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Anabelian
Anabelian geometry is a theory in number theory which describes the way in which the algebraic fundamental group ''G'' of a certain arithmetic variety ''X'', or some related geometric object, can help to recover ''X''. The first results for number fields and their absolute Galois groups were obtained by Jürgen Neukirch, Masatoshi Gündüz Ikeda, Kenkichi Iwasawa, and Kôji Uchida ( Neukirch–Uchida theorem, 1969), prior to conjectures made about hyperbolic curves over number fields by Alexander Grothendieck. As introduced in ''Esquisse d'un Programme'' the latter were about how topological homomorphisms between two arithmetic fundamental groups of two hyperbolic curves over number fields correspond to maps between the curves. A first version of Grothendieck's anabelian conjecture was solved by Hiroaki Nakamura and Akio Tamagawa (for affine curves), then completed by Shinichi Mochizuki. Formulation of a conjecture of Grothendieck on curves The "anabelian question" has been f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Grothendieck
Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory, and category theory to its foundations, while his so-called Grothendieck's relative point of view, "relative" perspective led to revolutionary advances in many areas of pure mathematics. He is considered by many to be the greatest mathematician of the twentieth century. Grothendieck began his productive and public career as a mathematician in 1949. In 1958, he was appointed a research professor at the Institut des Hautes Études Scientifiques, Institut des hautes études scientifiques (IHÉS) and remained there until 1970, when, driven by personal and political convictions, he left following a dispute over military funding. He receive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shinichi Mochizuki
is a Japanese mathematician working in number theory and arithmetic geometry. He is one of the main contributors to anabelian geometry. His contributions include his solution of the Grothendieck conjecture in anabelian geometry about hyperbolic curves over number fields. Mochizuki has also worked in Hodge–Arakelov theory and ''p''-adic Teichmüller theory. Mochizuki developed inter-universal Teichmüller theory, which has attracted attention from non-mathematicians due to claims it provides a resolution of the ''abc'' conjecture. Biography Early life Shinichi Mochizuki was born to parents Kiichi and Anne Mochizuki. When he was five years old, Shinichi Mochizuki and his family left Japan to live in the United States. His father was Fellow of the Center for International Affairs and Center for Middle Eastern Studies at Harvard University (1974–76). Mochizuki attended Phillips Exeter Academy and graduated in 1985. Mochizuki entered Princeton University as an undergradua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Class Field Theory
In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credited as one of pioneers of the notion of a class field. However, this notion was already familiar to Kronecker and it was actually Weber who coined the term before Hilbert's fundamental papers came out. The relevant ideas were developed in the period of several decades, giving rise to a set of conjectures by Hilbert that were subsequently proved by Takagi and Artin (with the help of Chebotarev's theorem). One of the major results is: given a number field ''F'', and writing ''K'' for the maximal abelian unramified extension of ''F'', the Galois group of ''K'' over ''F'' is canonically isomorphic to the ideal class group of ''F''. This statement was generalized to the so called Artin reciprocity law; in the idelic language, writing '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Esquisse D'un Programme
"Esquisse d'un Programme" (Sketch of a Programme) is a famous proposal for long-term mathematical research made by the German-born, French mathematician Alexander Grothendieck in 1984. He pursued the sequence of logically linked ideas in his important project proposal from 1984 until 1988, but his proposed research continues to date to be of major interest in several branches of advanced mathematics. Grothendieck's vision provides inspiration today for several developments in mathematics such as the extension and generalization of Galois theory. Brief history Submitted in 1984, the ''Esquisse d'un Programme'' was a proposal by Alexander Grothendieck for a position at the Centre National de la Recherche Scientifique. The proposal was not successful, but Grothendieck obtained a special position where, while keeping his affiliation at the University of Montpellier, he was paid by the CNRS and released of his teaching obligations. Grothendieck held this position from 1984 till 1988. Thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neukirch–Uchida Theorem
In mathematics, the Neukirch–Uchida theorem shows that all problems about algebraic number fields can be reduced to problems about their absolute Galois groups. showed that two algebraic number fields with the same absolute Galois group are isomorphic, and strengthened this by proving Neukirch's conjecture that automorphisms of the algebraic number field correspond to outer automorphisms of its absolute Galois group. extended the result to infinite fields that are finitely generated over prime fields. The Neukirch–Uchida theorem is one of the foundational results of anabelian geometry, whose main theme is to reduce properties of geometric objects to properties of their fundamental groups, provided these fundamental groups are sufficiently non-abelian. Statement Let K_1, K_2 be two algebraic number fields. The Neukirch–Uchida theorem says that, for every topological group isomorphism :\phi\colon\operatorname(\bar K_1/K_1)\xrightarrow\cong\operatorname(\bar K_2/K_2) of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jürgen Neukirch
Jürgen Neukirch (24 July 1937 – 5 February 1997) was a German mathematician known for his work on algebraic number theory. Education and career Neukirch received his diploma in mathematics in 1964 from the University of Bonn. For his Ph.D. thesis, written under the direction of Wolfgang Krull, he was awarded in 1965 the Felix-Hausdorff-Gedächtnis-Preis. He completed his habilitation one year later. From 1967 to 1969 he was guest professor at Queen's University in Kingston, Ontario and at the Massachusetts Institute of Technology in Cambridge, Massachusetts, after which he was a professor in Bonn. In 1971 he became a professor at the University of Regensburg. Contributions He is known for his work on the embedding problem in algebraic number theory, the Báyer–Neukirch theorem on special values of L-functions, arithmetic Riemann existence theorems and the Neukirch–Uchida theorem in birational anabelian geometry. He gave a simple description of the reciprocity maps i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Section Conjecture
In anabelian geometry, a branch of algebraic geometry, the section conjecture gives a conjectural description of the splittings of the group homomorphism \pi_1(X)\to \operatorname(k), where X is a complete smooth curve of genus at least 2 over a field k that is finitely generated over \mathbb, in terms of decomposition groups of rational points of X. The conjecture was introduced by in a 1983 letter to Gerd Faltings Gerd Faltings (; born 28 July 1954) is a German mathematician known for his work in arithmetic geometry. Education From 1972 to 1978, Faltings studied mathematics and physics at the University of Münster. Interrupted by 15 months of obligatory .... References * External links * Algebraic geometry Unsolved problems in geometry {{algebraic-geometry-stub Arithmetic geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grothendieck–Teichmüller Group
In mathematics, the Grothendieck–Teichmüller group ''GT'' is a group closely related to (and possibly equal to) the absolute Galois group of the rational numbers. It was introduced by and named after Alexander Grothendieck and Oswald Teichmüller, based on Grothendieck's suggestion in his 1984 essay ''Esquisse d'un Programme'' to study the absolute Galois group of the rationals by relating it to its action on the Teichmüller tower of Teichmüller groupoids ''T''''g'',''n'', the fundamental groupoids of moduli stacks of genus ''g'' curves with ''n'' points removed. There are several minor variations of the group: a discrete version, a pro-''l'' version, a ''k''-pro-unipotent version, and a profinite version; the first three versions were defined by Drinfeld, and the version most often used is the profinite version. References General references * Translation in ''Leningrad Math. J.'' 2 (1991), no. 4, 829–860. * Further reading Relation to algebraic topology via the little ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Curve
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the unit hyperbola. Also, similarly to how the derivatives of and are and respectively, the derivatives of and are and respectively. Hyperbolic functions are used to express the angle of parallelism in hyperbolic geometry. They are used to express Lorentz boosts as hyperbolic rotations in special relativity. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, and fluid dynamics. The basic hyperbolic functions are: * hyperbolic sine "" (), * hyperbolic cosine "" (),''Collins Concise Dictionary'', p. 328 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Field
In mathematics, a field ''K'' is called a non-Archimedean local field if it is complete with respect to a metric induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. In general, a local field is a locally compact topological field with respect to a non-discrete topology. The real numbers R, and the complex numbers C (with their standard topologies) are Archimedean local fields. Given a local field, the valuation defined on it can be of either of two types, each one corresponds to one of the two basic types of local fields: those in which the valuation is Archimedean and those in which it is not. In the first case, one calls the local field an Archimedean local field, in the second case, one calls it a non-Archimedean local field. Local fields arise naturally in number theory as completions of global fields. While Archimedean local fields have been quite well known in mathematics for at least 250 years, the first examples of non-Archimedean local ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
German Mathematical Society
The German Mathematical Society (, DMV) is the main professional society of German mathematicians and represents German mathematics within the European Mathematical Society (EMS) and the International Mathematical Union (IMU). It was founded in 1890 in Bremen with the set theorist Georg Cantor as first president. Founding members included Georg Cantor, Felix Klein, Walther von Dyck, David Hilbert, Hermann Minkowski, Carl Runge, Rudolf Sturm, Hermann Schubert, and Heinrich Weber. The current president of the DMV is . Activities In honour of its founding president, Georg Cantor, the society awards the Cantor Medal. The DMV publishes two scientific journals, the ''Jahresbericht der DMV'' and ''Documenta Mathematica''. It also publishes a quarterly magazine for its membership the ''Mitteilungen der DMV''. The annual meeting of the DMV is called the ''Jahrestagung''; the DMV traditionally meets every four years together with the Austrian Mathematical Society (ÖMG) an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
