Wilhelm Johann Eugen Blaschke was an Austrian differential and integral geometer. His students included Luis Santaló, Gheorghe Gheorghiev and Emanuel Sperner. In 1916 Blaschke published one of the first books devoted to convex sets: Sphere. Drawing on dozens of sources, Blaschke made a thorough review of the subject with citations within the text to attribute credit in a classical area of mathematics. In 1933 Blaschke signed the Loyalty Oath of German Professors to Adolf Hitler and the National Socialist State. Wilhelm Blaschke joined the NSDAP in 1937. Kreis und Kugel, Veit 1916. Berlin, de Gruyter 1956 Vorlesungen über Differentialgeometrie, 3 vols. Springer, Grundlehren der mathematischen Wissenschaften 1921-1929 with G. Bol: Geometrie der Gewebe. Berlin: Springer 1938 Ebene Kinematik. Leipzig: B. G. Teubner 1938, 2nd expanded edn. with Hans Robert Müller, Oldenbourg, München 1956 Nicht-Euklidische Geometrie und Mechanik I, II, III. Leipzig: B. G. Teubner Zur Bewegungsgeometrie auf der Kugel. In: Sitzungsberichte der Heidelberger Akademie der Wissenschaften Einführung in die Differentialgeometrie.
Springer 1950, 2nd expanded edn. with H. Reichardt 1960 with Kurt Leichtweiß: Elementare Differentialgeometrie. Berlin: Springer Reden und Reisen eines Geometers. Berlin: VEB Deutscher Verlag der Wissenschaften Mathematik und Leben, Steiner 1951 Griechische und anschauliche Geometrie, Oldenbourg 1953 Projektive Geometrie, 3rd edn, Birkhäuser 1954 Analytische Geometrie, 2nd edn. Birkhäuser 1954 Einführung in die Geometrie der Waben, Birkhäuser 1955 Vorlesungen über Integralgeometrie, VEB, Berlin 1955 Kinematik und Quaternionen. Berlin: VEB Deutscher Verlag der Wissenschaften Gesammelte Werke, Essen 1985 A number of mathematical theorems and concepts is associated with the name of Blaschke. Blaschke selection theorem Blaschke–Lebesgue theorem Blaschke product Blaschke condition Blaschke–Santaló inequality Blaschke conjecture: "The only Wiedersehen manifolds in any dimension are the standard Euclidean spheres." Blaschke, W.. Reden und Reisen eines Geometers. East Berlin. O'Connor, John J.. Wilhelm Blaschke at the Mathematics Genealogy Project
Amalie Emmy Noether was a German mathematician who made important contributions to abstract algebra and theoretical physics. She invariably used the name "Emmy Noether" in her life and publications, she was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics. As one of the leading mathematicians of her time, she developed the theories of rings and algebras. In physics, Noether's theorem explains the connection between conservation laws. Noether was born to a Jewish family in the Franconian town of Erlangen, she planned to teach French and English after passing the required examinations, but instead studied mathematics at the University of Erlangen, where her father lectured. After completing her dissertation in 1907 under the supervision of Paul Gordan, she worked at the Mathematical Institute of Erlangen without pay for seven years. At the time, women were excluded from academic positions. In 1915, she was invited by David Hilbert and Felix Klein to join the mathematics department at the University of Göttingen, a world-renowned center of mathematical research.
The philosophical faculty objected and she spent four years lecturing under Hilbert's name. Her habilitation was approved in 1919. Noether remained a leading member of the Göttingen mathematics department until 1933. In 1924, Dutch mathematician B. L. van der Waerden joined her circle and soon became the leading expositor of Noether's ideas: Her work was the foundation for the second volume of his influential 1931 textbook, Moderne Algebra. By the time of her plenary address at the 1932 International Congress of Mathematicians in Zürich, her algebraic acumen was recognized around the world; the following year, Germany's Nazi government dismissed Jews from university positions, Noether moved to the United States to take up a position at Bryn Mawr College in Pennsylvania. In 1935 she underwent surgery for an ovarian cyst and, despite signs of a recovery, died four days at the age of 53. Noether's mathematical work has been divided into three "epochs". In the first, she made contributions to the theories of algebraic invariants and number fields.
Her work on differential invariants in the calculus of variations, Noether's theorem, has been called "one of the most important mathematical theorems proved in guiding the development of modern physics". In the second epoch, she began work that "changed the face of algebra". In her classic 1921 paper Idealtheorie in Ringbereichen Noether developed the theory of ideals in commutative rings into a tool with wide-ranging applications, she made elegant use of the ascending chain condition, objects satisfying it are named Noetherian in her honor. In the third epoch, she published works on noncommutative algebras and hypercomplex numbers and united the representation theory of groups with the theory of modules and ideals. In addition to her own publications, Noether was generous with her ideas and is credited with several lines of research published by other mathematicians in fields far removed from her main work, such as algebraic topology. Emmy's father, Max Noether, was descended from a family of wholesale traders in Germany.
At age 14, he had been paralyzed by polio. He regained mobility. Self-taught, he was awarded a doctorate from the University of Heidelberg in 1868. After teaching there for seven years, he took a position in the Bavarian city of Erlangen, where he met and married Ida Amalia Kaufmann, the daughter of a prosperous merchant. Max Noether's mathematical contributions were to algebraic geometry following in the footsteps of Alfred Clebsch, his best known results are the Brill -- AF+BG theorem. Emmy Noether was born on 23 March 1882, the first of four children, her first name was "Amalie", after her mother and paternal grandmother, but she began using her middle name at a young age. As a girl, Noether was well liked, she did not stand out academically although she was known for being friendly. She was talked with a minor lisp during childhood. A family friend recounted a story years about young Noether solving a brain teaser at a children's party, showing logical acumen at that early age, she was taught to cook and clean, as were most girls of the time, she took piano lessons.
She pursued none of these activities with passion. She had three younger brothers: The eldest, was born in 1883, was awarded a doctorate in chemistry from Erlangen in 1909, but died nine years later. Fritz Noether, born in 1884, is remembered for his academic accomplishments; the youngest, Gustav Robert, was born in 1889. Little is known about his life. Noether showed early proficiency in English. In the spring of 1900, she took the examination for teachers of these languages and received an overall score of sehr gut, her performance qualified her to teach languages at schools reserved for girls, but she chose instead to continue her studies at the University of Erlangen. This was an unconventional decision. One of only two wome
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
Otto Schreier was a Jewish-Austrian mathematician who made major contributions in combinatorial group theory and in the topology of Lie groups. He studied mathematics at the University of Vienna and obtained his doctorate in 1923, under the supervision of Philipp Furtwängler, he moved to the University of Hamburg. According to Hans Zassenhaus: O. Schreier's and Artin's ingenious characterization of formally real fields as fields in which –1 is not the sum of squares and the ensuing deduction of the existence of an algebraic ordering of such fields started the discipline of real algebra. Artin and his congenial friend and colleague Schreier set out on the daring and successful construction of a bridge between algebra and analysis. In the light of Artin-Schreier's theory the fundamental theorem of algebra is an algebraic theorem inasmuch as it states that irreducible polynomials over real closed fields only can be linear or quadratic. Nielsen–Schreier theorem Schreier refinement theorem Artin–Schreier theorem Schreier's subgroup lemma Schreier–Sims algorithm Schreier coset graph Schreier conjecture Schreier domain O'Connor, John J..
Otto Schreier at the Mathematics Genealogy Project
Julius Wilhelm Richard Dedekind was a German mathematician who made important contributions to abstract algebra, axiomatic foundation for the natural numbers, algebraic number theory and the definition of the real numbers. Dedekind's father was Julius Levin Ulrich Dedekind, an administrator of Collegium Carolinum in Braunschweig. Dedekind had three older siblings; as an adult, he never used the names Julius Wilhelm. He was born, lived most of his life, died in Braunschweig, he first attended the Collegium Carolinum in 1848 before transferring to the University of Göttingen in 1850. There, Dedekind was taught number theory by professor Moritz Stern. Gauss was still teaching, although at an elementary level, Dedekind became his last student. Dedekind received his doctorate in 1852, for a thesis titled Über die Theorie der Eulerschen Integrale; this thesis did not display the talent evident by Dedekind's subsequent publications. At that time, the University of Berlin, not Göttingen, was the main facility for mathematical research in Germany.
Thus Dedekind went to Berlin for two years of study, where he and Bernhard Riemann were contemporaries. Dedekind returned to Göttingen to teach as a Privatdozent, giving courses on probability and geometry, he studied for a while with Peter Gustav Lejeune Dirichlet, they became good friends. Because of lingering weaknesses in his mathematical knowledge, he studied elliptic and abelian functions, yet he was the first at Göttingen to lecture concerning Galois theory. About this time, he became one of the first people to understand the importance of the notion of groups for algebra and arithmetic. In 1858, he began teaching at the Polytechnic school in Zürich; when the Collegium Carolinum was upgraded to a Technische Hochschule in 1862, Dedekind returned to his native Braunschweig, where he spent the rest of his life, teaching at the Institute. He retired in 1894, but continued to publish, he never married. Dedekind was elected to the Academies of Berlin and Rome, to the French Academy of Sciences.
He received honorary doctorates from the universities of Oslo and Braunschweig. While teaching calculus for the first time at the Polytechnic school, Dedekind developed the notion now known as a Dedekind cut, now a standard definition of the real numbers; the idea of a cut is that an irrational number divides the rational numbers into two classes, with all the numbers of one class being greater than all the numbers of the other class. For example, the square root of 2 defines all the nonnegative numbers whose squares are less than 2 and the negative numbers into the lesser class, the positive numbers whose squares are greater than 2 into the greater class; every location on the number line continuum contains an irrational number. Thus there are gaps, or discontinuities. Dedekind published his thoughts on irrational numbers and Dedekind cuts in his pamphlet "Stetigkeit und irrationale Zahlen". Dedekind's theorem states that if there existed a one-to-one correspondence between two sets the two sets were "similar".
He invoked similarity to give the first precise definition of an infinite set: a set is infinite when it is "similar to a proper part of itself," in modern terminology, is equinumerous to one of its proper subsets. Thus the set N of natural numbers can be shown to be similar to the subset of N whose members are the squares of every member of N,: N 1 2 3 4 5 6 7 8 9 10... ↓ N2 1 4 9 16 25 36 49 64 81 100... Dedekind edited the collected works of Lejeune Dirichlet and Riemann. Dedekind's study of Lejeune Dirichlet's work led him to his study of algebraic number fields and ideals. In 1863, he published Lejeune Dirichlet's lectures on number theory as Vorlesungen über Zahlentheorie about which it has been written that: Although the book is assuredly based on Dirichlet's lectures, although Dedekind himself referred to the book throughout his life as Dirichlet's, the book itself was written by Dedekind, for the most part after Dirichlet's death; the 1879 and 1894 editions of the Vorlesungen included supplements introducing the notion of an ideal, fundamental to ring theory.
Dedekind defined an ideal as a subset of a set of numbers, composed of algebraic integers that satisfy polynomial equations with integer coefficients. The concept underwent further development in the hands of Hilbert and of Emmy Noether. Ideals generalize Ernst Eduard Kummer's ideal numbers, devised as part of Kummer's 1843 attempt to prove Fermat's Last Theorem. In an 1882 article and Heinrich Martin Weber applied ideals to Riemann surfaces, giving an algebraic proof of the Riemann–Roch theorem. In 1888, he published a short monograph titled Was sind und was sollen die Zahlen?, which included his definition of an infinite set. He proposed an axiomatic foundation for the natural numbers, whose primitive notions were the number one and the successor function; the next year, Giuseppe Peano, citing Dedekind, formulated an equivalent but simpler set of axioms, now the standard ones. Dedekind made other
In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series and functions. A ring is an abelian group with a second binary operation, associative, is distributive over the abelian group operation, has an identity element. By extension from the integers, the abelian group operation is called addition and the second binary operation is called multiplication. Whether a ring is commutative or not has profound implications on its behavior as an abstract object; as a result, commutative ring theory known as commutative algebra, is a key topic in ring theory. Its development has been influenced by problems and ideas occurring in algebraic number theory and algebraic geometry. Examples of commutative rings include the set of integers equipped with the addition and multiplication operations, the set of polynomials equipped with their addition and multiplication, the coordinate ring of an affine algebraic variety, the ring of integers of a number field.
Examples of noncommutative rings include the ring of n × n real square matrices with n ≥ 2, group rings in representation theory, operator algebras in functional analysis, rings of differential operators in the theory of differential operators, the cohomology ring of a topological space in topology. The conceptualization of rings was completed in the 1920s. Key contributors include Dedekind, Hilbert and Noether. Rings were first formalized as a generalization of Dedekind domains that occur in number theory, of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. Afterward, they proved to be useful in other branches of mathematics such as geometry and mathematical analysis; the most familiar example of a ring is the set of all integers, Z, consisting of the numbers …, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, …The familiar properties for addition and multiplication of integers serve as a model for the axioms for rings. A ring is a set R equipped with two binary operations + and · satisfying the following three sets of axioms, called the ring axioms R is an abelian group under addition, meaning that: + c = a + for all a, b, c in R. a + b = b + a for all a, b in R.
There is an element 0 in R such that a + 0 = a for all a in R. For each a in R there exists −a in R such that a + = 0. R is a monoid under multiplication, meaning that: · c = a · for all a, b, c in R. There is an element 1 in R such that a · 1 = a and 1 · a = a for all a in R. Multiplication is distributive with respect to addition, meaning that: a ⋅ = + for all a, b, c in R. · a = + for all a, b, c in R. As explained in § History below, many authors follow an alternative convention in which a ring is not defined to have a multiplicative identity; this article adopts the convention that, unless otherwise stated, a ring is assumed to have such an identity. A structure satisfying all the axioms except the requirement that there exists a multiplicative identity element is called a rng. For example, the set of integers with the usual + and ⋅ is a rng, but not a ring; the operations + and ⋅ are called multiplication, respectively. The multiplication symbol ⋅ is omitted, so the juxtaposition of ring elements is interpreted as multiplication.
For example, xy means x ⋅ y. Although ring addition is commutative, ring multiplication is not required to be commutative: ab need not equal ba. Rings that satisfy commutativity for multiplication are called commutative rings. Books on commutative algebra or algebraic geometry adopt the convention that ring means commutative ring, to simplify terminology. In a ring, multiplication does not have to have an inverse. A commutative ring such; the additive group of a ring is the ring equipped just with the structure of addition. Although the definition assumes that the additive group is abelian, this can be inferred from the other ring axioms; some basic properties of a ring follow from the axioms: The additive identity, the additive inverse of each element, the multiplicative identity are unique. For any element x in a ring R, one has x0 = 0 = 0x and x = –x. If 0 = 1 in a ring R R has only one element, is called the zero ring; the binomial formula holds for any commuting pair of elements. Equip the set Z 4 = with the following operat
In mathematics, a group is a set equipped with a binary operation which combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study. Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; the concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s.
After contributions from other fields such as number theory and geometry, the group notion was generalized and established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists study the different ways in which a group can be expressed concretely, both from a point of view of representation theory and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory; the modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4.
The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots. The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 gives the first abstract definition of a finite group. Geometry was a second field in which groups were used systematically symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884; the third field contributing to group theory was number theory.
Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae, more explicitly by Leopold Kronecker. In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers; the convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques. Walther von Dyck introduced the idea of specifying a group by means of generators and relations, was the first to give an axiomatic definition of an "abstract group", in the terminology of the time; as of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, more locally compact groups was studied by Hermann Weyl, Élie Cartan and many others.
Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley and by the work of Armand Borel and Jacques Tits. The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher and Smith in 2004; this project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification; these days, group theory is still a active mathematical branch, impacting many other fields. One of the most familiar groups is the set of integers Z which consists of the numbers... − 4, − 3, − − 1, 0, 1, 2, 3, 4... together with addition. The following properties of integer addition serve as a model for the group axioms given in the definition below.
For any two integers a and b, the sum a + b is an integer. That is, addition of integers always yields an integer; this property is known as closure under addition. For all integers a, b and c, + c = a +. Expressed in words