Emil Artin was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century, he is best known for his work on algebraic number theory, contributing to class field theory and a new construction of L-functions. He contributed to the pure theories of rings and fields. Emil Artin was born in Vienna to parents Emma Maria, née Laura, a soubrette on the operetta stages of Austria and Germany, Emil Hadochadus Maria Artin, Austrian-born of mixed Austrian and Armenian descent. Several documents, including Emil's birth certificate, list the father's occupation as “opera singer” though others list it as “art dealer.” It seems at least plausible. They were married in St. Stephen's Parish on July 24, 1895. Artin entered school in September 1904 in Vienna. By his father was suffering symptoms of advanced syphilis, among them increasing mental instability, was institutionalized at the established insane asylum at Mauer Öhling, 125 kilometers west of Vienna.
It is notable that neither wife nor child contracted this infectious disease. Artin's father died there July 20, 1906. Young Artin was eight. On July 15, 1907, Artin's mother remarried to a man named Rudolf Hübner: a prosperous manufacturing entrepreneur in the German-speaking city called Reichenberg, Bohemia. Documentary evidence suggests that Emma had been a resident in Reichenberg the previous year, in deference to her new husband, she had abandoned her vocal career. Hübner deemed a life in the theater unseemly unfit for the wife of a man of his position. In September, 1907, Artin entered the Volksschule in Strobnitz, a small town in southern Czechoslovakia near the Austrian border. For that year, he lived away from home; the following year, he returned to the home of his mother and stepfather, entered the Realschule in Reichenberg, where he pursued his secondary education until June, 1916. In Reichenberg, Artin formed a lifelong friendship with a young neighbor, Arthur Baer, who became an astronomer, teaching for many years at Cambridge University.
Astronomy was an interest the two boys shared at this time. They each had telescopes, they rigged a telegraph between their houses, over which once Baer excitedly reported to his friend an astronomical discovery he thought he had made—perhaps a supernova, he thought—and told Artin where in the sky to look. Artin tapped back the terse reply “A-N-D-R-O-M-E-D-A N-E-B-E-L.” Artin's academic performance in the first years at the Realschule was spotty. Up to the end of the 1911–1912 school year, for instance, his grade in mathematics was “genügend,”. Of his mathematical inclinations at this early period he wrote, “Meine eigene Vorliebe zur Mathematik zeigte sich erst im sechzehnten Lebensjahr, während vorher von irgendeiner Anlage dazu überhaupt nicht die Rede sein konnte.” His grade in French for 1912 was “nicht genügend”. He did rather better work in chemistry, but from 1910 to 1912, his grade for “Comportment” was “nicht genügend.” Artin spent the school year 1912–1913 away from home, in France, a period he spoke of as one of the happiest of his life.
He lived that year with the family of Edmond Fritz, in the vicinity of Paris, attended a school there. When he returned from France to Reichenberg, his academic work markedly improved, he began receiving grades of “gut” or “sehr gut” in all subjects—including French and “Comportment.” By the time he completed studies at the Realschule in June, 1916, he was awarded the Reifezeugnis that affirmed him “reif mit Auszeichnung” for graduation to a technical university. Now that it was time to move on to university studies, Artin was no doubt content but to leave Reichenberg, for relations with his stepfather were clouded. According to him, Hübner reproached him “day and night” with being a financial burden, when Artin became a university lecturer and a professor, Hübner deprecated his academic career as self-indulgent and belittled its paltry emolument. In October, 1916, Artin matriculated at the University of Vienna, having focused by now on mathematics, he studied there with Philipp Furtwängler, took courses in astrophysics and Latin.
Studies at Vienna were interrupted when Artin was drafted in 1918 into the Austrian army. Assigned to the K.u. K. 44th Infantry Regiment, he was stationed northwest of Venice at Primolano, on the Italian front in the foothills of the Dolomites. To his great relief, Artin managed to avoid combat by volunteering for service as a translator—his ignorance of Italian notwithstanding, he did know French, of course, some Latin, was a quick study, was motivated by a rational fear in a theater of that war that had proven a meat-grinder. In his scramble to learn at least some Italian, Artin had recourse to an encyclopedia, which he once consulted for help in dealing with the cockroaches that infested the Austrian barracks. At some length, the article described a variety of technical methods, concluding with—Artin laughingly recalled in years—“la caccia diretta". Indeed, “la caccia diretta” was the straightforward method he and his fe
Ernst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics. Kummer was born in Brandenburg, he was awarded a PhD from the University of Halle in 1831 for writing a prize-winning mathematical essay, published a year later. Kummer was married in 1840 to Ottilie Mendelssohn, daughter of Nathan Mendelssohn and Henriette Itzig. Ottilie was a cousin of Felix Mendelssohn and his sister Rebecca Mendelssohn Bartholdy, the wife of the mathematician Peter Gustav Lejeune Dirichlet, his second wife, Bertha Cauer, was a maternal cousin of Ottilie. Overall, he had 13 children, his daughter Marie married the mathematician Hermann Schwarz. Kummer retired from teaching and from mathematics in 1890 and died three years in Berlin. Kummer made several contributions to mathematics in different areas; the Kummer surface results from taking the quotient of a two-dimensional abelian variety by the cyclic group. Kummer proved Fermat's last theorem for a considerable class of prime exponents.
His methods were closer to p-adic ones than to ideal theory as understood though the term'ideal' was invented by Kummer. He studied what were called Kummer extensions of fields: that is, extensions generated by adjoining an nth root to a field containing a primitive nth root of unity; this is a significant extension of the theory of quadratic extensions, the genus theory of quadratic forms. As such, it is still foundational for class field theory. Kummer, Ernst Eduard, André, ed. Collected papers. Volume 1: Contributions to Number Theory, New York: Springer-Verlag, ISBN 978-0-387-06835-0, MR 0465760 Kummer, Ernst Eduard, André, ed. Collected papers. Volume II: Function theory and miscellaneous, New York: Springer-Verlag, ISBN 978-3-540-06836-5, MR 0465761 Kummer configuration Kummer's congruence Kummer series Kummer theory Kummer's theorem, on prime-power divisors of binomial coefficients Kummer's function Kummer ring Kummer sum Kummer variety Eric Temple Bell, Men of Mathematics and Schuster, New York: 1986.
R. W. H. T. Hudson, Kummer's Quartic Surface, rept. 1990. "Ernst Kummer," in Dictionary of Scientific Biography, ed. C. Gillispie, NY: Scribners 1970–90. O'Connor, John J.. "Ernst Kummer", MacTutor History of Mathematics archive, University of St Andrews. Works by or about Ernst Kummer at Internet Archive Biography of Ernst Kummer Ernst Kummer at the Mathematics Genealogy Project
Teiji Takagi was a Japanese mathematician, best known for proving the Takagi existence theorem in class field theory. The Blancmange curve, the graph of a nowhere-differentiable but uniformly continuous function, is called the Takagi curve after his work on it, he was born in the rural area of the Gifu Japan. He began learning mathematics in middle school, reading texts in English since none were available in Japanese. After attending a high school for gifted students, he went on to the University of Tokyo, at that time the only university in Japan. There he learned mathematics from such European classic texts as Salmon's Algebra and Weber's Lehrbuch der Algebra. Aided by Hilbert, he studied at Göttingen. Aside from his work in algebraic number theory he wrote a great number of Japanese textbooks on mathematics and geometry. During World War I, he was isolated from European mathematicians and developed his existence theorem in class field theory, building on the work of Heinrich Weber; as an Invited Speaker, he presented a synopsis of this research in a talk Sur quelques théoremes généraux de la théorie des nombres algébriques at the International Congress of Mathematicians in Strasbourg in 1920.
There he found little recognition of the value of his research, since algebraic number theory was studied in Germany and German mathematicians were excluded from the Congress. Takagi published his theory in the same year in the journal of the University of Tokyo. However, the significance of Takagi's work was first recognized by Emil Artin in 1922, was again pointed out by Carl Ludwig Siegel, at the same time by Helmut Hasse, who lectured in Kiel in 1923 on class field theory and presented Takagi's work in a lecture at the meeting of the DMV in 1925 in Danzig and in his Klassenkörperbericht in the 1926 annual report of the DMV. Takagi was internationally recognized as one of the world's leading number theorists. In 1932 he was vice-president of the International Congress of Mathematicians in Zurich and in 1936 was a member of the selection committee for the first Fields Medal, he was instrumental during World War II in the development of Japanese encryption systems. Sigekatu Kuroda - son-in-law.
Mathematician. S.-Y. Kuroda - grandson. Mathematician and Chomskyan linguist. Takagi, Iyanaga, Shokichi, ed. Collected papers, Springer Collected Works in Mathematics, Springer-Verlag, doi:10.1007/978-4-431-54995-6, ISBN 978-4431549949, MR 1129240 Media related to Teiji Takagi at Wikimedia Commons O'Connor, John J.. Takagi Lectures by the Mathematical Society of Japan
Spencer Janney Bloch is an American mathematician known for his contributions to algebraic geometry and algebraic K-theory. Bloch is a R. M. Hutchins Distinguished Service Professor Emeritus in the Department of Mathematics of the University of Chicago, he is a member of the U. S. National Academy of Sciences and a Fellow of the American Academy of Arts and Sciences and of the American Mathematical Society. At the International Congress of Mathematicians he gave an invited lecture in 1978 and a plenary lecture in 1990, he was a visiting scholar at the Institute for Advanced Study in 1981-82. He received a Humboldt Prize in 1996. Bloch's formula Bloch group Bloch–Kato conjecture Bloch's higher Chow group James D. Lewis, Rob de Jeu and Algebraic Cycles: A Celebration in Honour of Spencer J. Bloch. Fields Institute Communications series, 2009, American Mathematical Society. ISBN 0-8218-4494-6 Spencer Bloch personal webpage, Department of Mathematics, University of Chicago Spencer Bloch at the Mathematics Genealogy Project
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Leopold Kronecker was a German mathematician who worked on number theory and logic. He criticized Georg Cantor's work on set theory, was quoted by Weber as having said, "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk". Kronecker was a student and lifelong friend of Ernst Kummer. Leopold Kronecker was born on 7 December 1823 in Prussia in a wealthy Jewish family, his parents and Johanna, took care of their children's education and provided them with private tutoring at home – Leopold's younger brother Hugo Kronecker would follow a scientific path becoming a notable physiologist. Kronecker went to the Liegnitz Gymnasium where he was interested in a wide range of topics including science and philosophy, while practicing gymnastics and swimming. At the gymnasium he was taught by Ernst Kummer, who noticed and encouraged the boy's interest in mathematics. In 1841 Kronecker became a student at the University of Berlin where his interest did not focus on mathematics, but rather spread over several subjects including astronomy and philosophy.
He spent the summer of 1843 at the University of Bonn studying astronomy and 1843–44 at the University of Breslau following his former teacher Kummer. Back in Berlin, Kronecker studied mathematics with Peter Gustav Lejeune Dirichlet and in 1845 defended his dissertation in algebraic number theory written under Dirichlet's supervision. After obtaining his degree, Kronecker did not follow his interest in research on an academic career path, he went back to his hometown to manage a large farming estate built up by his mother's uncle, a former banker. In 1848 he married his cousin Fanny Prausnitzer, the couple had six children. For several years Kronecker focused on business, although he continued to study mathematics as a hobby and corresponded with Kummer, he published no mathematical results. In 1853 he wrote a memoir on the algebraic solvability of equations extending the work of Évariste Galois on the theory of equations. Due to his business activity, Kronecker was financially comfortable, thus he could return to Berlin in 1855 to pursue mathematics as a private scholar.
Dirichlet, whose wife Rebecka came from the wealthy Mendelssohn family, had introduced Kronecker to the Berlin elite. He became a close friend of Karl Weierstrass, who had joined the university, his former teacher Kummer who had just taken over Dirichlet's mathematics chair. Over the following years Kronecker published numerous papers resulting from his previous years' independent research; as a result of this published research, he was elected a member of the Berlin Academy in 1861. Although he held no official university position, Kronecker had the right as a member of the Academy to hold classes at the University of Berlin and he decided to do so, starting in 1862. In 1866, when Riemann died, Kronecker was offered the mathematics chair at the University of Göttingen, but he refused, preferring to keep his position at the Academy. Only in 1883, when Kummer retired from the University, was Kronecker invited to succeed him and became an ordinary professor. Kronecker was the supervisor of Kurt Hensel, Adolf Kneser, Mathias Lerch, Franz Mertens, amongst others.
His philosophical view of mathematics put him in conflict with several mathematicians over the years, notably straining his relationship with Weierstrass, who decided to leave the University in 1888. Kronecker died on 29 December 1891 in several months after the death of his wife. In the last year of his life, he converted to Christianity, he is buried in the Alter St Matthäus Kirchhof cemetery in Berlin-Schöneberg, close to Gustav Kirchhoff. An important part of Kronecker's research focused on number algebra. In an 1853 paper on the theory of equations and Galois theory he formulated the Kronecker–Weber theorem, without however offering a definitive proof, he introduced the structure theorem for finitely-generated abelian groups. Kronecker studied elliptic functions and conjectured his "liebster Jugendtraum", a generalization, put forward by Hilbert in a modified form as his twelfth problem. In an 1850 paper, On the Solution of the General Equation of the Fifth Degree, Kronecker solved the quintic equation by applying group theory.
In algebraic number theory Kronecker introduced the theory of divisors as an alternative to Dedekind's theory of ideals, which he did not find acceptable for philosophical reasons. Although the general adoption of Dedekind's approach led Kronecker's theory to be ignored for a long time, his divisors were found useful and were revived by several mathematicians in the 20th century. Kronecker contributed to the concept of continuity, reconstructing the form of irrational numbers in real numbers. In analysis, Kronecker rejected the formulation of a continuous, nowhere differentiable function by his colleague, Karl Weierstrass. Named for Kronecker are the Kronecker limit formula, Kronecker's congruence, Kronecker delta, Kronecker comb, Kronecker symbol, Kronecker product, Kronecker's method for factorizing polynomials, Kronecker substitution, Kronecker's theorem in number theory, Kronecker's lemma, Eisenstein–Kronecker numbers. Kronecker's finitism made him a forerunner of intuitionism in foundations of mathematics.
Kronecker was elected as a member of several academies: Prussian Academy of Sciences French Academy of Sciences Roy
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (. Sometimes referred to as the Princeps mathematicorum and "the greatest mathematician since antiquity", Gauss had an exceptional influence in many fields of mathematics and science, is ranked among history's most influential mathematicians. Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick, in the Duchy of Brunswick-Wolfenbüttel, to poor, working-class parents, his mother was illiterate and never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the Feast of the Ascension. Gauss solved this puzzle about his birthdate in the context of finding the date of Easter, deriving methods to compute the date in both past and future years, he was christened and confirmed in a church near the school he attended as a child. Gauss was a child prodigy. In his memorial on Gauss, Wolfgang Sartorius von Waltershausen says that when Gauss was three years old he corrected a math error his father made. Many versions of this story have been retold since that time with various details regarding what the series was – the most frequent being the classical problem of adding all the integers from 1 to 100.
There are many other anecdotes about his precocity while a toddler, he made his first groundbreaking mathematical discoveries while still a teenager. He completed his magnum opus, Disquisitiones Arithmeticae, in 1798, at the age of 21—though it was not published until 1801; this work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day. Gauss's intellectual abilities attracted the attention of the Duke of Brunswick, who sent him to the Collegium Carolinum, which he attended from 1792 to 1795, to the University of Göttingen from 1795 to 1798. While at university, Gauss independently rediscovered several important theorems, his breakthrough occurred in 1796 when he showed that a regular polygon can be constructed by compass and straightedge if the number of its sides is the product of distinct Fermat primes and a power of 2. This was a major discovery in an important field of mathematics. Gauss was so pleased with this result that he requested that a regular heptadecagon be inscribed on his tombstone.
The stonemason declined, stating that the difficult construction would look like a circle. The year 1796 was productive for both Gauss and number theory, he discovered a construction of the heptadecagon on 30 March. He further advanced modular arithmetic simplifying manipulations in number theory. On 8 April he became the first to prove the quadratic reciprocity law; this remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic. The prime number theorem, conjectured on 31 May, gives a good understanding of how the prime numbers are distributed among the integers. Gauss discovered that every positive integer is representable as a sum of at most three triangular numbers on 10 July and jotted down in his diary the note: "ΕΥΡΗΚΑ! num = Δ + Δ' + Δ". On 1 October he published a result on the number of solutions of polynomials with coefficients in finite fields, which 150 years led to the Weil conjectures. Gauss remained mentally active into his old age while suffering from gout and general unhappiness.
For example, at the age of 62, he taught himself Russian. In 1840, Gauss published his influential Dioptrische Untersuchungen, in which he gave the first systematic analysis on the formation of images under a paraxial approximation. Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its cardinal points and he derived the Gaussian lens formula. In 1845, he became an associated member of the Royal Institute of the Netherlands. In 1854, Gauss selected the topic for Bernhard Riemann's inaugural lecture "Über die Hypothesen, welche der Geometrie zu Grunde liegen". On the way home from Riemann's lecture, Weber reported that Gauss was full of excitement. On 23 February 1855, Gauss died of a heart attack in Göttingen. Two people gave eulogies at his funeral: Gauss's son-in-law Heinrich Ewald, Wolfgang Sartorius von Waltershausen, Gauss's close friend and biographer. Gauss's brain was preserved and was studied by Rudolf Wagner, who found its mass to be above average, at 1,492 grams, the cerebral area equal to 219,588 square millimeters.
Developed convolutions were found, which in the early 20th century were suggested as the explanation of his genius. Gauss was a Lutheran Protestant, a member of the St. Albans Evangelical Lutheran church in Göttingen. Potential evidence that Gauss believed in God comes from his response after solving a problem that had defeated him: "Finally, two days ago, I succeeded—not on account of my hard efforts, but by th