Mathematics

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

Principalization (algebra)

In the mathematical field of algebraic number theory, the concept of principalization refers to a situation when, given an extension of algebraic number fields, some ideal of the ring of integers of the smaller field isn't principal but its extension to the ring of integers of the larger field is. Its study has origins in the work of Ernst Kummer on ideal numbers from the 1840s, who in particular proved that for every algebraic number field there exists an extension number field such that all ideals of the ring of integers of the base field become principal when extended to the larger field. In 1897 David Hilbert conjectured that the maximal abelian unramified extension of the base field, called the Hilbert class field of the given base field, is such an extension; this conjecture, now known as principal ideal theorem, was proved by Philipp Furtwängler in 1930 after it had been translated from number theory to group theory by Emil Artin in 1929, who made use of his general reciprocity law to establish the reformulation.

Since this long desired proof was achieved by means of Artin transfers of non-abelian groups with derived length two, several investigators tried to exploit the theory of such groups further to obtain additional information on the principalization in intermediate fields between the base field and its Hilbert class field. The first contributions in this direction are due to Arnold Scholz and Olga Taussky in 1934, who coined the synonym capitulation for principalization. Another independent access to the principalization problem via Galois cohomology of unit groups is due to Hilbert and goes back to the chapter on cyclic extensions of number fields of prime degree in his number report, which culminates in the famous Theorem 94. Let K be an algebraic number field, called the base field, let L / K be a field extension of finite degree. Let O K, I K, P K and O L, I L, P L denote the ring of integers, the group of nonzero fractional ideals and its subgroup of principal fractional ideals of the fields K, L respectively.

The extension map of fractional ideals ι L / K: I K → I L, a ↦ a O L is an injective group homomorphism. Since ι L / K ⊆ P L, this map induces the extension homomorphism of ideal class groups j L / K: I K / P K → I L / P L, a P K ↦ P L. If there exists a non-principal ideal a ∈ I K whose extension ideal in L is principal we speak about principalization or capitulation in L / K. In this case, the ideal a and its class a P K are said to principalize or capitulate in L; this phenomenon is described most conveniently by the principalization kernel or capitulation kernel, the kernel ker of the class extension homomorphism. More let m = m 0 m ∞ be a modulus in K, where

Jean-Pierre Serre

Jean-Pierre Serre is a French mathematician who has made contributions to algebraic topology, algebraic geometry, algebraic number theory. He was awarded the Fields Medal in 1954 and the inaugural Abel Prize in 2003. Born in Bages, Pyrénées-Orientales, France, to pharmacist parents, Serre was educated at the Lycée de Nîmes and from 1945 to 1948 at the École Normale Supérieure in Paris, he was awarded his doctorate from the Sorbonne in 1951. From 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique in Paris. In 1956 he was elected professor at the Collège de France, a position he held until his retirement in 1994, his wife, Professor Josiane Heulot-Serre, was a chemist. Their daughter is the former French diplomat and writer Claudine Monteil; the French mathematician Denis Serre is his nephew. He practices skiing, table tennis, rock climbing. From a young age he was an outstanding figure in the school of Henri Cartan, working on algebraic topology, several complex variables and commutative algebra and algebraic geometry, where he introduced sheaf theory and homological algebra techniques.

Serre's thesis concerned the Leray–Serre spectral sequence associated to a fibration. Together with Cartan, Serre established the technique of using Eilenberg–MacLane spaces for computing homotopy groups of spheres, which at that time was one of the major problems in topology. In his speech at the Fields Medal award ceremony in 1954, Hermann Weyl gave high praise to Serre, made the point that the award was for the first time awarded to a non-analyst. Serre subsequently changed his research focus. In the 1950s and 1960s, a fruitful collaboration between Serre and the two-years-younger Alexander Grothendieck led to important foundational work, much of it motivated by the Weil conjectures. Two major foundational papers by Serre were Faisceaux Algébriques Cohérents, on coherent cohomology, Géometrie Algébrique et Géométrie Analytique. At an early stage in his work Serre had perceived a need to construct more general and refined cohomology theories to tackle the Weil conjectures; the problem was that the cohomology of a coherent sheaf over a finite field couldn't capture as much topology as singular cohomology with integer coefficients.

Amongst Serre's early candidate theories of 1954–55 was one based on Witt vector coefficients. Around 1958 Serre suggested that isotrivial principal bundles on algebraic varieties – those that become trivial after pullback by a finite étale map – are important; this acted as one important source of inspiration for Grothendieck to develop étale topology and the corresponding theory of étale cohomology. These tools, developed in full by Grothendieck and collaborators in Séminaire de géométrie algébrique 4 and SGA 5, provided the tools for the eventual proof of the Weil conjectures by Pierre Deligne. From 1959 onward Serre's interests turned towards group theory, number theory, in particular Galois representations and modular forms. Amongst his most original contributions were: his "Conjecture II" on Galois cohomology. In his paper FAC, Serre asked whether a finitely generated projective module over a polynomial ring is free; this question led to a great deal of activity in commutative algebra, was answered in the affirmative by Daniel Quillen and Andrei Suslin independently in 1976.

This result is now known as the Quillen–Suslin theorem. Serre, at twenty-seven in 1954, is the youngest to be awarded the Fields Medal, he went on to win the Balzan Prize in 1985, the Steele Prize in 1995, the Wolf Prize in Mathematics in 2000, was the first recipient of the Abel Prize in 2003. He has been awarded other prizes, such as the Gold Medal of the French National Scientific Research Centre, he has received many honorary degrees. In 2012 he became a fellow of the American Mathematical Society. Serre has been awarded the highest honors in France as Grand Cross of the Legion of Honour and Grand Cross of the Legion of Merit. List of things named after Jean-Pierre Serre Nicolas Bourbaki Groupes Algébriques et Corps de Classes, translated into English as Algebraic Groups and Class Fields, Springer-Verlag Corps Locaux, Hermann, as Local Fields, Springer-Verlag Cohomologie Galoisienne Collège de France course 1962–63, as Galois Cohomology, Springer-Verlag Algèbre Locale, Multiplicités Collège de France course 1957–58, as Local Algebra, Springer-Verlag "Lie algebras and Lie groups" Harvard Lectures, Springer-Verlag.

Algèbres de Lie Semi-simples Complexes, as Complex Semisimple Lie Algebras, Springer-Verlag Abelian ℓ-Adic Representations and Elliptic Curves, CRC Press, reissue. Addison-Wesley. 1989. Cours d'arithmétique, PUF, as A Course in Arithmetic, Springer-Verlag Représentations linéaires des groupes finis, Hermann, as Linear Represent

David Hilbert

David Hilbert was a German mathematician and one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, foundations of mathematics. Hilbert warmly defended Georg Cantor's set theory and transfinite numbers. A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems that set the course for much of the mathematical research of the 20th century. Hilbert and his students contributed to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic, as well as for being among the first to distinguish between mathematics and metamathematics.

Hilbert, the first of two children of Otto and Maria Therese Hilbert, was born in the Province of Prussia, Kingdom of Prussia, either in Königsberg or in Wehlau near Königsberg where his father worked at the time of his birth. In late 1872, Hilbert entered the Friedrichskolleg Gymnasium. Upon graduation, in autumn 1880, Hilbert enrolled at the University of Königsberg, the "Albertina". In early 1882, Hermann Minkowski, returned to Königsberg and entered the university. Hilbert developed a lifelong friendship with the gifted Minkowski. In 1884, Adolf Hurwitz arrived from Göttingen as an Extraordinarius. An intense and fruitful scientific exchange among the three began, Minkowski and Hilbert would exercise a reciprocal influence over each other at various times in their scientific careers. Hilbert obtained his doctorate in 1885, with a dissertation, written under Ferdinand von Lindemann, titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen. Hilbert remained at the University of Königsberg as a Privatdozent from 1886 to 1895.

In 1895, as a result of intervention on his behalf by Felix Klein, he obtained the position of Professor of Mathematics at the University of Göttingen. During the Klein and Hilbert years, Göttingen became the preeminent institution in the mathematical world, he remained there for the rest of his life. Among Hilbert's students were Hermann Weyl, chess champion Emanuel Lasker, Ernst Zermelo, Carl Gustav Hempel. John von Neumann was his assistant. At the University of Göttingen, Hilbert was surrounded by a social circle of some of the most important mathematicians of the 20th century, such as Emmy Noether and Alonzo Church. Among his 69 Ph. D. students in Göttingen were many who became famous mathematicians, including: Otto Blumenthal, Felix Bernstein, Hermann Weyl, Richard Courant, Erich Hecke, Hugo Steinhaus, Wilhelm Ackermann. Between 1902 and 1939 Hilbert was editor of the Mathematische Annalen, the leading mathematical journal of the time. "Good, he did not have enough imagination to become a mathematician".

Around 1925, Hilbert developed pernicious anemia, a then-untreatable vitamin deficiency whose primary symptom is exhaustion. Those forced out included Hermann Weyl, Emmy Noether and Edmund Landau. One who had to leave Germany, Paul Bernays, had collaborated with Hilbert in mathematical logic, co-authored with him the important book Grundlagen der Mathematik; this was a sequel to the Hilbert-Ackermann book Principles of Mathematical Logic from 1928. Hermann Weyl's successor was Helmut Hasse. About a year Hilbert attended a banquet and was seated next to the new Minister of Education, Bernhard Rust. Rust asked whether "the Mathematical Institute suffered so much because of the departure of the Jews". Hilbert replied, "Suffered? It doesn't exist any longer, does it!" By the time Hilbert died in 1943, the Nazis had nearly restaffed the university, as many of the former faculty had either been Jewish or married to Jews. Hilbert's funeral was attended by fewer than a dozen people, only two of whom were fellow academics, among them Arnold Sommerfeld, a theoretical physicist and a native of Königsberg.

News of his death only became known to the wider world six months. The epitaph on his tombstone in Göttingen consists of the famous lines he spoke at the conclusion of his retirement address to the Society of German Scientists and Physicians on 8 September 1930; the words were given in response to the Latin maxim: "Ignoramus et ignorabimus" or "We do not know, we shall not know": Wir müssen wissen. Wir werden wissen. In English: We mus

Philipp Furtwängler

Friederich Pius Philipp Furtwängler was a German number theorist. Furtwängler wrote an 1896 doctoral dissertation at the University of Göttingen on cubic forms, under Felix Klein. Most of his academic life, from 1912 to 1938, was spent at the University of Vienna, where he taught for example Kurt Gödel, who said that Furtwängler's lectures on number theory were the best mathematical lectures that he heard. Furtwängler was paralysed and, without notes, lectured from a wheelchair while his assistant wrote equations on the blackboard; some of Furtwängler's doctoral students were Wolfgang Gröbner, Nikolaus Hofreiter, Henry Mann, Otto Schreier, Olga Taussky-Todd. Through these and others, he has over 3000 academic descendants, he is now best known for his contribution to the principal ideal theorem in the form of his Beweis des Hauptidealsatzes für Klassenkörper algebraischer Zahlkörper. Philipp Furtwängler was a grandson of the organ builder Philipp Furtwängler and a second cousin of the conductor Wilhelm Furtwängler.

With Helmut Hasse and W. Jehne: Allgemeine Theorie der algebraischen Zahlen. Vol. 8. Teubner, 1953. "Philipp Furtwängler". In: Österreichisches Biographisches Lexikon 1815–1950. Vol. 1, Austrian Academy of Sciences, Vienna 1957, p. 383. Nikolaus Hofreiter, "Furtwängler, Friedrich Pius Philipp", Neue Deutsche Biographie, 5, Berlin: Duncker & Humblot, pp. 740–740 Literature by and about Philipp Furtwängler in the German National Library catalogue http://bibliothek.bbaw.de/kataloge/literaturnachweise/furtwaen/literatur.pdf

Algebraic number theory

Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, function fields; these properties, such as whether a ring admits unique factorization, the behavior of ideals, the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations. The beginnings of algebraic number theory can be traced to Diophantine equations, named after the 3rd-century Alexandrian mathematician, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers x and y such that their sum, the sum of their squares, equal two given numbers A and B, respectively: A = x + y B = x 2 + y 2. Diophantine equations have been studied for thousands of years.

For example, the solutions to the quadratic Diophantine equation x2 + y2 = z2 are given by the Pythagorean triples solved by the Babylonians. Solutions to linear Diophantine equations, such as 26x + 65y = 13, may be found using the Euclidean algorithm. Diophantus' major work was the Arithmetica. Fermat's last theorem was first conjectured by Pierre de Fermat in 1637, famously in the margin of a copy of Arithmetica where he claimed he had a proof, too large to fit in the margin. No successful proof was published until 1995 despite the efforts of countless mathematicians during the 358 intervening years; the unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. One of the founding works of algebraic number theory, the Disquisitiones Arithmeticae is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. In this book Gauss brings together results in number theory obtained by mathematicians such as Fermat, Euler and Legendre and adds important new results of his own.

Before the Disquisitiones was published, number theory consisted of a collection of isolated theorems and conjectures. Gauss brought the work of his predecessors together with his own original work into a systematic framework, filled in gaps, corrected unsound proofs, extended the subject in numerous ways; the Disquisitiones was the starting point for the work of other nineteenth century European mathematicians including Ernst Kummer, Peter Gustav Lejeune Dirichlet and Richard Dedekind. Many of the annotations given by Gauss are in effect announcements of further research of his own, some of which remained unpublished, they must have appeared cryptic to his contemporaries. In a couple of papers in 1838 and 1839 Peter Gustav Lejeune Dirichlet proved the first class number formula, for quadratic forms; the formula, which Jacobi called a result "touching the utmost of human acumen", opened the way for similar results regarding more general number fields. Based on his research of the structure of the unit group of quadratic fields, he proved the Dirichlet unit theorem, a fundamental result in algebraic number theory.

He first used the pigeonhole principle, a basic counting argument, in the proof of a theorem in diophantine approximation named after him Dirichlet's approximation theorem. He published important contributions to Fermat's last theorem, for which he proved the cases n = 5 and n = 14, to the biquadratic reciprocity law; the Dirichlet divisor problem, for which he found the first results, is still an unsolved problem in number theory despite contributions by other researchers. Richard Dedekind's study of Lejeune Dirichlet's work was what 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." 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. David Hilbert unified the field of algebraic number theory with his 1897 treatise Zahlbericht, he resolved a significant number-theory problem formulated by Waring in 1770. As with the finiteness theorem, he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers, he had little more to publish on the subject.

Marvin Greenberg

Marvin Jay Greenberg was an American mathematician. Greenberg studied at Columbia University where he received his bachelor's degree in 1955 and received his doctorate 1959 from Princeton University under Serge Lang with the thesis Pro-Algebraic Structure on the Rational Subgroup of a P-Adic Abelian Variety. From 1955 Greenberg was an assistant at Princeton, from 1958 an assistant at the University of Chicago and in 1958 and 1959, an instructor at Rutgers University. From 1959 to 1964 he was an Assistant Professor at the University of California, two years of which time he spent on National Science Foundation postdoctoral fellowships at Harvard University and the Institut des Hautes Études Scientifiques in Paris. From 1965 to 1967 he was an associate professor at Northeastern University and from 1967 he worked as an associate professor, full professor, at the University of California, Santa Cruz, he moved back to Berkeley. He was known for his books on Algebraic topology. Greenberg was a passionate golfer and a founding member of the Shivas Irons Society.

Marvin Greenberg's faculty page at the University of California, Santa Cruz Marvin Greenberg at the Mathematics Genealogy Project