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Shafarevich
Igor Rostislavovich Shafarevich (russian: И́горь Ростисла́вович Шафаре́вич; 3 June 1923 – 19 February 2017) was a Soviet and Russian mathematician who contributed to algebraic number theory and algebraic geometry. Outside mathematics, he wrote books and articles that criticised socialism and other books which were (controversially) described as anti-semitic. Mathematics From his early years, Shafarevich made fundamental contributions to several parts of mathematics including algebraic number theory, algebraic geometry and arithmetic algebraic geometry. In particular, in algebraic number theory, the Shafarevich–Weil theorem extends the commutative reciprocity map to the case of Galois groups, which are central extensions of abelian groups by finite groups. Shafarevich was the first mathematician to give a completely self-contained formula for the Hilbert pairing, thus initiating an important branch of the study of explicit formulas in numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aleksei Parshin
Aleksei Nikolaevich Parshin (russian: Алексей Николаевич Паршин; 7 November 1942 – 18 June 2022) was a Russian mathematician, specializing in arithmetic geometry. He is most well-known for his role in the proof of the Mordell conjecture. Education and career Parshin entered the Faculty of Mathematics and Mechanics of Moscow State University in 1959 and graduated in 1964. He then enrolled as a graduate student at the Steklov Institute of Mathematics, where he received his '' Kand. Nauk'' (Ph.D.) in 1968 under Igor Shafarevich. In 1983, he received his ''Doctor Nauk'' (doctorate of sciences) from Moscow State University. Parshin became a junior research fellow at the Steklov Institute of Mathematics in Moscow in 1968, later becoming a senior and leading research fellow. He became the head of its Department of Algebra in 1995. He also taught at Moscow State University. Research In his 1968 thesis, Parshin proved that the Mordell conjecture is a logical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shafarevich–Weil Theorem
In algebraic number theory, the Shafarevich–Weil theorem relates the fundamental class of a Galois extension of local or global fields to an extension of Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...s. It was introduced by for local fields and by for global fields. Statement Suppose that ''F'' is a global field, ''K'' is a normal extension of ''F'', and ''L'' is an abelian extension of ''K''. Then the Galois group Gal(''L''/''F'') is an extension of the group Gal(''K''/''F'') by the abelian group Gal(''L''/''K''), and this extension corresponds to an element of the cohomology group H2(Gal(''K''/''F''), Gal(''L''/''K'')). On the other hand, class field theory gives a fundamental class in H2(Gal(''K''/''F''),''I''''K'') and a reciprocity law map from ''I''''K'' t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grothendieck–Ogg–Shafarevich Formula
In mathematics, the Grothendieck–Ogg–Shafarevich formula describes the Euler characteristic of a complete curve with coefficients in an abelian variety or constructible sheaf In mathematics, a constructible sheaf is a sheaf of abelian groups over some topological space ''X'', such that ''X'' is the union of a finite number of locally closed subsets on each of which the sheaf is a locally constant sheaf. It has its origin ..., in terms of local data involving the Swan conductor. and proved the formula for abelian varieties with tame ramification over curves, and extended the formula to constructible sheaves over a curve . Statement Suppose that ''F'' is a constructible sheaf over a genus ''g'' smooth projective curve ''C'', of rank ''n'' outside a finite set ''X'' of points where it has stalk 0. Then :\chi(C,F) = n(2-2g) -\sum_(n+Sw_x(F)) where ''Sw'' is the Swan conductor at a point. References * * * * {{DEFAULTSORT:Grothendieck-Ogg-Shafarevich formula Elliptic curves ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golod–Shafarevich Theorem
In mathematics, the Golod–Shafarevich theorem was proved in 1964 by Evgeny Golod and Igor Shafarevich. It is a result in non-commutative homological algebra which solves the class field tower problem, by showing that class field towers can be infinite. The inequality Let ''A'' = ''K''⟨''x''1, ..., ''x''''n''⟩ be the free algebra over a field ''K'' in ''n'' = ''d'' + 1 non-commuting variables ''x''''i''. Let ''J'' be the 2-sided ideal of ''A'' generated by homogeneous elements ''f''''j'' of ''A'' of degree ''d''''j'' with :2 ≤ ''d''1 ≤ ''d''2 ≤ ... where ''d''''j'' tends to infinity. Let ''r''''i'' be the number of ''d''''j'' equal to ''i''. Let ''B''=''A''/''J'', a graded algebra. Let ''b''''j'' = dim ''B''''j''. The ''fundamental inequality'' of Golod and Shafarevich states that :: b_j\ge nb_ -\sum_^ b_ r_i. As a consequence: * ''B'' is infinite-dimensional if ''r''''i'' ≤ ''d''2/4 for all ''i'' Applications This result has important applicatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shafarevich's Theorem On Solvable Galois Groups
In mathematics, Shafarevich's theorem states that any finite solvable group is the Galois group of some finite extension of the rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...s. It was first proved by , though Alexander Schmidt later pointed out a gap in the proof, which was fixed by . References * * Galois theory Solvable groups {{abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Néron–Ogg–Shafarevich Criterion
In mathematics, the Néron–Ogg–Shafarevich criterion states that if ''A'' is an elliptic curve or abelian variety over a local field ''K'' and ℓ is a prime not dividing the characteristic of the residue field of ''K'' then ''A'' has good reduction if and only if the ℓ-adic Tate module ''T''ℓ of ''A'' is unramified. introduced the criterion for elliptic curves. used the results of to extend it to abelian varieties, and named the criterion after Ogg, Néron and Igor Shafarevich Igor Rostislavovich Shafarevich (russian: И́горь Ростисла́вович Шафаре́вич; 3 June 1923 – 19 February 2017) was a Soviet and Russian mathematician who contributed to algebraic number theory and algebraic geometry. ... (commenting that Ogg's result seems to have been known to Shafarevich). References * * * Abelian varieties Elliptic curves Theorems in algebraic geometry Arithmetic geometry {{Abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Evgeny Golod
Evgenii Solomonovich Golod (russian: Евгений Соломонович Голод, 21 October 1935 – 5 July 2018) was a Russian mathematician who proved the Golod–Shafarevich theorem on class field tower A tower is a tall structure, taller than it is wide, often by a significant factor. Towers are distinguished from masts by their lack of guy-wires and are therefore, along with tall buildings, self-supporting structures. Towers are specifi ...s. As an application, he gave a negative solution to the Kurosh–Levitzky problem on the nilpotency of finitely generated nil algebras, and so to a weak form of Burnside's problem. Golod was a student of Igor Shafarevich. As of 2015, Golod had 39 academic descendants, most of them through his student Luchezar L. Avramov. Selected publications * * References {{DEFAULTSORT:Golod, Evgeny S. 1935 births 2018 deaths Russian mathematicians Scientists from Moscow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yuri Manin
Yuri Ivanovich Manin (russian: Ю́рий Ива́нович Ма́нин; born 16 February 1937) is a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics. Moreover, Manin was one of the first to propose the idea of a quantum computer in 1980 with his book ''Computable and Uncomputable''. Life and career Manin gained a doctorate in 1960 at the Steklov Mathematics Institute as a student of Igor Shafarevich. He is now a Professor at the Max-Planck-Institut für Mathematik in Bonn, and a professor emeritus at Northwestern University. Manin's early work included papers on the arithmetic and formal groups of abelian varieties, the Mordell conjecture in the function field case, and algebraic differential equations. The Gauss–Manin connection is a basic ingredient of the study of cohomology in families of algebraic varieties. He wrote a book on cubic surfaces and cubic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yuriy Drozd
Yuriy Drozd ( uk, Юрій Анатолійович Дрозд; born October 15, 1944) is a Ukrainian mathematician working primarily in algebra. He is a Corresponding Member of the National Academy of Sciences of Ukraine and head of the Department of Algebra and Topology at the Institute of Mathematics of the National Academy of Sciences of Ukraine. Biography Yiriy Drozd graduated from Kyiv University in 1966, pursuing a postgraduate degree at the Institute of Mathematics of the National Academy of Sciences of Ukraine in 1969. His PhD dissertation ''On Some Questions of the Theory of Integral Representations'' (1970) was supervised by Igor Shafarevich. From 1969 to 2006 Drozd worked at the Faculty of Mechanics and Mathematics at Kyiv University (at first as lecturer, then as associate professor and full professor). From 1980 to 1998 he headed the Department of Algebra and Mathematical Logic. Since 2006 he has been the head of the Department of Algebra and Topology (until 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boris Delaunay
Boris Nikolayevich Delaunay or Delone (russian: Бори́с Никола́евич Делоне́; 15 March 1890 – 17 July 1980) was a Soviet and Russian mathematician, mountain climber, and the father of physicist, Nikolai Borisovich Delone. The spelling ''Delone'' is a straightforward transliteration from Cyrillic he often used in later publications, while ''Delaunay'' is the French version he used in the early French and German publications. Biography Boris Delone got his surname from his ancestor French Army officer de Launay, who was captured in Russia during Napoleon's invasion of 1812. De Launay was a nephew of the Bastille governor marquis de Launay. He married a woman from the Tukhachevsky noble family and stayed in Russia. When Boris was a young boy his family spent summers in the Alps where he learned mountain climbing. By 1913, he became one of the top three Russian mountain climbers. After the Russian Revolution, he climbed mountains in the Caucasus and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Igor Dolgachev
Igor V. Dolgachev (born 7 April 1944) is a Russian–American mathematician specializing in algebraic geometry. He has been a professor at the University of Michigan since 1978. He introduced Dolgachev surfaces in 1981. Dolgachev completed his Ph.D. at Moscow State University M. V. Lomonosov Moscow State University (MSU; russian: Московский государственный университет имени М. В. Ломоносова) is a public research university in Moscow, Russia and the most prestigious ... in 1970, under the supervision of Igor Shafarevich. References External links *Dolgachev's website at University of Michigan 1944 births Living people 20th-century American mathematicians 21st-century American mathematicians Algebraic geometers University of Michigan faculty Russian mathematicians Moscow State University alumni Soviet mathematicians {{Russia-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
