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Yuri I. Manin
Yuri Ivanovich Manin (; 16 February 1937 – 7 January 2023) was a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics. Life and career Manin was born on 16 February 1937 in Simferopol, Crimean ASSR, Soviet Union. He received a doctorate in 1960 at the Steklov Mathematics Institute as a student of Igor Shafarevich. He became a professor at the Max-Planck-Institut für Mathematik in Bonn, where he was director from 1992 to 2005 and then director emeritus. He was also a Trustee Chair Professor at Northwestern University from 2002 to 2011. He had over the years more than 40 doctoral students, including Vladimir Berkovich, Mariusz Wodzicki, Alexander Beilinson, Ivan Cherednik, Alexei Skorobogatov, Vladimir Drinfeld, Mikhail Kapranov, Vyacheslav Shokurov, Ralph Kaufmann, Victor Kolyvagin, Alexander A. Voronov, and Hà Huy Khoái. Manin died on 7 Januar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simferopol
Simferopol ( ), also known as Aqmescit, is the second-largest city on the Crimea, Crimean Peninsula. The city, along with the rest of Crimea, is internationally recognised as part of Ukraine, but controlled by Russia. It is considered the capital of the Autonomous Republic of Crimea. Since 2014 it has been under the ''de facto'' control of Russia, which Annexation of Crimea by the Russian Federation, annexed Crimea that year and regards Simferopol as the capital of the Republic of Crimea (Russia), Republic of Crimea. Simferopol is an important political, economic and transport hub of the peninsula, and serves as the administrative centre of both Simferopol Municipality and the surrounding Simferopol District. Its population was After the 1784 Annexation of Crimea by the Russian Empire, annexation of the Crimean Khanate by the Russian Empire, the Russian empress decreed the foundation of a city named Simferopol on the location of the Crimean Tatars, Crimean Tatar town of Old ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Victor Kolyvagin
Victor Alexandrovich Kolyvagin (, born 11 March, 1955) is a Russian mathematician who wrote a series of papers on Euler systems, leading to breakthroughs on the Birch and Swinnerton-Dyer conjecture, and Iwasawa's conjecture for cyclotomic fields. His work also influenced Andrew Wiles's work on Fermat's Last Theorem. Career Kolyvagin received his Ph.D. in Mathematics in 1981 from Moscow State University, where his advisor was Yuri I. Manin. He then worked at Steklov Institute of Mathematics in Moscow until 1994. Since 1994 he has been a professor of mathematics in the United States The United States of America (USA), also known as the United States (U.S.) or America, is a country primarily located in North America. It is a federal republic of 50 U.S. state, states and a federal capital district, Washington, D.C. The 48 .... He was a professor at Johns Hopkins University until 2002 when he became the first person to hold the Mina Rees Chair in mathematics at the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hasse–Witt Matrix
In mathematics, the Hasse–Witt matrix ''H'' of a Algebraic curve#Singularities, non-singular algebraic curve ''C'' over a finite field ''F'' is the matrix (mathematics), matrix of the Frobenius mapping (''p''-th power mapping where ''F'' has ''q'' elements, ''q'' a power of the prime number ''p'') with respect to a basis for the differentials of the first kind. It is a ''g'' × ''g'' matrix where ''C'' has genus (mathematics), genus ''g''. The rank of the Hasse–Witt matrix is the Hasse or Hasse–Witt invariant. Approach to the definition This definition, as given in the introduction, is natural in classical terms, and is due to Helmut Hasse and Ernst Witt (1936). It provides a solution to the question of the ''p''-rank of the Jacobian variety ''J'' of ''C''; the ''p''-rank is bounded by the rank of a matrix, rank of ''H'', specifically it is the rank of the Frobenius mapping composed with itself ''g'' times. It is also a definition that is in principle algorithmic. There has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss–Manin Connection
In mathematics, the Gauss–Manin connection is a connection on a certain vector bundle over a base space ''S'' of a family of algebraic varieties V_s. The fibers of the vector bundle are the de Rham cohomology groups H^k_(V_s) of the fibers V_s of the family. It was introduced by for curves ''S'' and by in higher dimensions. Flat sections of the bundle are described by differential equations; the best-known of these is the Picard–Fuchs equation, which arises when the family of varieties is taken to be the family of elliptic curves. In intuitive terms, when the family is locally trivial, cohomology classes can be moved from one fiber in the family to nearby fibers, providing the 'flat section' concept in purely topological terms. The existence of the connection is to be inferred from the flat sections. Intuition Consider a smooth morphism of schemes X\to B over characteristic 0. If we consider these spaces as complex analytic spaces, then the Ehresmann fibration theorem te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ADHM Construction
In mathematical physics and gauge theory, the ADHM construction or monad construction is the construction of all instantons using methods of linear algebra by Michael Atiyah, Vladimir Drinfeld, Nigel Hitchin, Yuri I. Manin in their paper "Construction of Instantons." ADHM data The ADHM construction uses the following data: * complex vector spaces ''V'' and ''W'' of dimension ''k'' and ''N'', * ''k'' × ''k'' complex matrices ''B''1, ''B''2, a ''k'' × ''N'' complex matrix ''I'' and a ''N'' × ''k'' complex matrix ''J'', * a real moment map \mu_r = _1,B_1^\dagger _2,B_2^\daggerII^\dagger-J^\dagger J, * a complex moment map \displaystyle\mu_c = _1,B_2IJ. Then the ADHM construction claims that, given certain regularity conditions, * Given ''B''1, ''B''2, ''I'', ''J'' such that \mu_r=\mu_c=0, an anti-self-dual instanton in a SU(''N'') gauge theory with instanton number ''k'' can be constructed, * All anti-self-dual instantons can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Of Abelian Varieties
In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back to the studies of Pierre de Fermat on what are now recognized as elliptic curves; and has become a very substantial area of arithmetic geometry both in terms of results and conjectures. Most of these can be posed for an abelian variety ''A'' over a number field ''K''; or more generally (for global fields or more general finitely-generated rings or fields). Integer points on abelian varieties There is some tension here between concepts: ''integer point'' belongs in a sense to affine geometry, while ''abelian variety'' is inherently defined in projective geometry. The basic results, such as Siegel's theorem on integral points, come from the theory of diophantine approximation. Rational points on abelian varieties The basic result, the Mordell–Weil theorem in Diophantine geometry, says that ''A''(''K''), the group of points on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manin–Drinfeld Theorem
In mathematics, the Manin–Drinfeld theorem, proved by and , states that the difference of two cusps of a modular curve has finite order in the Jacobian variety In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelia .... References * * Modular forms Theorems in number theory {{numtheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manin Triple
In mathematics, a Manin triple (\mathfrak, \mathfrak, \mathfrak) consists of a Lie algebra ''\mathfrak'' with a non-degenerate invariant symmetric bilinear form, together with two isotropic subalgebras ''\mathfrak'' and ''\mathfrak'' such that ''\mathfrak'' is the direct sum of ''\mathfrak'' and ''\mathfrak'' as a vector space. A closely related concept is the (classical) Drinfeld double, which is an even dimensional Lie algebra which admits a Manin decomposition. Manin triples were introduced by Vladimir Drinfeld in 1987, who named them after Yuri Manin. In 2001 classified Manin triples where ''\mathfrak'' is a complex reductive Lie algebra. Manin triples and Lie bialgebras There is an equivalence of categories between finite-dimensional Manin triples and finite-dimensional Lie bialgebras. More precisely, if (\mathfrak, \mathfrak, \mathfrak) is a finite-dimensional Manin triple, then ''\mathfrak'' can be made into a Lie bialgebra by letting the cocommutator map \mathfrak \to \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manin Obstruction
In mathematics, in the field of arithmetic algebraic geometry, the Manin obstruction (named after Yuri Manin) is attached to a variety ''X'' over a global field, which measures the failure of the Hasse principle for ''X''. If the value of the obstruction is non-trivial, then ''X'' may have points over all local fields but not over the global field. The Manin obstruction is sometimes called the Brauer–Manin obstruction, as Manin used the Brauer group of X to define it. For abelian varieties the Manin obstruction is just the Tate–Shafarevich group and fully accounts for the failure of the local-to-global principle (under the assumption that the Tate–Shafarevich group is finite). There are however examples, due to Alexei Skorobogatov Alexei Nikolaievich Skorobogatov () is a British-Russian mathematician and Professor in Pure Mathematics at Imperial College London specialising in algebraic geometry. His work has focused on rational points, the Hasse principle, the Manin obs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manin Matrix
In mathematics, Manin matrices, named after Yuri Manin who introduced them around 1987–88, are a class of matrices with elements in a not-necessarily commutative ring, which in a certain sense behave like matrices whose elements commute. In particular there is natural definition of the determinant for them and most linear algebra theorems like Cramer's rule, Cayley–Hamilton theorem, etc. hold true for them. Any matrix with commuting elements is a Manin matrix. These matrices have applications in representation theory in particular to Capelli's identity, Yangian and quantum integrable systems. Manin matrices are particular examples of Manin's general construction of "non-commutative symmetries" which can be applied to any algebra. From this point of view they are "non-commutative endomorphisms" of polynomial algebra ''C'' 'x''1, ...''x''n Taking (q)-(super)-commuting variables one will get (q)-(super)-analogs of Manin matrices, which are closely related to quantum groups. Manin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manin Conjecture
In mathematics, the Manin conjecture describes the conjectural distribution of rational points on an algebraic variety relative to a suitable height function. It was proposed by Yuri I. Manin and his collaborators in 1989 when they initiated a program with the aim of describing the distribution of rational points on suitable algebraic varieties. Conjecture Their main conjecture is as follows. Let V be a Fano variety defined over a number field K, let H be a height function which is relative to the anticanonical divisor and assume that V(K) is Zariski dense in V. Then there exists a non-empty Zariski open subset U \subset V such that the counting function of K-rational points of bounded height, defined by :N_(B)=\#\ for B \geq 1, satisfies :N_(B) \sim c B (\log B)^, as B \to \infty. Here \rho is the rank of the Picard group In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yuri Tschinkel
Yuri Tschinkel (Юрий Чинкель, born 31 May 1964 in Moscow) is a Russian-German-American mathematician, specializing in algebraic geometry, automorphic forms and number theory. Education and career Tschinkel attended from 1979, the Erweiterte Oberschule Heinrich-Hertz-Gymnasium in East Berlin and passed there in 1983 the Abitur. He graduated with honors from Lomonosov Moscow State University in 1990 and received his doctorate in 1992 from the Massachusetts Institute of Technology with thesis '' Rational points on algebraic surfaces'' under the supervision of Yuri Manin and Michael Artin. From 1992 to 1995 Tschinkel was a junior fellow at Harvard University. In 1995 he became an assistant professor at the University of Illinois at Chicago (UIC) and from 1999 to 2003 he was an associate professor there. From 2003 to 2008 he was a professor at the University of Göttingen. He has been a professor at the Courant Institute of Mathematical Sciences of New York University since ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
