil was the place where Berthollet and Laplace had their estate so Cauchy had the benefit to meet them and also many famous scientists who came to visit them. Lagrange was one of them who were impressed by his ability. Lagrange advised Cauchy to enrol at the liberal Ecole Centrale du Pantheon to study the humanities. At the age of 16, Cauchy entered the Ecole Polytechnique to become a civil engineer. By 1810 Cauchy became a qualified junior engineer, and he left Paris to work on the construction of a naval base at Cherbourg, the Port Napoleon under Count Mole. Meanwhile he remained at Cherbourg for almost 3 years and gained experience as an engineer, he researched mathematics and made some discoveries which attracted the attention of learned society in Paris.
In 1812 Cauchy had completed a study on symmetric functions including the germ of the fundamental ideas that eventually blossomed into group theory. He submitted his study on the calculation of definite integrals in 1814, which made him famous. In 1815 Cauchy published his memoir which shows that he made good use of Gauss’s method then generalise results in number theory. He also developed the theory of determinants. After several previous attempts, he was elected a member of Paris Academy in 1816 and soon Cauchy was appointed associate professor of analysis at the Ecole Polytechnique, before long he was promoted to full professor in analysis and mechanics. His discoveries placed him in the front rank of mathematicians of the period. Often in his lectures he introduced new ideas and more rigorous methods. Cauchy devised definitions of convergence of series and continuity of functions based on the concept of limit, and he gave a definition of the derivative of a function using the same idea. He had a sure instinct for what was true. The most important discoveries conducted by Cauchy in the fields of both pure and applied mathematics are without doubt his fundamental theorems in complex analysis. In addition to his professional work, Cauchy devoted himself to an organisation of ‘young Catholics of good families’ aiming at detesting faithlessness, irreligion and secularism. He dislikes liberalism in all its forms. In deep political turmoil Cauchy left for Switzerland on what began as sick leave but ended up an eight-year self-imposed exile. He lost his position in Paris but was appointed professor of mathematical physics in Turin. In 1833 he moved to Prague where he taught the juvenile Duke of Bordeaux, later returned to Paris. Cauchy was far from popular due to his self-absorbed and self-righteous mind. Had it not been for his intolerant political opinon he might have achieved more. His health, never robust, began to deteriorate and after accepting doctor’s advice, Cauchy left Paris for Sceaux on May 12, 1857 but died 11days later. In his life time Cauchy published a huge amount of mathematical work but vast of his collection was destroyed just before the Second World War. 50 years after the death of towering genius, Leibniz in Germany, a child was born in Brunswick who was known as ‘the prince of mathematics’. Johann Carl Friedrich Gauss was born on April 30, 1777. Neither of his parents had much education, but Gauss had tended to trace his genius to his mother rather than his father. Gauss had shown an extraordinary ability in counting and arithmetic form in an early age. At 7 when he started attending school, teachers recognised his abilities and persuaded his parents to allow him to proceed to the local gymnasium where he learnt Latin along with the official High German. Gauss was strongly attached to philological studies and even more strongly to mathematics. He continued his study at the Brunswick Collegium Carolinum, and when he was 15, he became one of the best at this progressive science-oriented institution. One of the works he studied was Newton’s Principia. By the age of 18 Gauss left for the University of Gottingen in Hanover to study in more science-oriented institution. After leaving Gottingen, he submitted his doctoral dissertation to the University of Helmstedt, which was accepted ‘in absentia’.
Gauss’s first major publication, Arithmetical investigations, had impressed many mathematicians who could understand it. This was the first attempt to organise the number theory. He was also interested in astronomy and calculated the orbit of the asteroid Ceres from extremely limited observational data. In his personal life, Gauss married Johanna Osthoff, the first of his two wives on October 9, 1805 and had a son Joseph and a daughter Minna but Johanna died after giving to another son who didn’t survive more than a few months. Through his second marriage to Friderica Wilhelmine Waldeck, he fathered three more children, Eugene, Wilhelm and Therese. The oldest son Joseph acted as his father’s assistant for a while, then became a railway engineer later. The other two sons emigrated to North America after extended conflicts with their father. Gauss was the recipient of many academic and other honours, and later he was addressed as Geheimrat Gauss. He was elected a foreign member of the Royal Society of London in 1804 and awarded the Copley Medal in 1838. In 1854 the Georgia Augusta celebrated his scientific achievements. Gauss carried on enormous scientific subjects. In mathematics the number theory came first and then extended to algebra, analysis, geometry, mechanics, celestial mechanics, probability, error theory and actuarial science. In pure and applied science his interests included observational astronomy, surveying, geodesy, capillarity, geomagnetism, electromagnetism, optics, and the design of scientific equipment. While Newton is regarded as a physicist first and only secondarily as a mathematician, in the case of Gauss it is the other way round. Gauss wrote over three hundred papers and many more unpublished notebooks which have been worked over by many generations. On February 23, 1855 the greatest German mathematician died in his sleep. Jean-Baptiste-Joseph Fourier was born on March 21, 1768 in Auxerre, France. His father Joseph, who was a master tailor originally from Lorraine, and his mother Edmie died before he was ten years old. Fourier started his education from Ecole Royale Militaire where he displayed special gift for Mathematics. He went on his study at the College Montagu and aimed to join either the artillery or the engineers but embarked his career in the church. He became a novice at the famous Benedictine Abbey of St. Benoit-sur-Loire where he taught elementary mathematics to the other novices. Later he returned to Auxerre to teach at the Ecole Militaire when he was 21 and had read a research paper of the Paris Academy.
During the first years of the Revolution, Fourier was arrested by order of the Committee for Public Safety in 1794 for his courageous defence of victims including Robespierre. After having been released he started his study at the Ecole Normale where he met some of the poremost mathematicians such as Lagrange, Laplace and Monge. The next year he was appointed assistant lecturer in the Ecole Polytechnique to support the teaching of Lagrange and Monge. In 1798 Fourier was selected to join Napoleon’s expedition to Egypt as a permanent secretary. This expedition set to work on studying antiques. During that time Fourier thought about mathematics and proposed to report on a historical preface describing the rediscovery of the wonders of the ancient civilisation. Napoleon had been impressed by Fourier’s capacity for administration and decided to appoint him prefect of the Department of Isere. During this time Fourier wrote his classic monograph on heat diffusion entitled ‘On the propagation of heat in solid bodies’ and presented to the Paris Academy in 1807. At first many mathematician rejected several of its features such as the central concept of trigonometric of Fourier series and so its publication was blocked, but it was not until 1822 that Fourier’s theory of heat diffusion was published. Fourier had excellent administrative achievements; securing the agreement of 37 different communities to the drainage of huge area to make valuable agricultural land, and the planning of a spectacular highway of French section between Grenoble and Turin. Napoleon recognised his excellent work and conferred on him the title of Baron. After experiencing unstable political situations, he resigned and returned to Paris and concentrated on scientific work. In 1817, Fourier achieved an effective insight into the relation between integral-transform solutions to differential equations and the operational calculus. His integral-transform solutions of several equations paved the way for Cauchy to develop a systematic theory to the calculus of residues. In 1822 he was elected to the powerful position of permanent secretary of Academy de Sciences. In 1827, like Laplace before him, he was elected to the literary Academy and also to the Royal Society of London. Early in May 1830 he suffered a collapse and his condition deteriorated until he died on May 16, at the age of 62. He used to say that profound study of nature is the most fertile source of mathematical discoveries. Laplace was born on March 23, 1749 in Beaumont-en-Auge, small town in lower Normandy of France. His father was in the cider business and also an official of the local parish. Laplace’s father wanted him to make a career in the Church, and in 1766 Laplace entered the University of Caen for theological training. However, his mathematical interests were apparent and encouraged to go Paris to meet d’Alembert. The story said that how d’Alembert gave Laplace difficult mathematical problems as a test of his ability and Laplace was able to solve them overnight. After that Laplace got a teaching position at the Ecole Militaire in Paris and lasted for the next seven years.
Although Laplace’s first contributions to mathematics were to solve difference equations using integral calculus, his main interest was to make the Newtonian World picture perfect. Laplace investigated the problems of celestial mechanics including the orbital eccentricities and the acceleration of the moon around the earth and resolved many of them. Laplace had extensive knowledge of other sciences and developing a reputation for arrogance, which made him unpopular with his fellow academicians. In May 1790 the Revolutionary Government charged the Academy of making recommendations for units of length, area, volume and mass with decimal subdivisions and multiples. It was the Laplace’s suggestion that the basic unit of length was named the metre. In 1796, Laplace published his first major work, the Explanation of the solar system, which is a scientific classic. It is believed that Laplace foresaw the concept of the black hole which were deduced much later in Einstein’s general theory of relativity. Although the main field of Laplace’s study was celestial mechanics, he also made significant contributions to the theory of probability and statistical inference. Laplace believed that mathematics could be brought to study on the social phenomenon through probability and suggested various applications. Laplace considered himself the best mathematician in France and as he grew older his arrogance increased. After his death an anonymous critic compared Laplace less than favourably to Euler and Lagrange. Laplace was notorious for his rapidity of his teaching and for his frequent use of the phrase ‘It is easy to see’, by which he skipped steps in his proof. In his own work he frequently neglected to acknowledge the source of his results and left ambiguous position of whether they were his own or not. Even so, Laplace became a senator and held the office of Chancellor. He was awarded France’s highest honours, the Grand Cross of the Legion of Honour and the Order of the Reunion. Laplace’s political opportunism allowed him to engage in his scientific work, but his beliefs and theories began to be superseded as new discoveries undermined them. During his last years Laplace lived mainly on his estate in Arcueil. All his life he was generally healthy and vigorous. After a short illness he died on March 5, 1827 at Arcueil. Lagrange was born in Turin on January 25, 1736. As a boy he intended to be a lawyer but gradually decided that he would prefer to study sciences. He accidentally read an article written by the British astronomer and mathematician Edmund Halley arguing the superiority of calculus over Greek mathematics. This accelerated his study and acquired a great success that by the time he was 19 he had been appointed professor of mathematics at the Royal School of Artillery in Turin.
In 1755 Lagrange applied the calculus of variations to mechanics, and it offered a general procedure for solving dynamical problems. He communicated with these results with Euler who was greatly impressed. Throughout 18th century in Europe, the scientific academies encouraged research into celestial mechanics and there were prizes for the answer to specific questions. In 1764 Lagrange entered a competition to determine the gravitational forces that caused the moon to present a relatively unchanging face to the earth. He was the winner and received the Grand prize. Two years later he won again for a partial solution to a more complicated gravitational problem involving the planet Jupiter. In 1766 Lagrange became director of mathematical physics at the Berlin Academy. Lagrange was not required to lecture, instead he was composing memoires nearly every month ranging from probability to the theory of equations. In number theory Lagrange solved some of the questions made by Fermat including the famous theorem that every positive integer is the sum of the squares of four integers. In 1772, his third Gran prize from the Paris Academy and in 1774 and in 1778 again he won the Grand prizes for the study on the sun, moon, earth and the perturbations of comets. In 1788 Lagrange published his masterpiece, the Mechanique Analitique (Analytical Mechanics). Lagrange was known for his gentle demeanour and his diplomatic skills. At the Berlin Academy he remained in favour with the king unlike Euler. When he first settled in Paris he was doted by Queen Marie-Antoinette, yet later he managed to have good terms with Bonaparte. He was appointed Senator, a count of the Empire, and Grand officer of the Legion of Honour. Although he never met Euler but it was Euler who influenced Lagrange most. Any study of his work must be preceded by or accompanied by the work of Euler. He died on April 11, 1813 at the age of 77. Leonhard Euler was born in Basel on April 15, 1707. His father Paul Euler was a minister of the Protestant Evangelical Reformed Church and his mother, Margarete Brucker, was also the daughter of a minister. In 1720, at the age of thirteen, Euler matriculated into the faculty of philosophy at the university and mastered all the available subjects and graduated in 1722. The next year Euler entered to the faculty of theology but he began to study mathematics seriously. At that time he was introduced to a famous professor Johann Bernoulli who gave him valuable advice to start reading more difficult mathematical books on his own and to study them fervently.
Euler received his master’s degree in 1724 at the age of seventeen. He wrote a thesis on comparing the natural philosophy of Descartes with that of Newton. Also he entered a prize competition sponsored by the Paris Academy. Later Euler became a premier mathematician in Imperial Russian Academy, and married Catharina Gsell, the daughter of a Swiss artist then was working in Russia. They lived in a house near the river Neva. His 14 years stay in St Petersburg, he was mainly occupied with mathematical research. Euler’s best known work of his formulation of the seven bridges of Konigsberg was made during this period. This marks the beginning of the branch of mathematical known as a graph theory. He entered for the annual prize offered by the Paris Academy and was the winner 12 times. In 1741, Euler accepted an invitation from the King of Prussia, Frederick the Great, to move to the Prussian Royal Academy of Sciences in Potsdam (later it is called the Berlin Academy). But the different perspectives on science between the King and Euler made the relationship impasse, eventually Euler left Berlin and returned to St Petersburg. Euler’s energies seemed inexhaustible. In pure mathematics his major fields were calculus, differential equations, analytic and differential geometry of curves and surfaces, number theory, infinite series, and the calculus of variations. In applied mathematics he created analytical mechanics, algebra, mathematical analysis, analytical geometry, differential geometry, and the calculus of variations. In mathematical physics he discovered the fundamental differential equations for the motion of an ideal fluid. He was the one of the few scientists of the 18th century to think the wave as opposed to the particle theory of light. He studied the propagation of sound and achieved many results on the refraction and dispersion of light. Euler wrote almost 900 papers, memoirs, books and other works. His influence on the development of mathematical sciences was not restricted to his era but to bring many outstanding mathematicians in 19th century. |
EnaI love reading books. Let's talk about anything we feel interest in. Archives
September 2016
Categories |